Temperature of a metallic nanoparticle embedded in a phase change media exposed to radiation

International Journal of Heat and Mass Transfer 93 (2016) 980–990

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Temperature of a metallic nanoparticle embedded in a phase change media exposed to radiation T.K. Tullius, Y. Bayazitoglu ⇑ Department of Mechanical Engineering, Rice University, 6100 Main St., Houston, TX 77005, United States

a r t i c l e

i n f o

Article history: Received 24 May 2015 Received in revised form 21 October 2015 Accepted 22 October 2015

Keywords: Thermal storage Phase change material Nanoparticle Plasmonics

a b s t r a c t Over the past few years, researchers have introduced nanoparticles of varying sizes within a phase change material which enhances the thermal energy and the efficiency of the thermal capacity. A numerical analysis of a metallic nanoparticle embedded in a phase change material exposed to radiation is presented. This analysis includes plasmonic properties of metals when exposed to solar frequencies. Simulations of the model are initiated with a single metallic nanoparticle in a solid medium. When the particle temperature exceeds the phase change temperature an insulating liquid film forms around the particle. The temperature profiles of the solid medium, liquid film, and particle of different materials and particle diameters submerged in different mediums are presented. A thermal interface resistance between the particle and liquid film is included. It is shown that the larger particle heats faster, developing a smaller surrounding film and may eventually have a temperature that surpasses the melting temperature of the material. A compromise must be made between the thermal resistance caused by the particle/film interface and film as well as the absorption from radiation to determine the proper particle type, size, and medium for thermal storage. Ó 2015 Elsevier Ltd. All rights reserved.

1. Introduction Thermal energy storage is captured when there is a change in internal energy of a material caused by latent heat, sensible heat, and/or heat caused by a thermal–chemical process. For the past ten years, latent heat storage has shown to be an attractive approach for thermal storage because large amounts of heat can be stored in small volumes with small temperature differences in the media. Thermal energy is accumulated in this material when a rise in temperature causes the phase change material (PCM) to change from a solid to liquid or liquid to vapor through the heat of fusion or heat of vaporization, respectively. Using a PCM provides a higher heat storage capacity with lower storage temperature. Over the past decade, many people have studied the use of PCM for thermal storage [1–6]. Materials that utilize the phase change between solid and liquid have proved to be most effective. The solid–liquid PCMs comprise of organic, inorganic, and eutectic materials. Paraffins and salt hydrates are the typical choice for the PCM and more information on these types of materials can be found in [1–3]. Paraffins are compounds composed of hydrogen ⇑ Corresponding author. Tel.: +1 713 348 6291; fax: +1 713 348 5423. E-mail address: [email protected] (Y. Bayazitoglu). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.10.038 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.

and carbon where all the atoms are linked by single bonds, i.e. CnH2n+2. All paraffins are colorless and odorless and are transparent in the visible range [7,8]. Paraffins between C5 and C15 are liquids and the rest are waxy solids at room temperature with melting temperatures dependent on the number of carbons it contains ranging from 23–67 °C. The melting temperature increases as the number of carbons increases [9]. The paraffin used in these calculations is n-octadecane (C18H38). This material is commonly used to make crayons, candles, and electrical insulation and is affordable. In 2007, Khodadadi and Hossenizadeh [10] proposed the idea of placing nanoparticles within the phase change material in order to improve thermal storage. In Khodadadi and Hossenizadeh’s paper [10], they placed copper (Cu) nanoparticles of various concentrations into water and numerically simulated the solidification of a nanofluid in a square storage model. Fan and Khodadadi [11,12] and Nabil and Khodadadi [13] provided experimental insight of cyclohexane and eicosane both mixed with copper oxide (CuO2) nanoparticles. This showed that the freezing rate was increased compared to the fluid with no nanoparticles because of the enhanced thermal conductivity. Hasadi and Khodadadi [14] numerically simulated the solidification of copper (Cu) nanoparticles in water. Dhaidan et al. [15] showed experimentally and numerically the increase of the thermal conductivity using the mixture of n-octadecane with copper oxide (CuO2) nanoparticles

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Nomenclature A a0 C abs c d F go hsl Iinc k ^ k m00 P p Q abs ~r ~ R r tot h

surface area (m2) molecular diameter (m) cross-sections (m2) specific heat (J/kg K) diameter (m) interfacial force (N) gravity (m/s2) latent heat of fusion (J/kg) incident intensity (W/m2) wavenumber (m
1) thermal conductivity (W/m K) mass flow rate (kg/m2 s) parameter for polarizability pressure (kg/m s2) heat (W) interface resistance (m2 K/W) resistance (K/W) radial direction (m) total tangential direction

Greek variables a polarizability (m3) a^ thermal diffusivity (m2/s) Dc0 interfacial energies (kg/s2) c^ damping term (s
1) e permittivity e1 permittivity bulk h tangential direction T temperature (K)

in a square enclosure subjected to a constant heat flux from one side. Aluminum nanoparticles and silica oxide (SiO2) nanoparticles were mixed with n-nonadecane, n-eicosane, n-heneicosane, and ndocosane in a molecular dynamic simulation to determine the size of the particle provided by Rao et al. [16]. Cingarapu et al. [17] studied the viscosity, thermal conductivity, and total heat absorption of silica encapsulated tin (Sn/SiO2) nanoparticles dispersed in a synthetic HFT Therminol 66 (TH66) fluid. Chieruzzi et al. [18] studied the effect of heat capacity when the base fluid was the salt mixture NaNO3–KNO3 with nanoparticles of silica (SiO2), alumina (Al2O3), titana (TiO2), and a mix of silica alumina (SiO2–Al2O3). A computational fluid dynamics program, FLUENT, was used by Jegadheeswaran and Pohekar [19] who explored the heat transfer characteristics of micron sized copper particles during both charging and discharging modes. Ho and Gao [20] experimentally investigated how the thermal physical properties such as density, viscosity, and thermal conductivity with alumina (Al2O3) nanoparticles in paraffin (n-octadecane) are affected. The melting/freezing characteristics of paraffin and copper nanoparticles were studied by Wu et al. [21,22]. Wu showed that the melting/freezing rates were enhanced due to adding the nanoparticles. Zeng et al. enhanced the thermal conductivity of paraffin (n-octadecane) using silver (Ag) particles [23], silver (Ag) nanowires [24], and multiwalled carbon nanotubes [25]. All of these papers presented above concentrated on the enhancement of the thermal properties of various combinations of mediums and particles. Around the particle, the temperature may exceed the melting temperature causing a liquid film to form surrounding the metal sphere and begin the two phase process, see Fig. 1. When considering solid/liquid phase change there are numerical and experimental analysis of freezing and melting paraffin wax and water in spherical enclosures [26–29]. Only a few papers have

t Vp

v

time (s) volume (m3) velocity (m/s)

Subscripts crit critical f film ii step in time i interface j type of medium l liquid m melting n step in space p particle plasma plasma r direction s solid sl solid/liquid interface q density (kg/m3) k wavelength (m) x frequency (s
1) Non-dimensional variables R radial direction s time H temperature ^ V velocity / film diameter TLR temperature loss ratio

concentrated on the single particle interaction between a solid and liquid film. Uhlmann and Chalmers [30] shows that there exists a force between the solid interface and the particle that prevents the particle to lay on the solid surface unless the force of gravity is greater than the buoyant force. Shangguan et al. [31] and Garvin and Udaykumar [32] provide numerical analysis of particles experiencing the interface forces against a planar solid surface. Bulunti and Arslanturk [33] numerically investigated an inward melting of a sphere subject to radiation and convection. The focus of this paper is to understand how one metallic nanoparticle of various sizes and compositions inside a phase change material is affected when exposed to radiation. 1.1. Plasmonic properties Mechanical engineers, biologists, physicists, and chemists have all taken interest in studying metallic nanoparticles because they have unique, tunable absorption and scattering properties when interacting with photons. Gold (Au), silver (Ag), aluminum (Al),

Fig. 1. Film formulations of one nanoparticle when it reaches the melting temperature.

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and copper (Cu) are just a few plasmonic particles that when interacting with an electric field, create strong optical resonances called surface plasmon resonances (SPR). On the surface of these metal structures, a collection of bound mobile electrons generate quantized waves or plasmons that excite from their equilibrium position and oscillate when the frequency of the incident light is in tune with their motion. The surface plasmon resonances combined with the size, shape, and composition alters the sensitivity and tunability of the optical properties [34,35]. The solid nanosphere, nanoshell, nanorod, nanocage, nanobelt, and nanohexapod structures are a few of the metallic particle shapes that have been created and tested. More information on plasmonic nanoparticles and the various shapes that have been considered can be found [36– 41]. In addition to changing the physical components of the metallic nanoparticle, the particle absorption is extremely sensitive to changes in the medium that the particles are in [42,43]. Mie theory, a solution to Maxwell’s equations for a sphere, provides the scattering and absorption coefficients [44]. The plasmonic properties are accounted for using Drude’s model or the Drude–Lorentz model which is further described later in Section 2.1. Fig. 2 gives an idea on how the absorption varies with different particle composition, size, medium, and also varying liquid film size around the particle. Fig. 2a, shows how the absorption can be very different (1) for the various types of particles of a 90 nm particle, (2) the different sizes of a Au particle in water, and (3) different mediums: water and n-octadecane. In Fig. 2b, a 90 nm Au particle was shown with varying sizes of liquid film surrounding the particle. The term (90, 100) implies a 90 nm diameter particle with a film diameter of 100 nm. Notice that when a liquid film is

added to a 90 nm particle, the particle absorption decreases as the size of the film increases. Metallic particles, particularly Au nanoparticles, are considered to contribute to biomedical applications such as photothermal therapy [45–50] and imaging [51,52], and solar heating [53–55]. Neumann et al. [53,55] utilized the plasmonic effects for a solar based direct steam generation application. A vapor film was formed around a SiO2/Au nanoparticle and due to buoyancy, the vapor bubble floats to the surface and steam escapes. If PCMs such as water and paraffins are transparent in the visible region, the metallic nanoparticles of size 30, 60, 90, and 120 nm can be tuned to absorb the solar light in the visible region, increasing the temperature surrounding the nanoparticles faster because of the plasmonic effects and causing the liquid film to form. The temperature profiles of the film and the medium are tabulated. This paper provides an in depth numerical investigation of a single nanoparticle composed of various metallic properties: Au, Ag, Cu, and Al subject to solar heating and experiencing phase change. Different mediums were also used: water and n-octadecane. Plasmonic properties were included in the analysis as well as the interface resistance between the particle and the film. A detailed mathematical formulation is described in the next section followed by the numerical analysis and conclusions. 2. Nanoparticle mathematical formulation The model used for the film formation is based on the derivations made from [56,57]. Initially the particle is in a solid medium heated through radiation. Once the temperature of the particle

Fig. 2. Absorption versus wavelength (a) for a particle with varying composition, particle size, and medium; (b) for a Au, 90 nm particle and its surrounding liquid film. Note: (90, 100) nm implies that the particle has a diameter of 90 nm and a surrounding film diameter of 100 nm.

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reaches the melting temperature, a film begins to form. At this stage, a velocity equation for the liquid film is necessary. This section provides the equations necessary to portray this process by giving the mass, momentum, and energy equations for the particle, liquid, and solid layers. Several assumptions were made such that there is a uniform surface temperature of the particle, thin liquid film compared to that size of the particle, and constant thermal properties within the films and the particle. Spherical symmetry for the program is considered. For simplicity, the particle is stationary, and it is the medium around the particle that is changing. For this problem, a 2-D simulation in the radial coordinate, r, and tangential direction, h, is conducted. See Fig. 3 for details.

^ [s
1 ] is the damping constant dependent is the frequency, and c on the Fermi velocity and the mean free path for the metal particle. Drude’s model is a simplification of the Lorentz model for optical properties. This is derived by treating electrons as simple harmonic oscillators subject to a driving force of electromagnetic fields [44]. For coated spheres, the polarizability changes to account for the material within the film were the subscript 1 represents the properties for inside the sphere, 2 represents the properties for the film, and 3 are the properties for outside the film [59].

a ¼ 3V p e3




e2 ea
e3 eb e2 ea þ 2e3 eb



ð5Þ

where 2.1. Energy equation for the particle The particle is heated using lumped sum analysis heated through radiation, implying that the whole particle has a uniform temperature, T p . The expression for the temperature of the particle is

qp V p cp

@T p ¼ Q abs @t

ð1Þ

The particle density, volume, and heat capacity for the particle are defined by qp [kg=m3 ], V p [m3 ], and cp [J=kg K]. The parameter, Q abs [W], is the power absorbed from the small particle, based on Rayleigh scattering [44], such that

Q abs ¼ kImðaÞIinc ¼ C abs Iinc

ð2Þ

where the wavenumber is k [m
1 ], Iinc [W=m2 ] is the incident intensity, and a is the polarizability. The parameter C abs [m2 ] is the absorption cross sections. Rayleigh scattering applies to spherical particles with a diameter smaller than the wavelength considered [44]. The electric field acts on the charged electrons in the particle which causes them to move at the same frequency and become a small radiating dipole. The polarizability is the vector sum of the dipole moments in a unit volume and is defined by [47]:

a ¼ 3V p es

ep
1 ep þ 2

ð3Þ

where ep and es are the permittivity of the particle and medium, respectively. The permittivity of the particle must include the surface plasmon effects caused by the excitation of the electrons resulting from the interaction with photons on the surface of the metal as discussed earlier. Drude’s model [47],

ep ðxÞ ¼ e1


x2plasma xðx þ ic^Þ

3

P ¼ 1
ðd1 =d2 Þ

For other geometries like nanorods, the absorption cross section is generally found by using discrete dipole approximation developed by Draine and Flatau [60] or using Rayleigh Gans theory, which is an extension of Mie theory [44,34]. The parameters used for the two phases and the particle are provided in Tables 1 and 2. 2.2. Energy equations for the solid and liquid medium The energy equation for the film and outer medium in spherical coordinates is given by

^j @ 2 T j 2 @T j k @T j þ ¼ @t qj cj @r2 r @r

!

ð6Þ

Table 1 Properties of metallic nanoparticles. Particle

Au

Ag

Cu

Al

qp [kg/m3]

19.32  103

10.49  103

8.94  103

2.7  103

cp [J/kg m3] xplasma [s
1] ~r [m2 K/W] c^ [s
1] ^p [W/m K] k

129 2.18  1015 15  10
9 6.42  1012 318

235 1.37  1015 5  10
9 4.34  1012 429

384 1.69  1015 5  10
9 4.03  1013 396

904 2.83  1015 2.5  10
9 19.79  1012 247

T m [K]

1337

1235

1358

993

ð4Þ

e1 is the bulk permittivity for the metal values which can be found using [58], x [s
1 ] includes the plasma frequency, x2plasma [s
1 ],

ea ¼ e1 ð3
2PÞ þ 2e2 P eb ¼ e1 P þ e2 ð3
PÞ

Table 2 Medium properties. Medium

H2O

n-Octadecane

qs [kg/m3]

917 2050 2.22

814 2150 0.358

es

3.18

2.149

Liquid ql [kg/m3] cl [J/kg m3] ^ [W/m K] k

1000 4216 0.68

744 2180 0.148

el

1.78

2.065

Other hsl [kJ/kg] ao [m] Dco [kg/s2] T m [K] T crit [K]

334 1  10
7 0.0317 273 580

244 1.2  10
11 0.0096 301 747

Solid cs [J/kg m3] ^s [W/m K] k

l

Fig. 3. Film formulation of one nanoparticle.

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Fig. 4. Temperature profile vs. time (a) for various nanoparticle materials, sizes, and mediums (b) a 90 nm Au particle in H2O with different intensities.

where the density, conductivity, and specific heat of the liquid/solid ^j [W=m K], q [kg=m3 ], and cj [J=kg K], respectively. is described by k j The subscript j references the particular layer. The parameter, T j , is the temperature. The boundary condition at the particle/film interface incorporates an interface resistance, ~r [61], which is provided by

Ti ¼ Tp


Q abs~r Ap

ð7Þ

The parameter, Ap [m2 ], is the surface area of the particle and ~r [m2 K=W] is dependent on the type of material the particle is in and is found in Table 1. Note: these values represent the interface resistance for the particles in water. Values for n-octadecane have not been tabulated and therefore an assumption was made to use the same value. The boundary condition at the film/medium interface is provided by

@T 1 @m00 ¼ h ^ @t sl @r k l

ð8Þ

Here, hsl is the latent heat [J=kg]. The mass conservation at the bubble interface, is defined by [56]

dm @v h ¼ ql v r þ ql dt @h 00

dm00 dt

2

ð9Þ

where [kg=m s] is the area specific mass flux per unit area, [m=s] is the radial velocity of the liquid film interface, and

vr vh

[m=s] is the tangential velocity, see Fig. 4 and the next section for more details. 2.3. Liquid solid interface velocity equations Once the temperature of the particle reaches the melting temperature, T m , a liquid film develops between the particle and the solid medium. The interface velocity between the liquid film and the solid medium in the radial direction is satisfied by the equation

ql

  Z @v r @v r v r v h 3F @p
dr þ vr
¼ 3 @t @r r r @r 4pdsl

ð10Þ

The parameters dsl ðt; rÞ [m] represent the diameter of the film. The term F [N] is the interfacial energies between the solid phase and liquid given by the Van der Waals forces similar to [31,32]

F ¼ pdp Dc0



a0 a0 þ 0:5ðdsl
dp Þ

2 ð11Þ

where dp [m] is the diameter of the particle. The parameter Dc0 ¼ csp
clp
csl where c [kg=s2 ] is the interfacial energies between the solid/liquid, liquid/particle, and the solid/liquid interfaces. The parameter a0 [m] is the molecular diameter of the two materials and 0:5ðdsl
dp Þ represents the thickness of the liquid layer. For water a0 ¼ 0:10  10
6 m and Dc0 ¼ 31:7  10
3 kg=s2 [62]

and

Dc0 ¼ 9:6  10

for
3

n-octadecane, 2

a0 ¼ 0:12  10
10 m

and

kg/s [63]. The Van der Waal forces describe the

T.K. Tullius, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 93 (2016) 980–990

force given by the molecular attraction between the molecules within each substance. The change in pressure can be calculated using the Clausius–Clapeyron equation [64]

dp hsl ql dT ¼ dr T dr

ð12Þ

This expression represents the influence of the external pressure on the melting solid. This implies that Eq. (10) can be rewritten as

ql

  @v r @ v r v 2r 3F hsl ql dT ¼
þ vr
3 Tdr @t @r r 4pdsl

The tangential interface velocity,

ð13Þ

v h , can be expressed by

  @v h @v h v r v h ql þ vh
¼ ðqs
ql Þg 0 sin h @t @h r

ð14Þ

The term on the right hand side represents the buoyancy/gravitational effects where g 0 ¼
9:8 m=s2 is the gravitational force. 2.4. Non-dimensionalized terms After describing the mathematical model, non-dimensional equations were used to describe the equations found in Sections 2.1–2.3. The following dimensionless parameters for the radial direction, time, temperature, radial velocity, tangential velocity, and diameter of the film is defined as

2r R¼ ; dp

s¼t

dp v h ; V^ h ¼ ^ 2a

^ 4a 2 dp

;

T H¼ ; T1

dp v r V^ r ¼ ; ^ 2a

dsl /¼ dp

ð15Þ

^ =q c is the thermal diffusivity. Eqs. (1)–(14) can be con^¼k Here a l l l verted into the dimensionless governing equations: The mass flow rate

^ @m00 2a ^ ql ^ ql @ V^ h 2a 2a V^ r þ ¼ dp @t dp dp @h

ð16Þ

The velocity interface equations between the liquid and solid mediums for the radial and tangential direction, respectively, are



^r @V @s



^h @V @s

Vr þ V^ r @@R
^

þ

^h V R

V^ 2h




3d Dc

¼ 32q pa^ 2 /03 l  ^h ^ r V^ h ðqs
ql Þ @V V
¼ @h R q R

l



a0 a0 þ0:5dp ð/
1Þ

d3p ^2 8a

2




d2p ^2 4a

hsl @ H H@R

ð17Þ

g 0 sinðhÞ

The film and solid temperature profile equations in nondimensional form and boundary conditions are

^ k @ Hj @ 2 Hj 2 @ Hj j ¼ þ @s qj cj a^ @R2 R @R Q ~r H ¼ Hp
abs T 1 Ap ^ hsl @m00 @ H
a ¼ ^ @t @R T1k l

!

ð18Þ

where the surrounding medium starts to melt, a film begins to form surrounding the particle. Initially, the medium and particle are considered to be the same temperature at T s ¼ T p ¼ 270 K and the reference temperature is T 1 ¼ 383 K. Incident radiation is assumed to be transparent to the solid medium, and the particle absorbs heat from the intensity of Iinc ¼ 1  108 W/m2. The absorption term is calculated for all frequencies in the solar range (k ¼ 150
2000 nm). Properties of the materials used for the medium and particle are found in Tables 1 and 2. The temperature profiles for the particle and medium are captured. When the temperature of the particle reaches the melting temperature of the medium, the velocity equations for the liquid film becomes nonzero, the film diameter begins to grow, and the absorption term is recalculated using the new film diameter. Once 89% of the critical temperature is reached, the program breaks [65]. This value was used to avoid a vapor phase change that might occur at this temperature. In order to track the movement of the film, the level set method was used. This method, first introduced by Othian and Sethian [66], is a technique that relies on the implicit formulation of the interface given by a time dependent partial differential equation found in Eq. (20). More information can be found in various texts including [67]. The equation used for the non-dimensional level set method is

^r þ V ^h @/ V ¼ @s 2

ð20Þ

The non-dimensional equations are calculated using the finite difference method for space and fourth order Runga Kutta method was used in time. For this simulation, the time and space steps are different for each stage. Vertical symmetry was assumed in the model. The Von Neumann stability criteria is considered for this program and is given by [68]

^ Dt k

qcDr2

6

1 2

ð21Þ

For the first stage before the temperature of the particle reaches the melting temperature of the medium, T m , the non-dimensional time step and space step are Ds1  10
4 and DR  10
2 , respectively. The second stage when the liquid film begins to grow, the time step is

Ds2  10
5 , the space step in the liquid film is DR  10
3 , and space step in the solid medium is DR  10
2 . For these simulations Dh  0:39 rads. The finite difference method for the medium temperature is given by ! m
1 ^ @ Hm k Hmþ1
2Hm 2 Hmþ1
Hm j iiþ1 ii ii þ Hii ii ii ð22Þ ¼ þ R @s qj cj a DR DR2 where m is the space step in the r direction and ii is the time step. 3.1. Validation

The non-dimensional equations for the particle temperature profile is given as

@ Hp Q abs ¼ ^ @s 4qp cp V p T 1 a

985

ð19Þ

No experimental data has been found at this time for this model. Therefore this analysis was validated using several different methods depending on the expression. The temperature of the particle is a simplistic ordinary differential equation and the solution to the non-dimensional Eq. (19) is

Hp ðsÞ ¼ Hi þ s 3. Numerical analysis The model described from Sections 2.1–2.4 simulates the two stages for when a single nanoparticle in a solid medium is heated through radiation. Once the particle reaches a temperature to

Q abs ^ qp V p c p T 1 4a

ð23Þ

Another validation was used for the temperature profile of the medium using separation of variables (SV). Both validations expressed that the algorithm had accurate calculations and are not presented here.

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3.2. Numerical results This simulation captured many characteristics. First, the nondimensional temperature profile for the particle, Hp , vs. the nondimensional time, s, is given in Fig. 4. Similar results were found within both mediums. Fig. 4a compares the temperature profiles of the particle made up of different materials and sizes. These materials are in a H2O medium and s is displayed in logarithmic scale. The program was stopped when the particle temperature reaches 89% of the critical temperature which is about Hp ¼ 1:38 for water because a vapor layer may begin to develop at this temperature [65]. This graph shows that the Au particle heats up faster than the others with Cu, Ag, and Al follows. Fig. 4a also conveys that when comparing different size particles, the larger the particle, the faster the particle heats up. Similar results are given for the other particle materials and the n-octadecane medium. The difference of the temperature profiles of the particles in the different mediums is also conveyed showing that a particle in n-octadecane heats up faster than when in water. Fig. 4b implies that the particle will take a longer time to heat up with a lower intensity, which is accurate with Eq. (19). The results of Fig. 4 are attributed to the absorption from the particles. A particle of size 30 nm has less absorption, compared to the particles of size 60–120 nm which have similar absorption spectrum, see Fig. 2a [37]. Also, the absorption of the Al nanoparticles is much less than that of Au and the absorption for a particle in n-octadecane is higher than water as seen in Fig. 2a. Based on this graph, a 120 nm Au particle in n-octadecane heats the fastest. However, these simulations stopped when another film, a vapor layer, may occur and possibly

the particle may reach its melting temperature affecting its integrity. Choosing a particle where the integrity would not be affected but still heats up faster implies that a 90 nm or 60 nm particle works better. Fig. 5 gives insight on how the film radius is affected with particle properties and incoming heat. Fig. 5a shows how the nondimensional variable of the film diameter, /, is varied by the material properties, size of the particle, and different mediums. The profile for / follow similar paths when varying the compositions. Note that the / values were taken at the bottom of the sphere. The difference due to gravity from the bottom of the particle to the top of the particle is about D/ ¼ 2  10
6 and therefore considered negligible. For Au, Ag, Cu, and Al, the profiles are / ¼ 1:07; 1:45; 1:12; 1:62 respectively, resulting in dsl ¼ 96:3; 130:5; 100:8; 145:8 nm respectively. This figure also shows that when the particle size is small, the film diameter grows larger in relation to the particle size. For the 30, 60, 90, 120 nm particles, the film diameter reaches / ¼ 1:90; 1:14; 1:07; 1:06, respectively, which implies that the film diameter is dsl ¼ 57; 68:4; 96:3; 127:2 nm, respectively. Fig. 5a also gives the film profiles for a 90 nm Au particle in both water and n-octadecane. When the particle is in n-octadecane, the film diameter is much smaller than that of water. Fig. 5b, gives an idea of how the film grows with intensity. The smaller the intensity the larger the film growth. To understand why these results occurred in Fig. 5, the particle with less heat or less absorption added resulted in the particle taking longer to heat and, in turn, allows for a larger film to grow. The Al particle has the least absorption, as well as the 30 nm particle and it resulted in a greater / size.

Fig. 5. The liquid film diameter vs. non dimensional time (a) for various nanoparticle properties (b) for different intensities for a 90 nm Au particle in H2O.

T.K. Tullius, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 93 (2016) 980–990

Fig. 6. The temperature profile of the particle, film, and surrounding medium vs. particle in H2O, and (d) a 90 nm Al particle in H2O.

987

s for (a) a 90 nm Au particle in H2O (b) a 90 nm Au particle n-octadecane, (c) a 30 nm Au

988

T.K. Tullius, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 93 (2016) 980–990

Having a larger film provides more thermal insulation to the incoming radiation, causing the particle to absorb less heat and it takes longer to reach the stopping temperature. This will further be shown in Fig. 6. Fig. 6 portrays the non-dimensional temperature profile for the particle, film, and surrounding medium vs. the non-dimensional ‘‘r” direction at specific times, s. In this graph, the values where R < 1 implies the particle, R = 1 is the interface at the particle surface, and R > 1 implies the film and medium the particle is in. The first line in the legend represents time when the film has not yet developed. The other lines give an evolution of the temperature as the film grows. In this figure, arrows are used to show the interface resistance between the particle and film, ~r , and the film temperature profile. Fig. 6a gives the profile for a 90 nm Au particle in water, Fig. 6b is the profile for a 90 nm particle in n-octadecane, Fig. 6c gives information for a Au 30 nm particle in water, and lastly Fig. 6d gives information for a 90 nm Al particle in water. These figures were chosen arbitrarily. Based on Vera et al. [61], the temperature jump at the interface is huge for Au compared to the other particles, see Table 1, and is portrayed in Fig. 6. This can be understood by viewing the relation used,

Q abs~r Ap T 1

DHi ¼

ð24Þ

For gold, both Q abs and ~r are larger than for the other particles resulting in a larger change in temperature for Au caused by the interface resistance. When changing the size of the particle Q abs j Ap 30 nm

6 QAabs j120 nm implies that the temperature jump for the p

smaller particle is less than that of the larger particle. To further analyze the internal resistance at the surface of the particle, a temperature loss ratio is established and derived here. Using Eq. (1), the temperature change for the spherical particle heated by radiation is

DHp ¼

Q abs Dt T 1 qp c p V p

ð25Þ

At the interface, there is that thermal resistance as shown in Fig. 6 and Eq. (24). The temperature loss ratio is defined as ratio of the temperature jump at the interface over the temperature jump of the particle:

TLR ¼

DHi Q abs~r qp cp V p T 1 ~r qp cp V p ¼ ¼ DHp Ap T 1 Q abs Dt A p Dt

ð26Þ

Because the particle is spherical, the simplified expression is defined as

TLR ¼

qp cp dp~r 6 Dt

ð27Þ

Notice that this ratio is dependent on time. This temperature jump occurs instantaneously, therefore, Dt ¼ 1  10
9 s. Also notice that this ratio is dependent on the type of the particle and the size of the particle. The physical interpretation of this ratio is describing the heat at the interface due to the heat flow of the particle caused by the incoming radiation from the particle. This ratio determines whether the particle resistance at the interface will be too large and whether it is necessary to include in the analysis. When the TLR < 0:2, the temperature jump do to the interface resistance, ~r, is small, refer to Fig. 6d. When the value is larger than 0.2, the temperature at the particle interface is large and should be included into the analysis, refer to Fig. 6a. The value 0.2 was found by analyzing the results for all particles. Having a value of <0.2 implies that the temperature jump at the interface is <15 K. There is also an additional temperature jump between the particle/film interface and the solid medium due to the film. The film acts as an insulation and for larger films, it results in a larger temperature jump. The total thermal resistance is calculated by:

~ tot ¼ R ~i þ R ~ f ¼ ~r þ /
1 R pd2p 4pk^l /dp

ð28Þ

Within this equation, the interface resistance is comparable to the resistance caused by the film. In the field of thermal storage, high heat transfer rates are valued without affecting the structure of the particle, i.e. a larger temperature difference within a smallest time change. Based on Figs. 2 and 4–6, a compromise must be made dependent upon how fast the particle heats and how fast the medium heats. The heating of the medium depends on the particle/film interface and liquid film thermal resistances. A particle with a smaller film and ~r will heat the medium faster; however, the smaller the film, the faster the particle will reach the ending temperature and eventually may surpass the melting temperature of the particle. Because of this reasoning, a Cu particle of size 90 nm or 60 nm in either water or noctadecane is ideal for thermal storage.

4. Conclusion This work describes the characteristics to a single nanoparticle submerged into a phase change material. No other work had been conducted for a single nanoparticle analysis, but rather efforts have been focused on multiple nanoparticles in the PCM as a whole. The temperature profile of the particle and medium around the particle as well as the liquid film are analyzed for different material and size of nanoparticles submerged in two different mediums for various incident intensities. A few observations and conclusions are discussed below.  The non-dimensional temperature profile for the particle is dependent on the plasmonic absorption. Al has less absorption and results in a longer time to heat. Similarly, a 30 nm particle has less absorption and also results in a longer time to heat in order to reach similar ending temperatures. These simulations stopped when a vapor film may develop and/or the integrity of the particle may be become affected.  For the non-dimensional parameter of the liquid film, /, the time it takes to reach a stabilized value differs depending on the type of particle used, the size of the particle, and the medium. Al takes the longest to develop a film and results in a larger / with Au taking the least time and have the smallest film. The smaller the particle, the larger the relative film size. A medium of water showed larger film diameters when compared to n-octadecane. Having a larger film provides more thermal insulation to the incoming radiation, causing the particle to absorb less heat and it takes longer to reach the stopping temperature.  When decreasing the intensity of the particle, the film diameters of the liquid increase drastically. The lesser the intensity, the greater the impact there is on these parameters and the larger the films. This is because the film has more time to grow until the particle reaches the phase change temperature.  The temperature profiles in the medium convey a temperature jump that occurs at the interface. This temperature jump depends on the particle type and size, as seen by the TLR. This interface resistance was found to be necessary to include when TLR > 0.2. There is also a resistance due to liquid film. When the film is larger, this temperature jump is more pronounced. The total resistance caused by the film and interface is found by Eq. (28). The thermal storage application desires a particle with least total resistance (both interface and caused by the film) and also a particle that heats at a faster rate due to the absorption but does not affect the metal integrity. This implies that a Cu particle of size 90 nm or 60 nm is the most optimal.

T.K. Tullius, Y. Bayazitoglu / International Journal of Heat and Mass Transfer 93 (2016) 980–990

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