Numerical Investigation of the Microscopic Heat Current Inside a

Technical Note Cite This: Anal. Chem. 2018, 90, 4297−4302

pubs.acs.org/ac

Numerical Investigation of the Microscopic Heat Current Inside a Nanofluid System Based on Molecular Dynamics Simulation and Wavelet Analysis Tao Jia*,† and Di Gao‡ †

Department of Thermal Engineering, Taiyuan University of Technology, Taiyuan, 030024, China Department of Statistics, North Dakota State University, Fargo, North Dakota 58102, United States

‡

ABSTRACT: Molecular dynamics simulation is employed to investigate the microscopic heat current inside an argon−copper nanofluid. Wavelet analysis of the microscopic heat current inside the nanofluid system is conducted. The signal of the microscopic heat current is decomposed into two parts: one is the approximation part; the other is the detail part. The approximation part is associated with the low-frequency part of the signal, and the detail part is associated with the high-frequency part of the signal. Both the probability distributions of the high-frequency and the low-frequency parts of the signals demonstrate Gaussian-like characteristics. The curves fit to data of the probability distribution of the microscopic heat current are established, and the parameters including the mean value and the standard deviation in the mathematical formulas of the curves show dramatic changes for the cases before and after adding copper nanoparticles into the argon base fluid.

N

technology was employed to detect the concentration characterization of the water−copper nanofluid. The ultrasonic velocity in the nanofluid was greater than that in the base fluid.6 Experimental investigation of the thermal performance of the nanofluid flowing in an equilateral triangular duct was carried out. Nanoparticles made of Al2O3 and SiO2 were added to distilled water and then were dispersed with the ultrasonic vibration method. Finally a significant increase of heat transfer was found under the condition of the employment of the nanofluids.7 The effect of using water−silver nanofluid on the thermal performance of an inclined heat pipe was experimentally investigated. Four different inclination angles including 0°, 30°, 60°, and 90° were considered. As the concentration of the silver nanoparticle increased, the thermal resistance of the heat pipe decreased. The thermal conductivity of the heat pipe reached a maximum at the angle of 60°.8 Al2O3 nanoparticles were distributed inside ethylene glyol to form a nanofluid, and an improved transient-hot-wire method was used to measure the thermal conductivity of the nanofluid. The influences of the nanoparticle volume fraction and size distribution on the thermal conductivity were investigated. The experiment showed that the influence of particle size was over that of particle volume fraction on the thermal conductivity under the condition of volume fraction less than 0.25%.9 The character of TiO2−water nanofluid was experimentally investigated under the conditions of laminar and mixed flows. Brownian motion and thermophoresis were

anofluid is a homogeneous mixture of nanoparticles and conventional fluid, which is often termed base fluid, in which the nanoparticles are uniformly dispersed with low concentration in the base fluid, and the diameter of the nanoparticle is usually less than 100 nm. Heat transfer is a crucial issue in thermal management in industry. A traditional method to augment heat transfer was to enlarge the surface of heat transfer. However, once the equipment is built and installed on site, the surface is almost impossible to be changed. Nanofluids provide an effective and easy-to-implement solution to enhance the heat transfer without increasing the heat transfer surface. The thermal performance of a nanofluid composed of carbon nanotubes and water was experimentally investigated, and it was found that the thermal conductivity of the nanofluid increased as the length of the nanotube decreased.1 ZnO nanoparticle and propylene glycol were mixed to construct a nanofluid, and the technologies of ultraviolet−visible spectroscopy and light scattering were employed to study the nanoparticle size distribution and the stability of the suspension of the nanoparticle in the base fluid.2 The effects of temperature and nanoparticle concentration on the viscosity of nanofluids were investigated, and the result showed that the viscosity decreased with temperature and increased with the nanoparticle concentration.3 In the base fluid composed of 60 mass % propylene glycol and 40 mass % water, different nanoparticles made of Al2O3, ZnO, TiO2, and SiO2 were injected, and the nanofluids exhibited lower surface tension, compared to the surface tension of the base fluid.4 The thermal performance of water−silver nanofluids were experimentally studied. The nanofluid was formed by the two-step method, and the experiment results demonstrated that the thermal performance increased about 28% by using the nanofluid.5 Ultrasound-based © 2018 American Chemical Society

Received: December 21, 2017 Accepted: March 19, 2018 Published: March 19, 2018 4297

DOI: 10.1021/acs.analchem.7b05350 Anal. Chem. 2018, 90, 4297−4302

Technical Note

Analytical Chemistry

thermal conductivity of the nanofluid. So the investigation of the stochastic characteristics of the microscopic heat current can help us understand the microscopic thermal fluctuation inside the nanofluid system.25,26 Wavelets-based technology29−32 has found its application in a wide range of engineering fields, and it provides us a powerful tool to qualitatively represent a variety of signals. The basic idea of wavelet is to decompose the original signal into approximate and detail parts. The approximation part reflects the general trend of the original signal, and it is associated with high scale and low frequency. The detail part reflects local difference in the original signal, and it is associated with low scale and high frequency. The characteristics of the signal of the microscopic heat current inside a nanofluid system is a direct representation of the microscopic thermal transport inside the nanofluid system.25−27 In this paper, the wavelet analysis of the microscopic heat current signal is conducted with the aim to find some certain patterns as the nanoparticle volume fraction changes.

assumed to be responsible for the free movement of the nanoparticle in the base fluid.10 Contrast-enhanced video microscopy technology was used to find the growth of fingering patterns in dewetting nanofluids.11 Both temperature-sensitive and nontemperature-sensitive magnetic nanofluids were discussed, and the particle rotation and the influence of the torque on the particle by an externally magnetic field were thought to be the explanation of the effective viscosity of the nanofluid.12 Experiments showed that the employment of highly thermophilic particles resulted in an active control of the heat transferred by smart nanofluids.13 It was found that there was a dramatic increase in heat transfer by using nanofluid under the conditions of turbulent flow in a numerical investigation based on Euler and Lagrangian methods. Two different kinds of nanofluids including Cu and Al2O3 nanoparticles were investigated.14 The shadowgraph images were used to discover the pattern of power law scaling in the early stage of solutal convection in a nanofluid under conditions far from equilibrium.15 Nanofluid has been widely used in boiling heat transfer which is a high efficient heat transfer technique.16,17 The investigation of the thermal performance of nanofluids in convective boiling flows is experimentally conducted. It was found that adding alumina and copper oxide nanoparticles and multiwalled carbon nanotubes to water did enhance the whole thermal performance of the boiling flow system.16 Experimental investigation of the heat transfer coefficient of Titana nanofluids was carried out in the boiling heat transfer process. It was shown that the heat transfer coefficient of the nanofluid system is higher than that of the base fluid, and the nucleation site density was increased after adding the nanoparticles.18 It was found that the heat transfer coefficient of a copper-made heat sink with a rectangular microchannel was increased in the regime of laminar flow after using the nanofluid composed of silver nanoparticles and deionized water as the coolant.19 The green synthesis method is employed to produce silver nanoparticles. The produced nanoparticles were shown to have good stability when they are dispersed in deionized water.20 The influences of stirring, sonication, and surface active agents on the stability of cupric oxide nanoparticles dipersed in water, ethylene glycol, and a mixture of water and ethylene glycol were experimentally investigated. The experimental results showed that the ethylene glycol worked best to disperse the cupric oxide nanoparticles.21 The thermal performances of three kinds of coolants: gallium, nanofluid composed of cupric oxide nanoparticles and water, and water in experiments to cool CPU were investigated. It was found that gallium had the best performance.22 A carbon nanotube aqueous nanofluid was used as a coolant to flow inside a rectangular microchannel of a copper-made sink, and it was found that as the mass concentration of the nanoparticle increased, the heat transfer coefficient increased significantly.23 An experimental study of the thermal performance of the nanofluid composed of alumina nanoparticles and water under the process of flow boiling heat transfer was conducted. The experimental results were that the heat transfer coefficient of the thermal system increased during short time period between 0 and 60 min, and it decreased in the extended time period between 60 and 1000 min.24 Molecular dynamics simulation serves as an effective method to explore the microscopic behavior. The information on each individual microscopic particle including its velocity and momentum can be calculated based on the Verlet algorithm.25−27 The microscopic heat current is defined based on the information on all particles, and it is directly linked to the

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SIMULATION The numbers of the argon atoms are 4000, 3859, 3799, 3775, 3751, 3748, and the numbers of copper atoms are 137, 381, 456, 528, 601 for the cases of copper nanoparticle volume fraction of 0%, 2%, 2.5%, 3%, 3.5%, 4%, respectively. On each side of the simulation box, periodical boundary condition is applied to ensure the configuration that the nanoparticles are uniformly distributed in the base fluid. At the beginning of the simulation, the copper nanoparticle is arranged at the center of the simulation box. Figure 1 illustrates that copper nano-

Figure 1. Copper nanoparticles distributed in argon fluid.

particles are distributed in the argon base fluid. The large particles with black color represent the copper nanoparticle and the smaller particles with blue color represent the argon atoms. Different cases of nanoparticle volume fractions are investigated. As to the cases of nanoparticle volume fraction of 0%, 2%, 2.5%, 3%, 3.5%, 4%, the numbers of argon atoms are 4000, 3859, 3799, 3775, 3751, 3748, and the numbers of copper atoms are 137, 381, 456, 528, 601, respectively. To describe the interaction between the atoms including the interaction between argon atoms, the interaction between copper atoms, and the interaction between argon and copper atoms, the Lennard-Jones potential function27,28 is employed. The 4298

DOI: 10.1021/acs.analchem.7b05350 Anal. Chem. 2018, 90, 4297−4302

Technical Note

Analytical Chemistry potential well depth is 1.67 × 10−21 J, and the characteristic length is 0.3405 nm for the interactions between argon atoms. The potential well depth is 65.63 × 10−21 J, and the characteristic length is 0.233 nm for the interaction between copper atoms. The potential well depth is 10.415 × 10−21 J, and the characteristic length is 0.287 nm for the interaction between the argon atom and copper atom based on the Berthlot mixing rule.27,28 To get the thermal conductivity of the nanofluid, the Green−Kubo relation27,28 is employed. The Green−Kubo relation is widely used to calculate the macroscopic thermal transport coefficient from the microscopic quantities. The underlying mechanism is the fluctuation−dissipation theorem,27,28 which was first discovered by Einstein to mathematically relate the macroscopic transport coefficient of a material to the microscopic properties inside the material. To describe the transport inside a system, the phenomenological relationship is employed:28 Flux = − Transport Coefficient × Gradient

equivalent.27,28 The calculation of the thermal conductivity through Green−Kubo relation is the following:27,28

(2)

[E(x , t ) − E(x , 0)]2 dx

(7)

As t → ∞, we have t →∞

1 2

β

Nk

Nl

β

Nk

∑ hk ∑ v ik k=α

i=1

(12)

where ma is the argon atom mass, σaa is the characteristic length of the argon atom, and εaa is the potential well depth of argon atom. The time-evolution of the signal of the nondimensionalized microscopic heat current is shown in Figure 2. The approximation and detail parts of the signals of the microscopic heat currents inside the nanofluids with different nanoparticle volume fractions are plotted in Figure 3 and Figure 4, respectively. Verlet algorithm27 is chosen to conduct the simulation. It is rooted on Taylor series which decomposes a function into an infinite sum of the function’s derivatives at a reference point. As to the position of one atom, r, the Taylor series r(t + Δt) and r(t − Δt) are expressed as follows:

where L is the length of the one-dimensional domain. The combination of eq 5 and eq 6 results in the following: ⟨[E(x , t ) − E(x , 0)]2 ⟩ = 2Dt

2

J * = J ma /εaa /(εaaσaa2 )

(6)

lim

Nk

there are two components including argon and copper in the nanofluid system, α and β denote copper and argon atoms, respectively, vki is the velocity of atom i of species k(k = α or β), Nα is the number of atoms of species α, Nβ is the number of atoms of species β, hk is the mean partial enthalpy, and I is a unit tensor. In order to simplify the analysis of the microscopic heat current signal, we have conducted the nondimensionalization of the signal. It is rooted on four fundamental quantities including argon atom mass, the characteristic length of argon atom, and the potential well depth of argon atom.

To obtain the link between the microscopic property and the macroscopic property, the mean square displacement of the energy is used as the following: L

(10)

j

(11)

(5)

∫0

∑ uij

⎡ ∂ϕkl ⎤ ij kl kl ⎥ k ⎢ ∑ ∑ ∑ ∑ ⎢r ij kl − ϕij I⎥v i − ∂r ij k=α l=α i=1 j=1 ⎣ ⎦

(4)

1 L

(9)

i=1

∑ ∑ 1 mik(vik)2 v ik − β

If originally, the energy E0 is located at the position x = 0 at the time t = 0, then the solution of eq 4 is the following:

⟨[E(x , t ) − E(x , 0)]2 ⟩ =

∑ riEi

1 1 mi |vi|2 + 2 2

β

J=

Combining the above two equations, we get the following diffusion equation:

2 E0 e−x /4Dt 2 πDt

(8)

where mi is the atom mass, vi is the atom velocity, and uij is the potential energy between two atoms. The microscopic heat current in the nanofluid system is determined as the following:26

(3)

∂E ∂ 2E =D 2 ∂t ∂x

⟨J(0)J(t )⟩ dt

N

k=α i=1

∂(Ex)̇ ∂E + =0 ∂t ∂x

∞

where ri is atom position vector, Ei is the site energy of atom i, and N is the total atom number:

(1)

where E = E(x,t) is the energy at the position x and at time t. dx x ̇ = dt is the derivative of the position with respect to time, that is the velocity at the position x, D is the diffusion coefficient, Eẋ is the flux, which is a measure of the magnitude and direction of the flow of the energy. The conservation of energy leads to the following:

E (x , t ) =

d dt

J=

Ei =

∂E ∂x

∫0

where k is the thermal conductivity, kB is Boltzmann constant (1.38 × 10−23 J/K), T and V are the temperature and the volume of the system respectively, and J is the vector the microscopic heat current. To make the argon be in a liquid state, the nanofluid system temperature is kept as 86 K during the simulation. The microscopic heat current is determined as the following:

Based on the above, we have the following

Ex ̇ = −D

1 3VkBT 2

k=

⟨[E(x , t ) − E(x , 0)]2 ⟩ =D 2t

The above is Einstein relation,27,28 which calculates the macroscopic transport coefficient based the time average of the microscopic properties. Green−Kubo relation can be obtained from Einstein relation; the two are proved to be 4299

DOI: 10.1021/acs.analchem.7b05350 Anal. Chem. 2018, 90, 4297−4302

Technical Note

Analytical Chemistry r(t + Δt ) = r(t ) +

d r (t ) 1 d2r(t ) 2 1 d3r(t ) Δt + Δt + 2 dt 2 dt 3! dt 3

Δt 3 + ο(Δt 4) r(t − Δt ) = r(t ) −

(13)

d r (t ) 1 d2r(t ) 2 1 d3r(t ) Δt + Δ − t dt 2 dt 2 3! dt 3

Δt 3 + ο(Δt 4)

(14)

where t is time and Δt is the time-step in the simulation. The sum of eq 10 and eq 11 results in the following: r(t + Δt ) = 2r(t ) − r(t − Δt ) +

d2r(t ) 2 Δt + ο(Δt 4) dt 2 (15)

Based on the first-order central difference estimator, we have the velocity of the atom at time t as the following: v (t ) =

Figure 2. Microscopic heat currents inside nanofluids.

r(t + Δt ) − r(t − Δt ) 2Δt

(16)

and at the next time-step the velocity of the atom is determined as the following: v(t + Δt ) = v(t ) +

d2r Δt dt 2

(17)

Haar wavelet is employed here to analyze the microscopic heat current. The mother wavelet function of Haar wavelet is mathematically described as the following: ⎧ 1 0 ≤ x < 0.5 ⎪ φ(x) = ⎨−1 0.5 ≤ x < 1 ⎪ ⎩0 otherwise

(18)

and the scaling function of Haar wavelet is mathematically described as the following: ⎧1 0 ≤ x < 1 ϕ(x) = ⎨ ⎩ 0 otherwise

(19)

The Haar wavelet has been extensively used in the approximation of signal. The basic idea of the Haar-waveletbased analysis is to decompose the original signal into two parts: one is to represent the trend of the signal which manifestates the global character of the original signal; the other is to represent the detail in local the domain which serves as the local signature of the original signal. We give a simple example here. As to a signal composed of two numbers, {10, 15}, the number 12.5 = (10 + 15)/2 represents the trend of the signal, and the number −2.5 = (10 − 15)/2 represents local detail of the signal.

Figure 3. Approximation parts of the signals of the microscopic heat currents inside the nanofluids.

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RESULTS AND DISCUSSION The curve fitting of the microscopic heat current signal is conducted based on the least-squares method. It is found that both the probability distribution of the approximation and the probability distribution of detail parts of the microscopic heat current signal can be described as the following: y = I exp((x − c)2 /(2σ 2))

(20)

where the parameter I represents the amplitude of the signal, the parameter c represents the position of the line x = c with respect to which the curve is symmetric, the parameter σ is the standard deviation, the larger is σ, the wider range that the

Figure 4. Detail parts of the signals of the microscopic heat currents inside the nanofluids. 4300

DOI: 10.1021/acs.analchem.7b05350 Anal. Chem. 2018, 90, 4297−4302

Technical Note

Analytical Chemistry Table 1. Parameters for Curve Fitting to Approximate Part of Microscopic Heat Current NVF (%)

I (approximation part)

Sig (approximation part)

C (approximation part)

R2

0 2 2.5 3 3.5 4

0.055 0.07 0.065 0.063 0.063 0.075

0.12 0.28 0.29 0.3 0.31 0.32

0 0 0 0 0 0

0.9927 0.9916 0.9980 0.9969 0.9971 0.9973

The relationship between the microscopic heat current and the nanoparticle volume fraction reflects the relationship between the microscopic and macroscopic characters of the nanofluid system.

probability distribution covers. Table 1 and Table 2 give the data of the above parameters under the different conditions of Table 2. Parameters for Curve Fitting to Detail Part of Microscopic Heat Current NVF (%)

I (detail part)

Sig (detail part)

C (detail part)

R2

0 2 2.5 3 3.5 4

0.06 0.07 0.062 0.062 0.07 0.073

0.002 0.015 0.017 0.019 0.02 0.024

0 0 0 0 0 0

0.9947 0.9952 0.9974 0.9985 0.9958 0.9962

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Tao Jia: 0000-0003-1489-3068 Notes

The authors declare no competing financial interest.

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nanoparticle volume fractions. The R squared analysis for the data in Table 1 and Table 2 has been conducted. One main source of the uncertainty of the data obtained from the molecular dynamics simulation is that the Verlet integration method used to solve the mathematical equations describing the motion of the atoms is precise to third-order, which means the local trunction error varies as (Δt)427,28 when conducting the numerical integration. For the approximation part of the microscopic heat current, the value of the parameter I is the minimum under the condition that the nanoparticle volume fraction is zero, which means there is no nanoparticle in the base fluid, and it is the maximum in the case that the nanoparticle volume fraction is 4%. The same trend appears for the value of the parameter σ. The parameter σ increases with the increase of the nanoparticle volume fraction. As to the detail part of the microscopic heat current, the value of the parameter I has its minimum value in the case that there is no nanoparticle in the base fluid and reaches its maximum under the condition that the nanoparticle volume fraction is 4%. As the nanoparticle volume fraction increases, the parameter σ changes accordingly.

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CONCLUSION The microscopic thermal transport inside an argon−copper nanofluid is studied based on equilibrium molecular dynamics simulation. The signal of the microscopic heat current is analyzed based on the Haar wavelet, and the signal is decomposed into an approximation part and detail part. The probability distributions of the approximation and the detail parts of microscopic heat current signal are investigated, and it is found that the probability distributions show Gaussian-like character. The parameter of the standard deviation of the Gaussian-like curve of the probability distribution increases with the increase of the nanoparticle volume fraction. The signal of the microscopic heat current is a microscopic parameter that reflects the microscopic thermal fluctuation inside the nanofluid system, and the nanoparticle volume fraction is a macroscopic parameter of the nanofluid system. 4301

DOI: 10.1021/acs.analchem.7b05350 Anal. Chem. 2018, 90, 4297−4302

Technical Note

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DOI: 10.1021/acs.analchem.7b05350 Anal. Chem. 2018, 90, 4297−4302