case study iii effective conductivity of vapor -filled
TRANSCRIPT
Massoud [email protected]
Heat Transfer Physics Laboratory, University of MichiganDepartment of Mechanical Engineering and Applied Physics Program
Case Study III
Effective Conductivity of Vapor-Filled Nanogap and Nanocavity Using Nonequilibrium Molecular Dynamics
Outline
1. Effective Gas Thermal Conductivity and Gap Size Effect2. Nonequilibrium Molecular Dynamics (NEMD) Simulations
2.1 Algorithm and Interatomic Potentials2.2 Modeling Confining Solids2.3 Ar Phase Diagram and Adsorption2.4 Knudsen-Flow-Regime Surface Accommodation Coefficients
3. Kinetic Theory on Effective Thermal Conductivity in Nanogap 4. NEMD Results and Comparisons with Kinetic Theory Results
4.1 Nanogap4.2 Nancavity
5. Summary
1. Effective Thermal Conductivity
Figure 1. Schematic drawing of heat transfer in gas-filled nanogap both at high and low temperatures. At low temperatures, the
adsorbed layers are also shown.
• Thermal conduction by the Fourier Law is
,k gf
qk
T z= −
∂ ∂
• The thermal conductivity (bulk) from kinetic theory of gases is [1]
,13f f v f f fk n c u λ=
where nf is the fluid particle number density, cv,f is the specific heat per particle, uf is the average velocity of particles, λf is the mean free path.
• Thermal conductivity is predicted using NEMD, and interfacial phenomena, e.g., thermal and velocity accommodation coefficients and adsorption are also investigated.
[1] M. Kaviany, “Heat Transfer Physics”, Cambridge, 2008.
2.1 NEMD Simulations: Algorithm and Interatomic Potential
( )12 6
-4ij ij ij cut offij ij
r r rr rσ σϕ ε
= − <
• Interatomic potential for Ar-Ar and Pt-Ar is the Lennard-Jones potential [3]
• Equation of motions and the velocity Verlet algorithm [2]
( )ij ij ijF r ϕ= −∇ 21( ) ( ) ( ) ( )2
x t t x t u t t a t t+ ∆ = + ∆ + ∆1( ) ( ) ( )
2 21( ) ( ) ( )
2 2
tu t u t a t t
tu t t u t a t t t
∆+ = + ∆
∆+ ∆ = + + + ∆ ∆
• Interatomic potential for Pt-Pt is harmonic potential
( ) ( ) 212ij ij ij or r rϕ = Γ −
2,
1 1 1 1 1
1 1 ( )2
N N N N N
k g i i i ij i ij i ii i j i j
q m u u u uV
ϕ= = > = >
= + − ⋅
∑ ∑∑ ∑∑ r F
• The heat flux (ensemble average) is
[2] D. Frenkel and B. smit, “Understanding Molecular Simulation”, Academic Press, 2002
• NVT ensemble (N, V, T are fixed), and energy is transferred by the temperature gradient.
φij, meV σij, nm
Ar-Ar 10.4 0.341
Ar-Pt 6.8 0.309
[3] C.S. Wang, J.S. Chen, J. Shiomi, and S. Maruyama, “A study on the thermal resistance over solid-liquid-vapor interfaces in a finite-space by a molecular dynamics method”, International Journal of Thermal Sciences, 46, 1203-1210, 2007
2.2 Modeling Confining Solids• To save on the number of solid atoms used in the MD simulations, the Langevin thermal heat bath
with (111) plane of FCC Pt and phantom molecules, are used to control the temperature of the bounding solids [3, 4].
( )( )
( )
0.5
0.5
2
x re ph
y re ph
z re ph
F x x
F y y
F z z
= Γ −
= Γ −
= Γ −
( )( )
( )
3.5
3.5
2
x ph fix
y ph fix
z ph fix
F x x
F y y
F z z
= − Γ −
= − Γ −
= − Γ −
( ) ( ),6 6
B Di i i i i D
k Td m t m mdt
π πγ γ ω= − + = =
u F u F
21/ 2
1 1 1[ ( )] exp[ ( ( ) ) ](2 ) 2i i F
F
f t t σπ σ
= −F F
[4] J. Blomer and A.E. Beylich, Molecular dynamics simulation of energy accommodation of internal and translational degrees of freedom at gas-surface interfaces, surface Science, 423, 127-133, 1999.
1/ 2B2( )SF
k Tt
γσ =∆
Figure 2. The Langevin thermal heat bath using phantom molecules (k is the same as Г).
• Langevin equation for thermal heat bath • Treatment of phantom molecule
iF : steady, deterministic force on particle i
( )i tF : random (stochastic), time-dependent force on particle i
2.3 Ar Phase Diagram and Adsorption
• Temperature and pressure are calculated by ensemble averages as (used here for gas pressure only)
2
1
B
N
i ii
f
m uT
N k=
< >≡∑
B
1
1, ( )3
N N
ij iji j i
Nk TpV V = ≠
≡ + ⋅∑∑ x F
Figure 3. The NEMD simulated gas temperatures and pressures, marked in the Ar phase diagram [5]. Triple and
critical points are also shown.
• Adsorption: fluid particles interact with solid atoms by surfaces forces, which in turn results in a thin layer of the condensed fluid phase, i.e., gas adsorption.
• Strong surface forces - temperature - pressure control the thermodynamics states of the fluid particles.
• Phase change between gas (vapor) and liquid occurs when the operating condition is close to gas-liquid saturation curve.
[5] P.J. Linstrom and W.G Mallard, “NIST chemistry WebBook (NIST Standard Reference Database Number 69, July), 2001.
2.4 Knudsen-Flow-Regime Surface Accommodation Coefficients
• The momentum (velocity) accommodation coefficient is [1]
Figure 4. Fluid particle-surface interaction showing an impinging fluid particle with a z-component velocity colliding and reflecting from a solid surface.
, ,
,
f x f xu
f x
u ua
u
′−=
• The thermal accommodation coefficient is [1]
, ,
, , ( )f z f z
Tf z f z s
q qa
q q T
′−=
−
• For aT = 1, there is a complete thermal accommodation (fluid particle returns with the surface temperature), whereas for aT → 0 fluid particles retains its temperature after reflection.
• Thus, large effective thermal conductivity requires high thermal accommodation coefficients.
[6] P. Spijker, A. Markvoort, P. Hilbers, S. Nedea, Velocity correlations and accommodation coefficients for gas-wall interactions in nanochannels, AIP Conference Proceedings 1084 , 659–664, 2008.
1 1 1( ) ( )2 2f i i i i i i i ij ij
i i i jm
Vϕ
= ⋅ + + ⋅
∑ ∑ ∑∑q u u u u u F x
Momentum and Thermal Accommodation Coefficients
• Momentum and thermal accommodation coefficients strongly linearly increase as εs-f / kBT increases at high temperature, whereas they become nearly constant at low temperatures. These constant accommodation coefficients is in part affected by adsorption (surface coverage by fluid particles, screening the solid force field) at low temperature. The discrepancies may be due to surface roughness.
Figure 5. Predicted momentum accommodation coefficient au as a function of dimensionless inverse temperature. Other simulation
results are also shown [7].
Figure 6. Predicted thermal accommodation coefficient aT as a function of dimensionless inverse temperature. Experimental results
are also shown [8].
[7] G. Arya, H. Chang, E. McGinn, Molecular simulations of knudsen wall-slip: Effect of wall morphology, Molecular Simulations, 29, 697–709, 2003.[8] S. Borisov, S. Litvinenko, Y. Semenov, P. Suetin, Experimental investigation of the temperature dependence of the accommodation coefficients for the gases He, Ne, Ar, and Xe on a Pt surface, Journal of Engineering Physics and Thermophysics 34 (5) (1978) 603–606.
3. Kinetic Theory: Effective Thermal Conductivity of Nanogap
• Effective thermal conductivity in the free-molecular regime is [9, 10]
( )( )
, ,1 2
,1 ,1
2 11 1 21
f fm v f g
g
T T
k p c Rl MR T
a aπ
+=
+ −
,1 ,2 1, ,
,1 ,2 ,1 ,2
4 1(1 )15 Kn
T Tf t f fm
L T T T T
a ak k
a a a a−= +
+ −
• Effective thermal conductivity in the transition regime is [10]
• Effective thermal conductivity in the bulk regime is [11]
,54f f v f f fk n c u λ=
[10] G.S. Springer, “Heat Transfer in rarefied gases”, Advances in Heat Transfer, 163-218, 1971.[11] C.L. Tien and J.H. Lienhard, “Statistical Thermodynamics”, Holt, Rinehart and Winston, New York, 1971.
[9] E.H. Kennard, “Kinetic Theory of Gases”, McGraw-Hill, New York, 1938.
Figure 7. Classification of free-molecular flow, transition flow, and continuum (viscous) regimes for thermal conductivity of gas
occupying the gap between two parallel plates [1,7].
The above nanogap thermal conductivity regime diagram uses accommodation coefficient of the two bounding surfaces and inverse of Kundsen number [10].
4.1 Effective Thermal Conductivity: Nanogap
• At the high temperatures, MD results show agreement with the kinetic theory (within a 20% uncertainty).
• At the low temperatures, the high thermal accommodation coefficients result in high thermal conductivity tend to partly compensate for the low particle number density.
• MD results agree with the kinetic theory (within 15% uncertainty), the difference is larger at high pressures.
• The adsorbed layer (low temperatures) tends to slightly enhance the effective thermal conductivity.
Figure 6. Effective thermal conductivity of nanogap (using NEMD) as a function of pressure, for a few temperatures. Comparison with kinetic theory is also shown.
Temperature Distribution within Nanogap• Local temperature distributions within nanogap are shown in the figure below, for high temperatures (no
adsorption), at high (a) and low (b) pressures.• Temperature slips are found near the surfaces, due to low thermal accommodation coefficients and are
more pronounced at low density (b).• Moderate temperature gradient is observed in the gas phase at high pressure (a), whereas nearly constant
temperature distribution at low pressures (b).
Figure 7. Temperature distributions along the gap. Large temperature gradient is found near the walls, (a) Nf = 1,500, and (b) Nf = 600.(a) (b)
Visualization of NEMD in Nanogap
• High Temperature Nanogap (large number of gas particles)
• Low Temperature Nanogap (large number of adsorbed particles)
4.2 Effective Thermal Conductivity: Nanocavity
(i) Fluid particles desorb from the hot surface.
(ii) Then thermally diffuse due to the temperature gradient.
(iii) Adsorb on the cold surface.(iv) Return towards the hot surface by
surface diffusion (concentration gradient).
(i)-(iv) Repeat.
Figure 8. Possible circulation in a nanocavity, showing four steps in the circulation.
• Fluid particles are completely confined by solid surfaces in nanocavity and a collision with and diffusion along side wall may produce circulation and influence the effective thermal conductivity, with the following possible scenario for the circulation (also shown in the figure).
NEMD Results: Nanocavity
• At the high temperatures and pressures, MD results are close to the kinetic theory for no side walls (nanogap).
• Adsorption surface coverage is most significant at the edges, where the solid forces overlap.
• At low temperatures, MD results agree well with the kinetic theory. It shows unaltered mean free path, since the adiabatic side walls cause elastic (specular) reflections. Also, sluggish surface diffusion does not provide for a significant surface flow rate.
• For significant surface diffusion, large adsorption coverage is needed, and this would require low temperatures and reduces the number of gas-phase fluid particles in simulations (thus low pressure and effective thermal conductivity).
Figure 9. Prediction of effective thermal conductivity in nanocavity using NEMD as a function of pressure at given temperatures. Comparison between MD results and kinetic theory is also shown.
Visualization of NEMD in Nanocavity
• High Temperature Nanocavity (thin adsorbed layer on the surfaces)
• Low Temperature Nanocavity (thick adsorbed layer on the surfaces)
5. Summary
• For nanogap, momentum and thermal accommodation coefficients are obtained using NEMD and are in general agreement with existing experimental and MD results. These coefficients initially linearly increase as εs-f / kBT increases and the become nearly constant (and eventually reaching unity). This constant regime is affected by adsorption (screening of the solid particle fields).
• The MD results are in agreement with the kinetic theory effective thermal conductivity of rarefied gas, to within 20%. This agreement extends to the adsorption regime, since the thermal accommodation is already large at low temperatures.
• In nanocavity, the most adsorption coverage is in the edges where the surface forces overlap. This is not sufficient to produce significant surface diffusion for the surface return of the particles thus producing a circulation.
• The effective thermal conductivity is nearly the same as for the nanogaps, while the adiabatic side walls allow for elastic collisions without significant change in the fluid particle collision mean free path.
Acknowledgement
This presentation is based on the Ph.D. theses work of Dr. Gi Suk Hwang, and I am grateful for having the opportunity to present his work.