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Файл:Ординатура / Хирургия / Библиотека им академика М.И. Перельмана / Книга_39_библиотеки_им_акад_М_И_Перельмана
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172 6 Nanofoils in Dental Joining Practice
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Fig. 6.5 A schematic depiction of a self-propagating reaction propagating from left to right in a
multilayer foil. The unreacted foil is made up of alternating layers of elements A and B with an
intermixed area in between
Numerous investigations have been conducted [1–5
] of interface responses in
various multilayer systems. The apparent large free energy accessible for phase
formation, usually several tens of kJ/mol, is a feature shared by these multilayers.
As a result, we can infer that all possible product phases should be able to form from
the start of the reaction.
6.4.1 Numerical Modelling
The possibility is presented of Numerical Modelling using Computational Fluid
Dynamics (CFD) in the area of nano-foils [
(FVM) was used to answer t he governing equations. The computational domain
was discretised using a uniform Cartesian grid, with the corresponding number of
grid points along the x and y axes. Field variables were discretised at cell centres, and
the spatial and temporal derivatives were approximated with a second-order accurate numerical scheme. For the appropriate Al-Au nano-foil geometry and boundary
conditions, the time-evolution of temperature and concentration fields will be given,
26]. A Finite Volume Methodology
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6.4 CFD Analysis of Exothermic Reactions in Al-Au Nanofoils 173
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Density
Au 19320 130 318
Al 2700 910 237
Fig. 6.6 Schematic overview of Au-Al nano-foil, segment:
(up); boundary and initial conditions (middle) with physical properties for Au and Al (down)
Specific heat
Thermal conductivity
L = 2000nm, δ
Atomic Packing Factor
(A.P.F)
0.74
0.74
= δ
Au
= 100nm
Al
as well as the atomic diffusion coefficient. The following assumptions underpin the
formulation of the mathematical model used to explain self-propagating reactions in
multi-layered Al-Au nano-foil: (1) The effects of phase changes in the foil are ignored
(e.g., melting of the reactants and/or products); (2) Atomic diffusion is represented
by a single binary diffusion coefficient D; and (3) The physical properties of the foil
(e.g., density, thermal conductivity and specific heat c
) are assumed to be compo-
p
sition dependent. Last but not least, the breadth of the foil (z) is greater than the
thickness (y) but less than the length. (x). As a consequence, thermal and atomic
diffusion in a foil can be treated as a two-dimensional problem.
6.4.1.1 Governing Equations
Under the above conditions, atomic mixing is described by a time-dependent,
conserved scalar field C(x,y,t), where C = 1 for pure Au and C = 0 for pure Al.
C’s evolution is controlled by:
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∂C
∂t
∂
=
∂ x
j
∂C
D
∂x
j
(6.1)
According to [22, 23], the atomic diffusivity is considered to be composition
independent and to follow the Arrhenius temperature dependence:
D = D0exp−
E
RT
(6.2)
where D0 is the Arrhenius pre exponential, E is the activation energy and R is the
universal gas constant. The values D
= 6.80 × 10
0
used in the present study are taken from the work of Fouracre [
-4 m2
/s and E = 25.20kJ/mol as
19].
The time-evolution of the concentration field is coupled with the temperature
equation:
∂T
∂t
∂
=
∂x
ρc
j
λ
∂ T
∂x
p
j
∂
+
∂t
Q(C
ρc
)
p
(6.3)
where the physical properties (thermal conductivity λ, density ρ, and specific heat
) of any domain control volume are provided by the fraction of the phase (Au and
c
p
Al) in that volume. They can be calculated numerically as follows:
λ = λAuC + λ
1 − C),ρ = ρ
(
Al
Au
C + ρ
1 − C), c
(
Al
p
= c
p
Au
C + c
p
(
Al
1 − C
)
(6.4)
The rate of heat generation ((Q(C))t) is supposed to be proportional to the rate of
change in the composition of the foil. The linear relationship between the composition
and the energy released can be presumed in the simplest case. A parabola function,
on the other hand, may be more accurate:
Q(C ) =−ρc
T
p
2
C
− T
f 0
0
(6.5)
and was used in the present study.
T
and T0 are the adiabatic temperature of the reaction and the initial temperature
f 0
of the foil.
6.4.1.2 Geometry, Initial and Boundary Conditions
Figure 6.6 depicts the laminar and time-dependent thermal and species transfer in a
two-dimensional nano-foil.
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6.4 CFD Analysis of Exothermic Reactions in Al-Au Nanofoils 175
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Density ρ,
kg/m
Au 19,320 130 318 0.74
Al 2700 910 237 0.74
3
Specific heat cp,
J/kgK
Thermal conductivity λ,
W/mK
Atomic packing factor (A.P.F)
6.4.2 Numerical Procedure
The governing equations were solved with the Ansys CFX numerical code, which
uses standard finite volume methodology, with all variables defined at the centre of
the control volumes populating the physical area under consideration. Each equation
is integrated over the control volume to produce a discrete equation that links the
variable at the volume’s centre to its neighbours [
The computational space was discretised using a uniform Cartesian grid with
mesh size Δx = Δy along the x and y axes. N
corresponding amount of grid points. The second order accurate scheme, based on
central-differences, was used for spatial discretisation of diffusive terms, while the
temporal derivatives were estimated with second order accurate backward differences
–9
for a fixed time step Δt = 10
s.
26].
= 800 and Ny = 80 represent the
x
6.4.3 Results and Discussion
Figure
6.7 depicts the time development of the temperature and concentration fields.
The temperature (i.e. thermal) transport is seen to be basically unidirectional (mostly
in the axial direction) and happens parallel to the layering.
In contrast to previous studies [23
one-dimensional phenomenon (occurring normal to layering), the current research
demonstrates that atomic diffusion is a two-dimensional phenomenon (see Fig.
This is because the diffusion coefficient D, as described by Eq. (
temperature (in fact, it rises as the temperature increases) in the axial direction. Of
course, if the diffusion coefficient remains constant, concentration diffusion will
only occur in the streamwise path (due to the concentration gradient). Finally, the
concentration diffusion process was much faster in the current research, due to the
higher value of the diffusion coefficient D in comparison to the value of the thermal
diffusion coefficient
α.
In summary, the time-dependent temperature and concentration diffusion in the
multi-layered Al-Au nano-foil was investigated numerically in this research using
mathematical modelling and a finite volume-based CFD code. The atomic diffusion
coefficient D was assumed to be composition independent and to follow the Arrhenius temperature dependence, whereas the physical properties (and, thus, thermal
], which modelled atomic diffusion as a
6.5).
6.2), varies with
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t = 0 ns
t = 1 ns
t = 5 ns
Fig. 6.7 Time evolution of the temperature and concentration field
t = 50 ns
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References 177
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diffusion) in any control volume of the domain were assumed to be dependent on
the corresponding fraction of the phase (Au and Al) in that volume [
26].
Due to the variable diffusion coefficient [19] and concentration gradient, the
current numerical results indicate that temperature diffusion is essentially unidirectional (in the axial direction), whereas atomic diffusion occurs in both (i.e.
the axial and streamwise) directions. The latter is an important fact (influencing
the thickness of the intermixed area between the layers, as well as the heat release
in the process) that has been overlooked in the majority of existing analytical and
computational models.
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