Computer simulation of temperature profiles of a two-layer sample during heating by the electron beam
Table of contents: The Kazakh-American Free University Academic Journal №4 - 2012
Alontseva Darya, East Kazakhstan State Technical University in honor of D. Serikbayev, Kazakhstan
Krasavin Alexander, East Kazakhstan State Technical University in honor of D. Serikbayev, Kazakhstan
The efficiency of
advanced technology of getting protective coating by means of pulsing plasma
jet deposition of Ni-based powders onto steel items often falls due to the porosity of the received
coatings and their poor adhesion to the substrate [1,
2]. To eliminate these disadvantages the coatings are modified by the
plasma jet or electron beam . The processes of diffusion and formation of
new phases in materials under the influence of electron irradiation happen very
quickly, the temperature being one of the main factors influencing these
processes. However, the temperature measurement under irradiation conditions is
difficult and unreliable. Development of a mathematical model of temperature
distribution in a material depending on irradiation parameters makes it
possible to assume the kind of structures and phases that form in the material
during irradiation (on the basis of the received values of temperature and the
known phase diagrams). Based on this model, one can choose the parameters of
irradiation so as to develop sufficiently high temperatures on the boundary of
the coating to the substrate to accelerate the diffusion processes in order to
improve adhesion of the coating to the substrate. The sources devoted to the
development of such a model [3, 4] testify the relevance of this problem, but
they do not provide a comprehensive solution.
The aim of this work is to
propose a model of temperature distribution in two-layer metal absorbents
during irradiation by a direct current electron beam depending on the energy
and beam current density; on the basis of a simulation experiment on the calculation
of temperature profiles to recommend specific irradiation modes; to carry out
the exposure to radiation according to these modes.
Results and Discuss
Experiment and modeling
The protective coatings with a thickness of 150 to 300 microns were
formed on a substrate of quality carbon steel St3 (20x30x10 mm3 samples) using "Impulse-6"
plasma-detonation facility. They were deposited with the PG-10N-01 and PG-AN-33
(Russian standards) Ni-based powder alloys.
The irradiation of samples on the side of the surfaces according to
the calculated modes was carried out in vacuum by an electron beam on
"U-212" generator with an accelerating voltage of 30 kV. The scan is
sawlike; the beam travel speed in the horizontal direction is 360 mm/min; the
diameter of the electron beam on the sample is 10 mm; the current amperage is 20-30mA.
The need for a detailed explanation of the coating structure scheme
stems from the fact that in order to develop a mathematical model of
temperature distribution in the coating during irradiation we have to justify
the choice of material and thickness of the irradiated layers. Resting on
reliable experimental data [5-7] we proposed a layered scheme of the coating
structure . A thin layer (less than 5 microns) with mostly Cr oxides and
carbides forms on the coating surface. Then comes the main layer of the
Ni-based coating, 100-300 microns thick, then a layer of Fe (substrate), 10 000
microns thick. Because of the small thickness of the Cr layer on the surface,
this layer was neglected when calculating the temperature profile during
electron irradiation, and Ni-Fe double-layer coatings irradiated from the Ni
side were considered.
In order to formulate the problem of describing the heating of the
coated sample by a moving electron beam as a boundary problem of heat
conductivity theory, it is necessary to specify the density of heat sources in
a composite solid body. Since the thickness of the coating layer in which the
electron beam is almost completely absorbed is very small compared to the
thickness of the coating, and we are interested primarily in the temperature
field at the boundary surface between the coating and the substrate, we
simulate a moving beam of electrons by a moving flat normal-circular source of
a given power, i.e. we assume that the specific heat flux at a distance r from
the point of intersection of the symmetry axis of the beam with the sample
surface is given by expression (1) (without considering losses):
where qmax=kN/π (N- beam power, N=UkI, where Uk- cathode voltage, and I – the beam amperage), and the heat flux
concentration ratio k is correlated with the heating spot radius Rb (the beam radius) by the formula
k=1.125/Rb2. The analytical solution of the problem of
heating a plate of finite thickness with a moving normal-circular source
presented in the literature  makes it possible to roughly estimate the
maximum heat value of the points on the unheated sample surface. The corresponding
calculations for the given ranges of beam energies and the geometrical dimensions
of the sample show that the maximum heating (the difference between the maximum
temperature reached by a point and the initial temperature of the sample) for
the points on the ends and the "back" side of the plate does not
exceed 3° C. Thus, the nature of the heat exchange with the environment on the
unheated plane of the substrate and the ends of the sample has little effect on
the temperature distribution in the contact area of the substrate and coating;
and we simulate a sample by an infinite plate of thickness h lying on
the surface of the semi-infinite space filled with a material with desired
Introducing the Cartesian coordinates by the method indicated in
Fig. 1 (X and Y axes lie in the plane of the surface coating, Z axis
points into the sample), we believe that at the time t0=x0/v a normally circular source begins to operate at the surface, its center moves
uniformly with velocity v along the axis X, and switches off at
time t1=-t0 (and at time t=0 corresponds to
the passage of the beam center point with the coordinates O(000).
Since the heating occurs in vacuum, we believe that the only
mechanism of heat loss from the heated surface of the coating is the heat
emission described by the Stefan-Boltzmann equation
where ð - beam
surface power density [W/m2] s - the Stefan-Boltzmann constant, ε –
the emissivity factor for the coating material.
1. Schematic representation of a two-layer sample with a moving spot during heating
by the electron beam, indicating the choice of the coordinate system
Thus, we have the following problem of heat conductivity theory:
find function T1 (x,y,z,t) (temperature of the coating) and T2 (x,y,z,t) (substrate temperature), as defined in areas S1 and S2 respectively (area S1 is defined by
inequalities 0 ≤ z ≤ h, t0 ≤ t ≤ t1, while are S2 is
defined by the inequalities h ≤ z ≤ ∞ and t0 ≤
t ≤ t1, at that for both areas õÎ(-¥, ¥) and yÎ(-¥, ¥), that comply in these areas with the differential equations (3)
where l1=l1(T) the thermal conductivity of the
coating material, considered as a function of temperature and l2=l2(T) the thermal conductivity of the substrate material, also considered
as a function of temperature. In the calculations for computing the values of
the functions l1(T) and l2(T) we used polynomal interpolation on tabulated values of thermal
conductivity of nickel and iron, c1=c1(T) and c2=c2(T) – specific heat capacity of the coating and the substrate, respectively,
also considered as a function of temperature; r1 and r2 the density of the coating and
the substrate materials (the constants), when the initial and boundary
conditions described below are met: the initial conditions: T1(x,y,z,t0)=T0 and T2(x,y,z,t0)=T0, where T0 - the initial temperature of the sample set equal to T0=20°C; the boundary conditions (5), (6), (7) è
(8): at the boundary z=0 (the coating surface) – condition (5)
where P(x,y,0) – the point on the surface of the coating, and (T1)p=T1(x,y,0) and respectively,
the values of the temperature and the normal derivative of temperature at the
point , – thermal
conductivity of the coating material (depending on the temperature), – the distance
from point P to the center of the normally circular source (Xö(t)=X0+vt); at the boundary
between the coating and the substrate (plane z = h) must be met the two
conditions (6) and (7):
where the thermal
conductivity of the substrate material, considered as a function of temperature
that is, for all x, y and any t,
belonging to the interval (t,t0) at z tending to
infinity, the temperature tends to the initial temperature of the sample Ò0, condition (8)
The problem was being solved by the finite-difference method. We
used the data  for the values of the thermal conductivity, emissivity,
specific heat and density of Ni and Fe. Fig. 2 shows the dependence of the
temperature at the point with the coordinates (0,0,h) (the point that
lies at the boundary surface between the coating and the substrate) on the time
at the following design parameters: coating thickness h=300 µm, the beam power N=300W (cathode voltage Uk=30 kV , beam amperage I=20 mA),
beam radius Rb=5 mm, beam velocity v=0.004 m/s,
calculation time interval t1-t0=14s (t0=7.0s),
correspondingly x0=-28 mm. Fig. 2b displays the corresponding temperature dependence on the z coordinate for the point with the
coordinates x=0, y=0 at the time t=0 at the above
calculated parameters (the source switches on at the time t0=7.0 s,
time t=0 corresponds to the center of the source passing the point with
the coordinates (0,0,0)).
The samples of Ni-based coatings were additionally irradiated according
to the modes recommended in the result of numerical simulation calculation:
electron beam current density – 20 mA/cm2, accelerating voltage – 30
kV, in the continuous exposure regime.
For practical calculations at low electron energies we needed to turn
to the experimentally obtained patterns. The empirical evidence  suggests
that the 1-2 mm thick Ni layer at the energies of the electron beam of 30 keV
is being completely absorbed. Since the depth of the total absorption of
electrons is extremely small in comparison with the thickness of coatings, a
model of surface distributed sources of heat can be taken for the construction
of the temperature profile in the sample. In our proposed model not only a high
temperature in the boundary zone is achieved, but also long enough, the order
of several seconds, holding of the area in the high temperature diapason of
400° C is provided, which allows for diffusion processes. The model enabled to
choose low current density values, which allows one to save energy for further
processing, without penetration into the coating or substrate.
Dependence of the temperature of a sample point on the boundary of the substrate
and the coating on the time at the surface heating by a moving beam of electrons
(a) and the corresponding temperature dependence on the coordinate z (b)
Based on the model of temperature distribution in two-layer
absorbents with the surface distribution heat sources, the temperature profiles
were calculated according to the irradiation parameters and conditions. The
choice of materials and thicknesses of absorbent layers is based on the
experimentally developed scheme of the structure of thick plasma-detonation
powder coatings. Basing on the calculations we proposed the modes of exposure
leading to the formation of high temperatures in the coating - substrate
contact zone to accelerate diffusion processes.
This research was funded by the National Agency of Technology Development
of Kazakhstan for the projects No. 389, “Development of technology for surface
modification of by irradiation to produce nanostructured multifunctional protective
coatings with high performance properties.”
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Table of contents: The Kazakh-American Free University Academic Journal №4 - 2012