УДК 536.21, 537.32, 548.55
Конечно-элементное моделирование и аналитическая аппроксимация
процесса электрического нагрева и охлаждения корсетного образца из
монокристаллического сплава на никелевой основе
Савиковский Артем Викторович,
студент ФГАОУ ВО СПбПУ
Семенов Артем Семенович,
к. ф.-м. н., доцент, преподаватель кафедры
МПУ, ФГАОУ ВО СПбПУ
Гецов Леонид Борисович,
д.т.н., проф., ОАО «НПО ЦКТИ»
монокристаллических корсетных образцов, подвергающихся воздействию
неоднородному нагреву при различных температурных режимах. Введено
аналитическое приближение для процесса нестационарного нагрева корсетного
экспериментальными данными и аналитической моделью, которое показало
хорошую корреляцию с экспериментальными данными.
Ключевые слова: корсетный образец, монокристаллические сплавы на
термоусталостная прочность, конечно-элементное моделирование, ANSYS,
нестационарное уравнение теплопроводности.
The finite element modeling and an analytical approximation of thermoelectric heating and cooling processes of corset samples from
a single-crystal nickel - based superalloy
Semenov A.S., PhD
Getsov L.B., D.Eng.
Abstract. The results of computations of the thermal state of single-crystal corset
specimens subjected to the action of periodic electric current, leading to variable
inhomogeneous heating are presented for different temperature modes. An analytic
approximation was introduced for the process of nonstationary heating of corset
sample and a comparation of modeling results with experimental data and analytic
model was evaluated which showed a good correlation with experimental data.
Keywords: corset sample, single-crystal
nickel - based superalloys, nonstationary
thermoelectric boundary value problem, thermal fatigue strength,
modeling, ANSYS, unsteady heat conduction equation.
Single-crystal nickel-based superalloys  are used for production of gas turbine
engines (GTE). These materials have a pronounced anisotropy of properties and a
dependence properties on temperature and thermal fatigue strength of superalloys is
not studied very well. For the investigation of thermal fatigue durability under a wide
range of temperatures the experiments are carried out on different types of samples,
including corset (plane) specimen on the installation developed in NPO CKTI 
(see Fig. 1). Fixed in axial direction by means of two bolts with a massive foundation
the corset sample (see Fig. 2) is heated periodically by passing electric current
through it. During cycling the maximum and minimum temperatures are
automatically maintained constant.
Fig. 1. Installation for carrying out
Fig. 2. Geometry corset sample for thermal
experiments on thermal fatigue.
The aim of the study is to investigate numerically a process of heating and cooling of
corset sample and to request analytical approximation for this process for single
crystal superalloys using the results of finite element (FE) simulation of full-scale
experiments and results of analytical formulae. The results of simulation and their
verification are obtained for one single-crystal nickel-based superalloy: VZhM4.
Modeling of a heating process by an electric current and a cooling process without
an electric current of the corset sample was carried in the FE program ANSYS with
taking into account the temperature dependence of all material properties,
thermoelectric contacts between the sample and an equipment, nonstationary Joule
heating, the convective heat exchange and radiative heat transfer between the sample
and the environment. The full-scale FE model of experimentation object including
discrete models of the specimen and the setup is presented in Fig. 3.
Fig. 3. Finite-element model in thermoelectric problem
The problem was solved for one single-crystal nickel-based superalloy VZhM4.
Modeling of heating and cooling processes of sample was carried out for five
temperature regimes (modes): 100÷800, 150÷900, 250÷1000, 500÷1050 and
700÷1050 °C. There are nonstationary experimental data for four regimes: 100÷800
°C, a heating time is 19s, a cooling time is 46s, 150÷900 °C, a heating time is 42s, a
cooling time is 59,5s, 500÷1050 °C, a heating time is 14s, a cooling time is 10s,
700÷1050 °C, a heating time is 8s, a cooling time is 7s. In case of temperature mode
250÷1000 time of heating is 80 s. The used in FE simulations material properties for
the single crystal nickel superalloy sample and for the steel equipment were taken
from literature , - (see also Tables 1-2). While specifying the properties of
nickel alloy and steel the implementation of the Wiedemann-Franz’s law was
controlled: 𝜆 · 𝜌𝑒 = 𝐿𝑇, where 𝜆 is the thermal conductivity, e is the specific
electrical resistance, T is the temperature in K, L= 2.22 *10-8 W··K-2 is the
Thermo-electric properties of nickel superalloy used in simulations:
Thermo-electric properties of pearlitic steel used in simulations:
The coupled three-dimensional transient thermo-electrical analysis has been
performed. Due to the symmetry in respect to the xz and yz planes, a quarter of the
structure was considered. The thermal and electric contacts between the sample and
bolts, between the sample and the foundation were taken into account. The initial
temperature for the sample and the equipment was set to 30 °C. For the free surface
of sample the boundary condition of convective heat transfer is used:
𝑞𝑛 = ℎ(𝑇 − 𝑇0) ,
where n is the normal to body, qn is the heat flux density, ℎ = 20
coefficient of convective heat transfer, 𝑇0 is the ambient temperature. The condition
of radiative heat transfer was also set on the surfaces of central (high temperature)
part of the sample (10 mm length): 𝑞𝑛 = ε𝜎𝑆𝐵 (𝑇 4 − 𝑇04) ,
where ε = 0.8 is the black factor of the body, 𝜎𝑆𝐵 = 5.67 ∙ 10−8𝑊𝑚−2𝐾 −4 is the
coefficient of Stefan-Boltzmann.
In order to realize an analytical approximation for the curve of temperature change in
time, we consider the problem of mathematical physics of heating the sample with a
constant cross-section. For example, the sample has a length and a depth the same
with the corset sample 32.5 mm and 3 mm respectively, but the sample with a
constant cross section has width is equal to 10 mm (fig. 4).
Fig. 4. The statement of simplify thermal problem
The aim of our analogy is to simplify a task of heating the corset sample to onedimensional problem with equivalent boundary conditions. The boundary conditions
of a lack of heat flow were set on surfaces S3, S4, S5, S6. On the surface S2 was fixed
the temperature, on the surface S1 boundary condition of convection with a
convective heat transfer coefficient h is equal to 20
. The sample in the thermal
problem is heating by electric current that’s why boundary condition of heat
generation was set on the sample is equal to some constant Q, which does not depend
on time. The equation of unsteady thermal conductivity can be represented as
where T is the temperature, ∆ 𝑖s the laplassian operator, τ is the slow time and τ =
Q is the heat generation, 𝜆, 𝐶𝜌 , 𝜌 are the conductive coefficient, the specific heat and
the density respectively. Considering that boundary conditions in the axis y and z are
a lack of heat flux and overwriting the laplassian operator in Cartesian coordinates,
we come to the equation:
where x is the axial coordinate along the sample. Representing T as a sum of two
functions T1(x) and T2(x, τ), we come to two equations. One of these equations has
two variable, x and time:
Using Fourier method , we put two equations with variables X and 𝜴 respectively.
The equation with a variable X is
𝑋 ′′ + 𝛽𝑋 = 0,
where 𝛽 is the arbitrary constant. Boundary conditions for the equation (6) are a
convective boundary condition in the middle of the sample and temperature is equal
to zero on the edges. Also the equation with a variable 𝜴 is Ω′ + 𝛽Ω =0,
where 𝛽 is the arbitrary constant from equation (6). Finding a solution of equation (6)
as a sum of sinus and cosine with constants and substituting boundary conditions we
put a transcendental equation:
where ɣ𝑛 = √𝛽𝑛 l, h = 20
tgɣ𝑛 = -
τ) = C·X(x)· 𝑒
,l = 32.5 mm. General solution of equation (7) is Ω =
C𝑒 −𝛽𝑛𝜏 , where 𝛽𝑛 is the eigenvalue and 𝛽𝑛 =
. In general, we put solution T2(x,
, where ɣ𝑛 𝑐𝑎𝑛 be found from an equation (8). We use simple
approximation for temperature changing in time for a heating and cooling as one
exponential with exponent -
with constants. Returning to usual time t, we can
rewrite an analytical approximation for heating as:
𝑇 = 𝐴−𝐵·𝑒
ɣ𝑛 2 𝜆
𝑙 2 𝐶𝜌 𝜌
where A and B are constants, which are selected from conditions of equality in the
beginning of the heating to minimum temperature in the cycle and in the end of the
heating to maximum temperature in the cycle, ɣ𝑛 𝑐𝑎𝑛 be found from a transcendental
equation (8), l is the length of the sample. For a process of the cooling of the sample
similar analytical approximation was considered:
Materials constants 𝜆, 𝐶𝜌 , 𝜌 were
𝑇 = 𝐶+𝐷·𝑒
ɣ𝑛 2 𝜆
𝑙 2 𝐶𝜌 𝜌
respectively.Comparison of experimental data, computational results and analytical
approximation for temperature changing in time are presented in fig. 5 for
temperature modes 100÷800, 150÷900, 500÷1050 and 700÷1050 °C.
Fig. 5. Comparison of experimental data, simulation results and analytical
approximation for temperature modes: a) 100÷800, a heating time is 19s, a
cooling time is 46s, b) 150÷900 °C, a heating time is 42s, a cooling time is
59,5s, c) 500÷1050 °C, a heating time is 14s, a cooling time is 10s, d)
700÷1050 °C, a heating time is 8s, a cooling time is 7s.
Comparison of experimental data and computations for maximum temperature
and temperature distributions along the corset sample in different times for
temperature modes 150÷900, 250÷1000, 500÷1050 and 700÷1050 °C are
shown in fig. 6.
Fig.6. Comparison of experimental data and computations for maximum
temperature and temperature distributions along the corset sample in
different times for temperature modes a) 150÷900, a heating time is 42s, b)
250÷1000, a heating time is 80s c) 500÷1050, a heating time is 14s d)
700÷1050 a heating time is 8s
Temperature plane distributions for maximum temperature for temperature modes
100÷800, 150÷900, 250÷1000 and 700÷1050 °C are presented in fig. 7.
Fig.7. Temperature plane distributions for maximum temperature for temperature
modes a) 100÷800, b) 150÷900, c) 250÷1000 d) 500÷1050 °C
The results of the computations and the analytical approximations show a good
agreement with the experiment in the thermo-electric problem, which suggests that
the finite-element computations in combination with an analytical approximation can
be used to predict a dynamical behavior of a temperature and temperature distribution
along the corset sample for various nickel-based superalloy in wide range
The research is supported by the SIEMENS Scholarship Program and
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