Analysis Model and Numerical Simulation of Thermoelectric Response of CFRP Composites
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Abstract
An electric current generates Joule heating, and under steady state conditions, a sample exhibits a balance between the strength dissipated by the Joule effect and the heat exchange with the environment by radiation and convection. In the present paper, theoretical model, numerical FEM and experimental methods have been used to analyze the radiation and free convection properties in CFRP composite samples heated by an electric current. The materials employed in these samples have applications in many aeronautic devices. This study addresses two types of composite materials, UD [0]_{8} and QI [45/90/−45/0]_{S}, which were prepared for thermoelectric experiments. A DC electric current (ranging from 1A to 8A) was injected through the specimen ends to find the coupling effect between the electric current and temperature. An FE model and simplified thermoelectric analysis model are presented in detail to represent the thermoelectric data. These are compared with the experimental results. All of the test equipments used to obtain the experimental data and the numerical simulations are characterized, and we find that the numerical simulations correspond well with the experiments. The temperature of the surface of the specimen is almost proportional to the electric current. The simplified analysis model was used to calculate the balance time of the temperature, which is consistent throughout all of the experimental investigations.
Keywords
CFRP Thermoelectric Temperature Law joule AeronauticalNomenclature
 ᅟc_{p}
Specific heat capacity (J/kg.K)
 g
Gravity acceleration (g=9.81 m/s^{2})
 Gr
Average Grashof number
 Gr_{x}
Local Grashof number at height x
 h_{x}
Local heat transfer coefficient (W / m^{2}.K)
 I
Electric current intensity (A)
 k
Thermal conductivity (W/m.K)
 Nu
Average Nusselt number
 Nu_{x}
Local Nusselt number at position x
 Pr
Average Prandtl number
 Pr_{x}
Local Prandtl number at position x
 Ra
Average Rayleigh number
 Ra_{x}
Local Rayleigh number at position x
 R
Electric resistance (Ω)
 S
Area of plate (m^{2})
 q
Heat flow rate (W)
 q^{'}
Heat flow rate per unit length (W/m)
 q^{"}
Heat flux (W/m^{2})
 t
Time (s)
 T
Temperature (K)
 U
Electric voltage (V)
 u_{0}
Velocity of air (m/s)
 x
Coordinate from the base of the plate (m)
 y
Coordinate normal to the plate(m)
Greek Symbols
 α
Fluid thermal diffusivity(m^{2}/s)
 β
Fluid thermal expansion rate(1/K)
 δ
Thickness of air film (m)
 ε
Surface emissivity
 μ
Dynamic viscosity (kg/m.s)
 ρ
Density (kg/m^{3})
 σ^{E}
Electrical conductivity(S/m)
 σ
StefanBoltzmann constant 5.67 × 10–8 W/m^{2}.K^{4}
 λ
Thermal conductivity (W/m.K)
Subscripts
 elec
Electric
 conv
Convection
 rad
Radiation
 paroi
Surface of plate
 amb
Ambient
 t
Total
1 Introduction
Currently, fuselages, wings, and several other structures of the Chinese aircraft C919 are composed of CFRP (Carbon Fiber Reinforced Polymer) composites, which make this civil aircraft lighter in comparison to aircrafts made from aluminum. A solution for monitoring the damage of CFRP as a sensor is now possible, as the carbon fibers are electrical conductors which are contained in an insulating polymer matrix. However, the electric currents may lead to temperature changes in the aircraft composite structure. In this case, electrical resistance measurements—coupled with a material degradation model—are employed to assess the damage and development within the composite structure^{.1–3} However, the injected electric currents are of a very low magnitude (roughly 1 mA), so the direct impact of electric currents on the composite material cannot be appropriately accounted for.
The thermoelectric effect (Joule heating) may contribute to the material’s stiffness and strength degradation; therefore, a better understanding of the material aging phenomenon under coupled thermoelectromechanical solicitations is needed for the development of civil aeronautical structures. The relationship between electric resistance and mechanical damage in unidirectional CFRP when subjected to a tensile force and fatigue charge has been investigated by Scultet et al [1, 2] Xia et al. and Cutin [3, 4] have established an electrical and mechanical coupled model to describe changes in electrical resistance due to broken fibers in CFRP. Vavouliotis A. et al. [5] have undertaken fatigue experiments with the passage of an electrical current, the results from which have nearly the same tendency. This has been further confirmed by other reports^{.8–11} The electrical resistance may directly follow the fatigue damage of CFRP materials generated by cycle loading during the fatigue tests. However, quite few papers focus on the direct influence of thermoelectric effect (temperature fields) and electrical currents on the residual behaviour of CFRP materials and structures.
We aimed to investigate whether applied currents or electrical fields may have a direct effect on the longterm behavior of CFRPs, including their mechanical properties, stiffness strength, viscoelastic response, fatigue, thermal conductivity, expansion, and internal state of damage (all these effects are almost unknown). Preliminary experimental results show that a shortterm application of DC electric current leads to an increase in the impact resistance of the composites, whereas a prolonged application of the current induces significant heating in the electrified composites and has a detrimental effect [6]. In particular, the existing experimental evidence suggests that exposure of the CFRP material to an electromagnetic field leads to changes in the material’s strength and resistance to delamination^{.9} Robert L. Sierakowski, et al. [6, 7], conducted a series of low velocity impact tests to assess the influence of the electric current and the duration of its application on the impact response of CFRP. The results revealed an increase in the impact resistance of the composites in the presence of the DC electric current applied for a short period of time. There was also a considerable dependence of the impactinduced damage upon the intensity of the electrical field applied. It was further shown that a larger magnitude of the electrical current leads to a larger impact load that may be sustained by the composite.
This study focuses on the thermoelectricity response of UD and QI CFRP composite samples subject to DC electric currents of up to 8A. Preliminary thermoelectric tests were carried out on the CFRP samples. During the tests, the dimensions and electrodes of the sample were precisely chosen to achieve a homogeneous temperature field along the sample and to minimize the contact resistance. The electrical solicitations were selected carefully so that the sample temperature did not trespass the glass transition temperature T_{g} of the resin.
2 Experiments
The experimental data reported in this paper were obtained from a device that was being used for a more general study to characterize the thermal behavior of composite materials.
2.1 Materials and Samples
2.2 Equipment
An infrared camera was used to measure the temperature of the sample surface, which offers advanced thermal imagery and accurate temperature measurements. The spectra range was between 3.6 and 5.1 μm, the temperature range was between 20 and 1000 °C, and the number of pixels was 320 × 256 for each image.
The electrical voltage was provided by a generator, which was controlled by a computer. The computer controlled the electrical experimental current intensity, the measurement range frequency (1–2 kHz), and the maximum voltage (~10 mV).
3 Analysis Model
3.1 Natural Convection
3.2 Heat Flux
 P

is the flux exchange between the surface and the fluid (W)
 S

is the plate surface (m^{2})
3.3 Electrical Resistance
Two different types of fiber orientation samples, UD and QI, were used. The measurement of the electrical resistance was obtained by the equipment.
In this case, α = −0.0003 (1/°C). This indicates that the electrical resistance decreases with temperature, but the variation is relatively small. For a change in temperature of about 200 °C, the relative variation of the resistance is less than 5%.
3.4 Nusselt Number
When a fluid is in contact with a heated plate, its temperature increases and its density decreases as it moves along the heat plate. This flow of fluid along the heat plate causes convection, which is known as natural convection or free convection.
In natural convection, the velocity of the fluid or gas is indirectly dependent on the conditions of the problem. It is therefore necessary to determine an expression that characterizes this situation.
The flow of natural convection can be laminar or turbulent, depending on the distance to the leading edge, the properties of the fluid, the force of gravity, g, and the temperature difference ΔT between the surface and the fluid.
The temperature field in the natural gas film increases with position x; thus, the physical interpretation of the Nusselt number must be used.
The purpose of Eq. (9) is to calculate the Nusselt number to deduce the convection coefficient, h_{ x }. From experimental data and formulas (8) and (9), the dimensionless Nusselt number was calculated.
3.5 Heat Convection Coefficient
The reason for deducing Eq. (9) is to calculate the convection coefficient of the natural convection h_{ x }, which is not uniform over a surface. For a vertical flat plate, there are the local value of h_{ x } in different positions along the length of the plate and the average value obtained over the entire surface.
3.6 Grashof Number
3.7 Expansion Coefficient
3.8 Prandtl Number
3.9 Rayleigh Number
3.10 Reynolds Number
In order to calculate the convection coefficient, two different types of conditions are evaluated: uniform surface temperature and uniform surface flux.
3.11 Uniform Temperature
Equation (23) is used to calculate the local heat convection coefficient when the surface temperature is uniform.
3.12 Uniform Flux
All properties in Eq. (28) were evaluated at the local film temperature. The average heat transfer convection coefficient for constant heat flux may be obtained through a separate application of Eq. (13). Thus, for the laminar region, Eq. (28) was used to evaluateh_{ x }.
3.13 Modify the Formulas
The purple curve was calculated using the method of the uniform surface flux under the assumption that the heat flux was uniform over the surface of the specimen.
The blue curve was calculated using the uniform surface temperature equation, where the temperature of the plate surface is constant (Fig. 6).
Figures 5 and 6 show the experimental results for uniform heating of a vertical plate. It is seen that the local heat convection coefficients agree more with the Churchill and Ozoe correlations than with that of the isothermal vertical plate surface calculated by Chu.
4 FE Model
This section presents the numerical approaches and theories to establish the finite elemental model for composite laminate materials.
4.1 Geometric Model
Dimensions of the FE model
Length  Width  Thickness 

0.2 m  0.02 m  0.0014 m 
Initial conditions
U _{1}  U _{2}  T _{0} 

2.25 V  0 V  20 °C 
4.2 Model Properties
Material properties for anisotropic UD
CFRP UD  Density (kg/m^{3})  Thermal conductivity (W/m°C)  Electrical conductivity (S/m)  Specific heat (J/kg°C)  
ρ  k _{11}  k _{22}  k _{33}  σ_{11}  σ_{12}  σ_{22}  c  
1530  6.1  0.61  0.61  42,017  0  0  957  
Radiation exchange  Surface emissive coefficient  Ambient temperature  
ε = 0.99  T_{0} = 20 °C  
Convection exchange  Surface convection coefficient  Ambient temperature  
h_{ x } (Eq.30) = 10 W/m^{2} °C  T_{0} = 20 °C 
4.3 Theory Analysis
Joule heating arises when the energy dissipated by an electrical current flowing through a conductor is converted into thermal energy. Abaqus provides a fully coupled model for analyzing this type of phenomenon. In the electrical effect, the conductivity is temperaturedependent. In the thermal effect, the internal heat generated is a function of the electric current.
4.4 Equilibrium Time
5 Results and Discussion
This section focuses on the simulation results, which are compared with the experimental data.
5.1 Equilibrium Time
The equilibrium time for different temperatures
Percent %  Temperature  Analysis  Abaqus  Matlab  Error (%)  

T_{p} %  °C  Time (s)  Time (s)  Time (s)  %  
0.93  120  131.00  131.00  132.00  0.00  −0.76 
0.94  121  137.00  135.00  138.00  1.46  −0.73 
0.95  122  145.00  141.00  144.00  2.76  0.69 
0.95  123  154.00  147.00  151.00  4.55  1.95 
0.96  124  164.00  154.00  159.00  6.10  3.05 
0.97  125  176.00  162.00  168.00  7.95  4.55 
0.98  126  191.00  172.00  180.00  9.95  5.76 
0.98  127  211.00  184.00  196.00  12.80  7.11 
0.99  128  241.00  200.00  219.00  17.01  9.13 
1.00  129  300.00  223.00  264.00  25.67  12.00 
5.2 Equilibrium Temperature
6 Conclusions
Experimental data of the local convection heat transfer coefficients were obtained from composite samples heated by an electric current. In this paper, the experimental apparatus and the calculation procedure used to obtain the experimental data are described in detail.
An FE model of natural convection for the uniform flux vertical plate surface passing of the electric current intensity changing from 1A to 8A was established. The model considers the local convective heat transfer coefficient h_{ x } affected by coupling of the temperature difference ΔT and the position x.
Experimental results of heating and numerical studies reveal that the transient behavior of the local convection coefficient is highly dependent on the temperature difference ΔT and the position x on the surface and the surrounding air.
The correlations obtained from the literature based on many experiments have been proposed for calculating the natural convection and are simultaneously compared to our experimental results. They are shown to be suitable for all the tests in this paper.
Notes
Acknowledgements
All partners of the research are gratefully acknowledged and some supports from CAUC Tianjin are gratefully acknowledged (Projects of CAUC:2016SYCX04, and MHRD20160105).
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