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 Aeronautical 

Nomenclature

cp

Specific heat capacity (J/kg.K)

g

Gravity acceleration (g=9.81 m/s2)

Gr

Average Grashof number

Grx

Local Grashof number at height x

hx

Local heat transfer coefficient (W / m2.K)

I

Electric current intensity (A)

k

Thermal conductivity (W/m.K)

Nu

Average Nusselt number

Nux

Local Nusselt number at position x

Pr

Average Prandtl number

Prx

Local Prandtl number at position x

Ra

Average Rayleigh number

Rax

Local Rayleigh number at position x

R

Electric resistance (Ω)

S

Area of plate (m2)

q

Heat flow rate (W)

q'

Heat flow rate per unit length (W/m)

q"

Heat flux (W/m2)

t

Time (s)

T

Temperature (K)

U

Electric voltage (V)

u0

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(m2/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/m3)

σE

Electrical conductivity(S/m)

σ

Stefan-Boltzmann constant 5.67 × 10–8 W/m2.K4

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 thermo-electro-mechanical 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 long-term behavior of CFRPs, including their mechanical properties, stiffness strength, visco-elastic response, fatigue, thermal conductivity, expansion, and internal state of damage (all these effects are almost unknown). Preliminary experimental results show that a short-term 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 impact-induced 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 Tg 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

IMA T800carbon fibers were used as the reinforcement and epoxy M21E was used as the matrix. The geometry of the samples was 200 mm × 20 mm × 1.4 mm. Two types of samples were used: Unidirectional [0°]8 and QI Quasi-Isotropic [45/90/−45/0]S (Fig. 1). The two extremes of the samples were processed with copper electrodes for passing of the electrical current; the copper plates were pasted with silver conductors (Fig. 2). These copper electrodes were present to avoid local heating, which could result in a local rise in temperature.
Fig. 1

a The geometry of the sample. b The sample CFRP

Fig. 2

Specimens with copper electrodes

2.2 Equipment

The samples were inserted into a square frame with two electric spring poles to promote surface contact between the samples and the electrodes. The copper electric poles supplying force to ensure that the contact between the sample and the electrodes does not damage the specimens. An ampere gauge was used to measure the intensity of the electric current. Figure 3 shows the experimental frame for circulation of the electrical current.
Fig. 3

Schematic of circulation of electrical current

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

Natural convection or free convection is often defined as a convective process, where fluid motion results from buoyancy effects, see Fig. 4. For a vertical heated plate surface, there were two common conditions: the uniform temperature condition and the uniform flux condition. When an electrical current passed through the sample, the electrical strength was uniform in the sample surface; therefore, the uniform flux condition is more suitable for this research. Several important parameters in fluid mechanics are presented in this section. The Nusselt number is the ratio of convective to conductive heat transfer across the boundary, and is a dimensionless value. The Rayleigh number for a fluid is also a dimensionless value associated with buoyancy driven flow for natural convection. The Grashof number describes the relationship between buoyancy and viscosity within a fluid. The Prandtl number describes the relationship between momentum diffusivity and thermal diffusivity, etc.
Fig. 4

Schematic of the vertical flat plate, δ is the air film thickness

3.2 Heat Flux

The parameters most often used to characterize thermoelectricity is the heat flux strength, which is defined by \( {q}_{\mathrm{x},\mathrm{electrique}}^{"} \), with unit of W/m2. The dissipation of the surface strength is divided into two parts: convection and radiation, as shown by the following Eqs. (1) – (3) [8].
$$ {q}_{\mathrm{x},\mathrm{electrique}}^{"}={q}_{\mathrm{x},\mathrm{conv}}^{"}+{q}_{\mathrm{x},\mathrm{rayon}}^{"} $$
(1)
$$ {q}_{\mathrm{x},\mathrm{conv}}^{"}={h}_x\cdot \left({T}_{\mathrm{paroi}}-{T}_{\mathrm{ambiant}}\right)\kern0.5em \left(\mathrm{Convection}\right) $$
(2)
$$ {q}_{\mathrm{x},\mathrm{rayon}}^{"}=\varepsilon \cdot \sigma \cdot \left({T}_{\mathrm{paroi}}^4-{T}_{\mathrm{ambiant}}^4\right)\kern0.5em \left(\mathrm{Radiation}\right) $$
(3)
The unit of the temperature of the plate surface T paroi and the temperature of the flux film near the plate T ambient is K. The heat flux is dissipated in an electrical current circulation, with the current intensity I. This is a constant value if the current is continuous, or its root mean square if the current is alternative. It has the following expression, revealed by Eq. (4),
$$ P={RI}^2= UI=\frac{U^2}{R}\kern0.5em {q}_{\mathrm{x},\kern0.5em \mathrm{electrique}}^{"}=\frac{P}{S}=\frac{RI^2}{S} $$
(4)
where,
P

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

S

is the plate surface (m2)

These considerations allow for the development of a simple model capable of expressing the equilibrium temperature of the sample as a function of the injected electrical current. Taking into account the strength losses by radiation and convection, the relationship is expressed in Eq. (5) [9, 10, 11],
$$ \frac{RI^2}{S}=\varepsilon \cdot \sigma \cdot \left({T}_{\mathrm{paroi}}^4-{T}_{\mathrm{ambiant}}^4\right)+{h}_x\cdot \left({T}_{\mathrm{paroi}}^4-{T}_{\mathrm{ambiant}}\right) $$
(5)

3.3 Electrical Resistance

The total electric resistance of the sample R t includes three parts: the resistance of sample R s , the resistance of contact R e , and the resistance of the two copper extremes R c .
$$ {R}_t={R}_s+{R}_e+{R}_c $$
(6)

Two different types of fiber orientation samples, UD and QI, were used. The measurement of the electrical resistance was obtained by the equipment.

When the temperature of the samples increases, atom agitation increases, and the collisions between the electrons and atoms will limit electron movement. The resistance varies with temperature according to the nature of the conductor, except for certain materials that were made at an almost constant resistance in the entire temperature range. The total electrical resistance R t changes linearly according to the temperature of the conductor. This relationship is expressed in Eq. (7),
$$ R={R}_0\cdot \left(1+\alpha \Delta T\right) $$
(7)
where, R0 is the initial resistance, for example, the resistance measured at ambient temperature, and α is a coefficient to be identified.

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.

However, for practical applications, Eq. (8) is typically utilized,
$$ P={h}_x\cdot S\cdot \left({T}_{paroi}-{T}_{ambiant}\right)\kern0.5em \left(\mathrm{Law}\kern0.5em \mathrm{of}\kern0.5em \mathrm{Newton}\right) $$
(8)
Applying Buckingham’s theory for a problem of convection leads to two groups of dimensionless numbers, which are related by the following form (9),
$$ {Nu}_x=F\left({\Pr}_x\right){\left({Gr}_x\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}={F}_1\left({\Pr}_x\right){Ra_x}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$4$}\right.}\kern0.5em \left(\mathrm{Local}\ \mathrm{Nusselt}\ \mathrm{Number}\right) $$
(9)

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.

The determination of h x involves analyzing the relationship between the dimensionless numbers. This will be presented in the following section. From a semi-empirical equation, the Nusselt number is expressed by Eq. (10),
$$ {Nu}_x=\frac{x\cdot {h}_x}{\lambda } $$
(10)
The average value of the Nusselt number has the expression in Eq. (11),
$$ \overline{Nu}=\frac{\overline{h\cdot L}}{\lambda } $$
(11)
From Eq. (10), Eq. (12) can be deduced,
$$ {h}_x=\frac{\lambda \cdot {Nu}_x}{x} $$
(12)
The average value of the heat transfer convection coefficient has the expression in Eq. (13),
$$ \overline{h}=\frac{1}{L_x}\underset{0}{\overset{L}{\int }}{h}_x dx $$
(13)

3.6 Grashof Number

Another dimensionless number, the local Grashof number Gr x is the characteristic of natural convection. See Eq. (14):
$$ {Gr}_x=\frac{g\cdot \beta \cdot \Delta T\cdot {x}^3}{v^2}\kern0.5em \left(\mathrm{Grashof}\ \mathrm{Number}\right) $$
(14)

3.7 Expansion Coefficient

The expansion coefficient β, gives the relative increase in volume, which is a function of the temperature while the pressure remains constant. Note that β is often defined by the relationship in (15),
$$ \beta =\frac{1}{T_{\mathrm{film}}} $$
(15)
Here:
$$ {T}_{\mathrm{film}}=\frac{T_{\mathrm{paroi}}+{T}_{\mathrm{ambiant}}}{2}\kern0.5em \left(\mathrm{Temperature}\ \mathrm{of}\ \mathrm{air}\ \mathrm{film}\right) $$

3.8 Prandtl Number

The Prandtl Number is another dimensionless value, and represents the ratio of the momentum diffusivity v (or kinematics viscosity) and thermal diffusivity, defined as follows:
$$ \mathit{\Pr}=\frac{\mu \cdot {c}_{\mathrm{p}}}{\lambda }=\frac{v}{a}\kern0.5em \left(\mathrm{Prandtl}\ \mathrm{Number}\right) $$
(16)

3.9 Rayleigh Number

Another dimensionless number, the Rayleigh number, is used to characterize heat transfer in fluid mechanics. It can be defined as the product of the Grashof number and Prandtl number as follows:
$$ {Ra}_x={Gr}_x{\Pr}_x=\frac{g\cdot \beta \cdot \Delta T\cdot {x}^3}{v\cdot \alpha}\kern0.5em \left(\mathrm{Rayleigh}\ \mathrm{Number}\right) $$
(17)

3.10 Reynolds Number

The Reynolds Number is used to characterize the flow, in particular, the nature of the regime laminar, transitional, or turbulent flow. This is so if the introduced kinematics viscosity is \( v=\frac{\mu }{\rho } \). The Reynolds Number is defined as follows:
$$ {\operatorname{Re}}_x=\frac{\rho \cdot {u}_0\cdot x}{\mu }=\frac{u_0\cdot x}{v}\kern0.5em \left(\mathrm{Reynolds}\ \mathrm{Number}\right) $$
(18)

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

Many formulas are given by researchers in previous literature. Churchill and Chu [8, 12] determined a formula for the vertical flat plate, which can be calculated by the local Nusselt number. In the laminar regime (Ra <109), Eq. (19) was used to calculate the Grashof number of the plate surface, then Eq. (20) was used for the uniform heating temperature of the vertical plate.
$$ {Gr}_x=\frac{g\cdot \beta \cdot \Delta T\cdot {x}^3}{v^2} $$
(19)
Equation (20) was used to calculate the local Nusselt number,
$$ {Nu}_x^T=0.68+\frac{0.503\cdot {\left({Ra}_x\right)}^{0.25}}{{\left[1+{\left(\frac{0.492}{\Pr_x}\right)}^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$16$}\right.}\right]}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.}} $$
(20)
From Eqs. (19) to (20), and the relation of Ra x  = Nu x  ⋅ Pr x , Eq. (21) is of the following type,
$$ {Nu}_x=\frac{0.503\cdot {\left(\frac{g\cdot \beta \cdot \Delta T\cdot {x}^3}{v^2}\cdot {\Pr}_x\right)}^{0.25}}{{\left[1+{\left(\frac{0.492}{\Pr_x}\right)}^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$16$}\right.}\right]}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.}}+0.68 $$
(21)
From Eqs. (12) to (21), Eq. (22) can be used to calculate the heat convection coefficient.
$$ {h}_x^T=\frac{\lambda }{x}\cdot \left[0.68+\frac{0.503\cdot {\left(\frac{g\cdot \beta \cdot \Delta T\cdot {x}^3}{v^2}\cdot {\Pr}_x\right)}^{0.25}}{{\left[1+{\left(\frac{0.492}{\Pr_x}\right)}^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$16$}\right.}\right]}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.}}\right] $$
(22)
By bringing order to the parameters, Eq. (23) can be expressed by the following:
$$ {h}_x^T=\lambda \cdot \left[0.68+\frac{0.503\cdot {\left(\frac{g\cdot \beta \cdot \Delta T}{v^2}\cdot {\Pr}_x\right)}^{0.25}}{{\left[1+{\left(\frac{0.492}{\Pr_x}\right)}^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$16$}\right.}\right]}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.}}\right]\cdot \frac{1}{x^{0.25}} $$
(23)

Equation (23) is used to calculate the local heat convection coefficient when the surface temperature is uniform.

3.12 Uniform Flux

Extensive experiments have been reported [1, 2, 3] for free convection from the vertical plate surface under constant heat flux conditions. In such experiments, the results are presented in terms of a modified Grashof number. In the laminar regime (Ra <109), Eq. (25), analyzed by Churchill and Ozoe [4], was used for a vertical plate flux uniform. Equation (24) was then used to calculate the Grashof number, particularly for the uniform surface flux.
$$ {Gr}_x^{\ast }=\frac{g\cdot \beta \cdot {q}^{"}\cdot {x}^4}{\lambda \cdot {v}^2} $$
(24)
Equation (25) was used to calculate the local Nusselt number:
$$ {Nu}_x^F={F}_2\left({\Pr}_x\right){\left({Gr}^{\ast }{\Pr}_x\right)}^{1/5} $$
(25)
Here,
$$ {F}_2\left({\Pr}_x\right)={\left(\frac{\Pr_x}{4+9{\Pr}_x^{1/2}+10{\Pr}_x}\right)}^{1/5} $$
(26)
From Eqs. (12) to (26), Eq. (27) was deduced:
$$ {h}_x^F=\frac{\lambda }{x}\cdot \left[{\left(\frac{\Pr_x}{4+9{\Pr}_x^{1/2}+10{\Pr}_x}\right)}^{1/5}\cdot {\left(\frac{g\cdot \beta \cdot {q}^{"}\cdot {x}^4}{\lambda \cdot {v}^2}\cdot {\Pr}_x\right)}^{0.2}\right] $$
(27)
By bringing order of the parameters, Eq. (28) is of the following type,
$$ {h}_x^F=\lambda \cdot \left[{\left(\frac{\Pr_x}{4+9{\Pr}_x^{1/2}+10{\Pr}_x}\right)}^{1/5}\cdot {\left(\frac{g\cdot \beta \cdot {q}^{"}}{\lambda \cdot {v}^2}\cdot {\Pr}_x\right)}^{0.2}\right]\cdot \frac{1}{x^{0.2}} $$
(28)

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

Equations (29) and (30) were compared and adjusted to fit for composite samples UD. The variables were replaced with the experimental data. When an electrical current of 8 A was injected, the heat convection coefficient was calculated by two methods, namely formulas (29) and (30). These are shown in Fig. 5.
$$ {h}_x^T=\lambda \cdot \left[\frac{0.503\cdot {\left(\frac{g\cdot \beta \cdot \Delta T}{v^2}\cdot {\Pr}_x\right)}^{0.25}}{{\left[1+{\left(\frac{0.492}{\Pr_x}\right)}^{\raisebox{1ex}{$9$}\!\left/ \!\raisebox{-1ex}{$16$}\right.}\right]}^{\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$9$}\right.}}\right]\cdot \frac{1}{x^{0.25}}\kern0.5em \left(\mathrm{Uniform}\ \mathrm{temperature}\right) $$
(29)
$$ {h}_x^F=\lambda \cdot \left[{\left(\frac{\Pr_x}{4+9{\Pr}_x^{1/2}+10{\Pr}_x}\right)}^{1/5}\cdot {\left(\frac{g\cdot \beta \cdot {q}^{"}}{\lambda \cdot {v}^2}\cdot {\Pr}_x\right)}^{0.2}\right]\cdot \frac{1}{x^{0.2}}\kern0.5em \left(\mathrm{Uniform}\ \mathrm{flux}\right) $$
(30)
Fig. 5

Curves of the heat transfer coefficient, h x . The violet curve is the uniform flux, P = 10 W, as a function of position x; the blue curve is for the uniform temperature, Tparoi = 100 °C

Equation (31) was used to calculate the average value of the local heat convection coefficient, \( \overline{h} \). Figure 6 shows the two curves of \( \overline{h} \) as a function of I.
$$ \overline{h}=\frac{1}{L_x}\underset{0}{\overset{L}{\int }}{h}_x dx\kern0.5em \left(\mathrm{Average}\ \mathrm{value}\ \mathrm{of}\ \mathrm{the}\ \mathrm{local}\ \mathrm{heat}\ \mathrm{convection}\ \mathrm{coefficient}\right) $$
(31)
Fig. 6

The average value of the heat convection coefficient, h x . The purple curve is for the uniform surface flux and the blue curve for the uniform surface temperature

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

A 3-D model was established. An element of the type, DC3D8E, was used. This was specifically selected for thermal-electric coupling. There were a total of 469 elements in the model. Table 1 reveals the FE model size.
Table 1

Dimensions of the FE model

Length

Width

Thickness

0.2 m

0.02 m

0.0014 m

The load condition of the model is shown in Table 2. The electric potential V1 = 2.25 (V) was added at the two extremes of the sample to generate an electric current I = 6 (A).
Table 2

Initial conditions

U 1

U 2

T 0

2.25 V

0 V

20 °C

4.2 Model Properties

An anisotropic model was established. Table 3 displays the material properties of the composite samples. The input parameters were measured from the experiments, and the local heat convection coefficient h x was determined using Eq. (30).
Table 3

Material properties for anisotropic UD

CFRP UD

Density (kg/m3)

Thermal conductivity (W/m°C)

Electrical conductivity (S/m)

Specific heat (J/kg°C)

ρ

k 11

k 22

k 33

σ11

σ12

σ22

c

1530

6.1

0.61

0.61

42,017

0

0

957

Radiation exchange

Surface emissive coefficient

Ambient temperature

ε = 0.99

T0 = 20 °C

Convection exchange

Surface convection coefficient

Ambient temperature

h x (Eq.30) = 10 W/m2 °C

T0 = 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 temperature-dependent. In the thermal effect, the internal heat generated is a function of the electric current.

The constitutive relationship is linear, where it is assumed that the electric conductivity is independent of the electrical field. Introducing Ohm’s law, the governing conservation of charge is expressed by Eq. (32):
$$ \underset{V}{\int}\frac{\partial \varphi }{\partial \mathbf{x}}\cdot {\boldsymbol{\upsigma}}^E\cdot \frac{\partial \varphi }{\partial \mathbf{x}} dV=\underset{V}{\int}\delta \varphi \cdot {r}_c dV+\underset{V}{\int}\delta \varphi JdS $$
(32)
The flow of electrical current is described by Ohm’s law, the equation for which is (33):
$$ \mathbf{J}={\boldsymbol{\upsigma}}^E\cdot \mathbf{E}=\hbox{-} {\boldsymbol{\upsigma}}^E\cdot \frac{\partial \varphi }{\partial \mathbf{x}} $$
(33)
where:
$$ {\boldsymbol{\upsigma}}^E=\left[\begin{array}{ccc}{\sigma}_{11}& {\sigma}_{12}& {\sigma}_{13}\\ {}{\sigma}_{21}& {\sigma}_{22}& {\sigma}_{23}\\ {}{\sigma}_{31}& {\sigma}_{32}& {\sigma}_{33}\end{array}\right]=\left[\begin{array}{ccc}{\sigma}_{11}& 0& 0\\ {}0& 0& 0\\ {}0& 0& 0\end{array}\right]=\left[\begin{array}{ccc}18500& 0& 0\\ {}0& 0& 0\\ {}0& 0& 0\end{array}\right] $$
heat conduction is supposed by the law of Fourier:
$$ \mathbf{F}=\hbox{-} \mathbf{k}\cdot \frac{\partial T}{\partial \mathbf{x}} $$
(34)
The conductivity can be fully anisotropic, where k is described by the equation below:
$$ \mathbf{k}=\left[\begin{array}{ccc}{k}_{11}& {k}_{12}& {k}_{13}\\ {}{k}_{21}& {k}_{22}& {k}_{23}\\ {}{k}_{31}& {k}_{32}& {k}_{33}\end{array}\right]=\left[\begin{array}{ccc}{k}_{11}& 0& 0\\ {}0& {k}_{22}& 0\\ {}0& 0& {k}_{33}\end{array}\right]=\left[\begin{array}{ccc}6.1& 0& 0\\ {}0& 0.61& 0\\ {}0& 0& 0.61\end{array}\right] $$
Equation (35) was used to model the balance of thermal energy. This heat conduction behavior is described by the following relationship:
$$ \underset{V}{\int}\rho \dot{U}\cdot \delta T dV+\underset{V}{\int}\frac{\partial \delta T}{\partial \mathbf{x}}\cdot \mathbf{k}\cdot \frac{\partial T}{\partial \mathbf{x}} dV=\underset{V}{\int}\delta T\cdot r\mathrm{dV}+\underset{S}{\int}\delta T\cdot q\mathrm{d}S $$
(35)
Joule’s law describes the rate of electrical energy Pec, dissipated by a current flowing through a sample as Eq. (36):
$$ {P}_{ec}=\mathbf{E}\cdot J=\mathbf{E}\cdot {\boldsymbol{\upsigma}}^E\cdot \mathbf{E}\kern0.5em \left({\mathrm{Joule}}^{'}\mathrm{s}\ \mathrm{Law}\right) $$
(36)
It is assumed that there is surface energy loss by convection at ambient temperature T0 = 20 °C. The air film heat convection coefficient varies with temperature, as shown by the empirical relationship:
$$ {q}_{\mathrm{c}}={h}_x\cdot \left(T-{T}_0\right) $$
(37)
For radiation, Eq. (38) is used to describe the radiation flux per unit area,
$$ {q}_{\mathrm{r}}=\varepsilon \sigma S\cdot \left({\left(T-{T}_0\right)}^4-{\left(T-{T}_0\right)}^4\right) $$
(38)

4.4 Equilibrium Time

This section presents the simplified analysis to calculate the equilibrium time. Equation (39) is used to describe the electrical power dissipated by convection and radiation.
$$ {R}_s{I}^2= hS\left(T-{T}_0\right)+\varepsilon \sigma S\left({T}^4-{T}_0^4\right) $$
(39)
For different time, Eq. (39) is written as Eq. (40):
$$ {R}_s{I}^2= hS\cdot \left(T\left(\tau \right)-{T}_0\right)+\sigma \varepsilon S\cdot \left(T{\left(\tau \right)}^4-{T}_0^4\right) $$
(40)
From the second part of Eq. (40), it is possible to obtain an equivalent convection coefficient, as revealed below by Eq. (41):
$$ \varepsilon \sigma S\cdot \left(T{\left(\tau \right)}^4-{T}_0^4\right)={h}^{\ast }S\left({T}_{\mathrm{p}}\hbox{-} {\mathrm{T}}_0\right) $$
(41)
By taking the derivative of Eq. (41), Eq. (40) is modified to produce Eq. (42):
$$ {h}^{\ast }=\sigma \varepsilon \left({T}_{\mathrm{p}}+{\mathrm{T}}_0\right)\left({T}_{\mathrm{p}}^2+{T}_{\mathrm{amb}}^2\right) $$
(42)
Substituting the data in Eq. (41), the value of \( {h}^{\ast }=9.5\kern0.5em \mathrm{W}/\left({\mathrm{m}}^2\mathrm{K}\right) \) was calculated. Following this, h t was obtained using Eq. (43):
$$ {h}_t=h+{h}^{\ast }=15\kern0.5em \mathrm{W}/\left({\mathrm{m}}^2\mathrm{K}\right) $$
(43)
The Eq. (39) was modified to yield Eq. (44):
$$ {R}_s{I}^2={h}_tS\left(T-{T}_0\right) $$
(44)
After taking the derivative of Eq. (44), Eq. (45) was used to estimate the time equilibrium:
$$ \rho cV\frac{\mathrm{d}T}{\mathrm{d}\tau }={R}_s{I}^2-{h}_tS\left(T-{T}_0\right) $$
(45)

5 Results and Discussion

This section focuses on the simulation results, which are compared with the experimental data.

5.1 Equilibrium Time

For specimen QI [45/90/−45/0]S, when the equilibrium temperature of Tp = 129 °C was selected, the waiting time for a certain percentage of the equilibrium temperature Tp is shown in Table 4. Figure 7 reveals the curves of the equilibrium temperature estimated by the method analysis (purple curve) and by the numerical method (blue curve). The method can be used to predict and determine the time necessary for steady state equilibrium of the experiments in the next step.
Table 4

The equilibrium time for different temperatures

Percent %

Temperature

Analysis

Abaqus

Matlab

Error (%)

Tp %

°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

Equilibrium Temperature Tp 129 °C

Fig. 7

The curves of equilibrium temperature

5.2 Equilibrium Temperature

Three points were chosen in the center line along the longitude of the specimen: point-1 was in the center of the specimen, point-2 was located at the top of 37.6 mm from point-1, and point-3 was located 37.6 mm below point-1. Figure 8 shows the curves of temperature versus the intensity of the electric current. The red curve is the result of the numerical simulation. With an increase in the electric current intensity, the equilibrium temperature increased. The temperatures of the three different points differ slightly from one another. This is a result of the temperature gradient on the free surface. Three different values of the convective heat transfer coefficient h, are used to represent the experiments. We found that h = 10 W/m2 is most suitable for this type of materials.
Fig. 8

The equilibrium temperature T as function of the electric current intensity, I

The Fig. 9 shows the curves of the temperature versus time. The red curve is the result of an FEM simulation and the black curve is the experimental data of the specimen UD composite. The intensity electric current was changed from 1A to 8A.
Fig. 9

The temperature T of three points on function of time t

Figure 10 shows the FEM simulations and experimental temperatures as function of position, x. The specimen was mounted vertically in the assembly. In the experiments, the position started at the extreme top, and the temperature decreased in moving from 50 to 150 mm. Three lines on the surface of the specimen were chosen, specially, line-A at the center line of the specimen, line-B at the left side of 5.64 mm from line-A, and line-C at the right side of 5.64 mm from line-A. The lines of the temperature decrease from the top extreme until the base extreme. For the electric test of the vertical flat plate, the higher the position, the greater the temperature increase.
Fig. 10

The profile of FEM simulations and experimental data at temperature T as a function of position x. The dotted line is FEM and the black curves are the experimental data

Figure 11 shows the details of the temperature profiles for 6A, 7A and 8A of the UD composite samples. All the curve simulations reveal better correspondence with the experimental curves.
Fig. 11

The detailed temperature gradient profiles T of simulations and experiments as a function of position x for the UD sample

QI specimens were simulated by the same methods as UDs. In Figs. 12 and 13, the red curve is the result of FEM numerical simulation and the black curves are the experimental data. The temperature maximum was attained until 220 °C. It should be noted that the glass transition temperature of the epoxy T g was about 200 °C.
Fig. 12

The equilibrium temperature T as function of the electric current intensity I for sample QIs

Fig. 13

The curves of temperature T as a function of time t for the test QI

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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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Designs and Manufactures of AircraftsCivil Aviation University of ChinaTianjinChina

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