A Two-Temperature Model Based on Fin-Approximation for Transient Longitudinal Heat Transfer in Unidirectional Composite

The reliability of a two-temperature model is assessed in the case of longitudinal heat transfer in unidirectional composites. One interest is that it makes it possible to apply separate boundary conditions or source terms on the fibre and the matrix (emissivity for example), without necessitating an explicit description of the fibre and matrix domains. For the sake of simplicity, the model under study is based on a fin-approximation in both fibre and matrix, which implies a high interfacial thermal resistance. The range of validity of this assumption is investigated by comparing the model to an axisymmetric one-temperature model, using non-dimension variables and Dirichlet boundary conditions. It turns out that this range of validity is strongly dependent on the parameters.


Introduction
Heat transfer in composites is usually computed based on homogenization approaches, either at micro- [1][2][3], meso-or macro-scale [4]. High interfacial resistance between highly conductive fibre and low conductive matrix may result in a difference of temperature between the fibre and the surrounding matrix during fast transient heat transfer, which are neglected by the usual homogenisation methods. This can be accounted for at the macro-scale by two-temperature models, which consider the composite as a single-phase continuum whose state is described by a separate temperature field for each phase [5]. For instance, such approaches successfully describe heat transfer in phase-change materials [6]. Thanks to the high thermal conductivity and the low radius of the fibre, a fin-approximation is likely to be reliable in the fibre provided that the interfacial thermal resistance is high enough. For very high interfacial thermal resistance and high fibre volume fraction ( ), the fin-approximation is likely to be reliable even in the surrounding matrix. This assumption is not of general purpose, but can be of interest for some composites, for instance carbon-geopolymer composites with interfacial porosity, as illustrated in [7], which implies initial high interfacial resistance likely to increase with fibre/matrix decohesion in damaged composites. Moreover, such composites can be hardened by Joule heating thanks to the high electrical conductivity of the carbon fibres [8]. Solving this specific problem involving different heat sources in the fibre (Joule effect) and in the matrix (Joule effect and exothermic hardening [9]), together with temperature dependent kinetics and material properties (geopolymer electrical resistivity for instance [10]) would benefit from considering the temperature difference between fibre and matrix. Moreover, if the abovementioned assumptions are met, the additional computational cost would be low. This paper investigates the suitability of this model for longitudinal heat conduction in a unidirectional composite ( Fig. 1) in order to provide insight about its applicability to specific problems. assigned at both extremities of the domain (z=0 and z=Z). The initial temperature is in the whole domain. The matrix is isotropic, whereas the fibre is considered as an orthotropic material, with different conductivity in the axial and radial directions.
One-temperature two-dimension axisymmetric model (1T2D). The reference solution for this problem is obtained with a usual one-temperature model, involving a single temperature field ( , , ) defined in the whole domain ∪ . The first case study involves Dirichlet boundary conditions.
Heat equation. The heat balance is obtained by Eq. (1) in the fibre and the matrix, where is a volumetric heat source.
Initial conditions. The initial temperature is constant in the whole domain Eq. (2).
Dirichlet boundary conditions. The temperature is fixed at the extremities of the domain Eq. (3). At = 0, the temperature at = 0 is instantaneously set to .
Fibre-matrix interface. The heat flux density at the fibre-matrix interface is given in Eq. (4), where − and + are the radial coordinate of the interface ( = ) respectively in the fibre and the matrix. (1-3) reduce to Eq. (5-7) respectively, assuming temperature-independent conductivity. will be defined later. This non-dimension problem is schematized in Fig. 3.

Key Engineering Materials Vol. 926
The non-dimension radial conductivity of the fibre and the matrix are respectively � = It is noteworthy that in the non-dimension coordinate system, both fibre and matrix may have orthotropic conductivity.
Two-temperature one-dimension model (2T1D). The same problem is solved by a two-temperature 1D model, assuming that both fibre and matrix are thermally thin. This assumption is meaningful provided that the Biot number in the fibre, (Eq. (10)) and in the matrix, (Eq. (11)) are together low. This assumption implies that the 2D heat transfer described previously reduces to a 1D heat transfer, in which the radial heat flux at the fibre-matrix interface is instantaneously diffused into each subdomain and . It is therefore accounted for by a source term in the heat equation.
It is noteworthy that the definition of corresponds to the axisymmetric conditions described in Fig. 2, but is only a rough approximation in an actual configuration as described in Fig. 1.

Achievements and Trends in Material Forming
Heat equation. Under such assumptions, Eq. (1) writes as Eq. (12), where the cross-section averaged temperature field is denoted in the fibre and in the matrix.
Numerical application. The reference solution computed from 2T1D model is compared with the results of 1T2D model in order to assess the conditions in which the latter model is suitable. The following assumptions are considered in this section: -No volume heat sources are considered.
-The ratio is not expected to vary significantly for most material couple and is set to 1.
-The fibre volume fraction is set to = 60%, a usual order of magnitude inside a tow. The non-dimension outer matrix radius is � = 2 � .

Achievements and Trends in Material Forming
1T2D and 2T1D models were computed thanks to a 1D or 2D finite difference scheme for several and ̅ values, with = 1 (isotropic fibre) and = 0.02 or = 0.2 (representative of carbon/geopolymer composites assuming = 5 W.m -1 .K -1 and 0.1 < < 1 W.m -1 .K -1 ). The simulation duration, ̅ , was chosen in such a way to catch the maximum gap between fibre and matrix average temperature obtained with both models, Eq. (19).
Some examples of the results obtained with the following parameters are plotted in Fig. 4 and 5. In most cases, the discrepancy tends toward zero at the end of the computations. It turns out that the initial postulate stating that the 2T1D model would provide accurate results with low Biot numbers is not straightforward. Indeed, the assumption of thermally thin fibre and matrix is very well verified in Fig. 4.b, where the minimum and maximum temperatures computed are superposed with the average temperatures. Nonetheless, large differences are found between both models.
Actually, the most sensitive parameter is the length ̅ , as illustrated in Fig. 6, where the maximum gap in both fibre and matrix is reported as a function of and ̅ for = 0.2 and = 0.02.

Discussion
Fig . 6 provides the validity map of the model, mainly characterised by low ̅ values and depending on the other parameters. The length-scale, , and time-scale, , of the problem are very low for usual applications, typically lower than 1 mm and 1s respectively, and it may be concluded that the model would not be useful for usual application at the macro-scale. At a time-scale larger than a few , Fig.  4 and Fig. 5 show that both models provide the same results, actually because fibre and matrix temperatures are very close, which finally opens a new validity range of the 2T1D model. In this case, the proposed model could provide a new insight onto microscopic scale, for example the heat transfer between fibre and matrix, which is necessarily null in a homogenized approach. The following case study investigates the model capability to apply separate heat source on fibres and matrix.

Case study: hardening by Joule heating
Heating by Joule effect in conductive carbon fibers is a way toward geopolymer matrix hardening [8]. The electric potential gradient, , is assumed to be constant over the UD sample, i.e. the Joule volumetric heating rate is uniform and there is no heat transfer in the sample, except local heat exchange between fiber and matrix, assuming that the above-mentioned assumptions are met. In such With the initial condition ∆ ���� = 0 at ̅ = 0, Eq. (22) reduces to ∆ ���� = 1 − − ̅ , i.e. ∆ ���� monotonously increases and tends toward its maximal value ∆ ���� = 1. Therefore ∆ tends toward . In [8], of value 7.5, 15 and 30 V.m -1 of value 17 µΩ.m and of value 3.45 µm were considered.
is expected to change significantly during the hardening process, but a minimum value of the order of 1 Ω.m seems to be meaningful [10,11]. Assuming ≈ 1, ≪ 1 and With ≈ 30 V.m -1 , the maximum value of ∆ is of the order of 1 K if ℎ ≈ 45 W.m -2 .K -1 . Even if it is consistent with a low Biot number, this order of magnitude is very low and do not seem to be reliable to cases where both fibre and matrix are actually in contact [12], even with a high interfacial porosity. This order of magnitude would be representative of radiative heat transfer at 400 °C, or convective heat transfer at low pressure in damaged materials without any contact.

Conclusion
The domain of validity of a two-temperature model has been investigated in the case of longitudinal heat transfer in a unidirectional composite with Dirichlet boundary conditions. This domain is twofold. First at very small time-and length-scale, which are not the scope of usual composite applications. The second domain of validity actually correspond to cases where the temperature difference between fibre and matrix is very low. The low additional computational cost of the model could make it suitable to get a better accuracy in cases involving several temperature-dependent physics in a coupled manner, where neglecting a low temperature difference between fibre and matrix would lead to small errors likely to be amplified during the successive iterations. The case study described above will be extended to a more complex problem with boundary conditions involving heat transfer at the macro-scale, and considering electrical transfer in a matrix of changing resistivity, together with an exothermic hardening kinetics.
The interest of the model will also be investigated in cases involving convective or radiative boundary conditions, with possibly different values on the fibre and the matrix as the result of different emissivity for example.