Hybrid Thermal Treatment Based on Microwaves and Heating Resistance for Composite Materials

Abstract

Nowadays, it is well known that volumetric heating through microwaves inhibits certain surface properties from being achieved. Similarly, exclusively heating via thermal radiation is neither deep nor homogeneous enough for short periods of time. But combining both approaches can alleviate such issues. In fact, this kind of hybrid heating has been used for many years in real processes for thermal treatment of composite materials. Nonetheless, many questions remain unsettled. In this manuscript, we discuss the modeling and simulation of such a hybrid system, when heating a heterogeneous load composed of a solid core with three concentric spherical shells. The heating sources are given by electromagnetic waves in the microwave range, and by constant thermal radiation over one of the outer hemispheres. Only the core is considered to absorb the energy transported by the electromagnetic waves (high dielectric loss material). Hence, shells are transparent to microwaves (low dielectric loss materials). The thermophysical properties were considered constant with position. For all cases, peak temperature was observed in the geometrical center of the system, as has been shown by experimentation. Furthermore, simulation results revealed that this hybrid heating strategy has a drastic effect on the temperature profiles generated with only microwave, although the surface temperature homogeneity can be improved using an external electrical resistance.

1 Introduction

The mathematical foundations of conductive heat transfer are well established. But a need has existed since its beginning: Forecasting temperature profiles in one, two and three dimensions, in homogenous and heterogeneous materials. A particular case of the latter is given by a solid with several layers around it, which exhibits different thermophysical properties. Nowadays, literature contains some works in such regard [1,2,3,4]. There is also a similar interest in simulating processes where the material is treated with electromagnetic fields, especially in the range of microwaves. Nonetheless, only a practical purpose material with certain electrical and magnetic properties interacts with such fields, transforming electromagnetic radiation into thermal energy within the material. An example of such properties is a high imaginary part in either, its complex electrical permittivity, or its complex magnetic permeability. It is common to assume that most materials heavily interact with electric fields and very weakly with magnetic fields, within the microwave range. However, this fails when the material is an electromagnetic absorber, whose magnetic permeability is quite high. Even so, it is usual to find many simulations where the main interest rests on determining the interaction between material and electric field, highlighting the contribution of the following properties: $\varepsilon^{\prime}$, $\varepsilon^{\prime\prime}$ and ${\text{tan}}\left( \delta \right)$. Such dielectric properties chiefly depend upon frequency, purity, chemical state and temperature.

Employing microwaves in industrial processes, such as the thermal treatment of materials, is a crystal-clear alternative today. Microwaves can operate independently of or along conventional heating (i.e., thermal radiation and conduction). Literature contains plenty of successful examples, particularly about systems composed of layers of different materials [5,6,7,8,9,10,11,12,13,14]. Many of these processes, especially new ones, require basic knowledge and experimentation before moving on to a laboratory scale. And, much more experience is required before taking them out to a pilot plant scale and, subsequently, to an industrial scale. During these stages, modeling and simulation need to start anew. The reason: Microwave heating is not scalable since field distribution depends on the dimensions of the cavity they are being contained into. Hence the process always begins at the physical foundations, using invaluable tools such as Maxwell equations coupled with the first law of thermodynamics (the energy balance), along with the constitutive equations. For the thermal treatment of materials through electromagnetic waves, results have been noticeably impressive at the industrial level, e.g. for aluminum and ceramic alloys. Traditional aluminum alloys have an acceptable mechanical strength and hardness, low density and excellent anticorrosive properties. Hence, they are a feasible target for study. Loganathana et al. developed an experimental study about the effect of using microwaves for treating an Al-6061 alloy [15]. The authors focused on improving some properties and used a multimodal microwave source at 2.45 GHz and 850 W. They reported an evident improvement on mechanical properties and even remarked that sheets of this alloy remained unbent throughout processing; an effect unachievable with conventional heating. Likewise, there have been technological advancements related to the hybrid thermal treatment of composites with different nature, in processes of carburizing, carbonitriding, chromizing and boronizing [16]. For these cases, the authors achieved a uniform heat transfer on all surfaces, since their hybrid system allowed testing the effectiveness of both volumetric and thermal radiation heating. A more in-depth analysis of the literature over the past ten years reveals a growing interest in composite structure preparation for various applications [17,18,19,20]. In the present work, we disregard a distinction about which property was responsible for heating a four-layered system. Instead, we focus on how heating behaves. An external electrical resistance provides energy to heat the upper hemisphere of the shell through thermal radiation and the heat is transferred by conduction. We assume that there are no air currents, nor any other fluid. This means that we exclusively solved the Fourier equation in spherical coordinates when heat conduction is present in a heterogeneous medium (a composite structure made of three layers of different materials and one core). Furthermore, we analyze the transient state, which includes the simultaneous heating from thermal radiation (upper hemisphere) and from electromagnetic radiation (inner sphere). We assume that only the inner sphere absorbs microwave energy. The external shell receives thermal radiation from the electrical resistance. Finally, and for simulation purposes only, we consider all thermal properties as constant, including permittivity.

This article is organized as follows: After this brief introduction, the problem definition and the mathematical model governing the thermal process are presented. Subsequently, initial and boundary conditions are displayed by describing how the model is analytically and numerically developed. Then, simulation results and their analysis are detailed. Finally, this manuscript ends with some of the most relevant conclusions.

2 Problem Definition and Mathematical Method

2.1 Heat Conduction Problem

The thermal system for this work has been defined as an internal solid sphere coated by three spherical shells. The core absorbs the energy from the surrounding electromagnetic field (EMF) due to its electrical properties $\varepsilon^{\prime}$, $\varepsilon^{^{\prime\prime}}$ and ${\text{tan}}\left( \delta \right)$. The EMF is within the microwave range (i.e., 2.45 GHz). The spherical shells are considered to be transparent to such kind of radiation. It is also assumed that the thermal resistance between shells (interface regions), also known as contact resistances, are negligible. It means that all surfaces are perfectly smooth. Likewise, it is considered that thermophysical properties are invariant with respect to position and temperature. All the system is suspended within an electromagnetic applicator (oven cavity) where a magnetron is laid to radiate, at constant power, electromagnetic energy over the combined sphere. Furthermore, because of process considerations, there exists an external electrical resistance that uniformly radiates thermal energy over a hemisphere of the external shell.

2.2 Heat Transfer Model

The process under analysis is the heat transfer due to conduction in a series of spherical shells, both contacting an inner solid sphere. Hence, the entire system undergoes a hybrid thermal treatment. The heating setup comprises a microwave generator and an electrical resistance, which radiates over a hemisphere of the external shell. The other hemisphere is kept at a constant temperature (a Dirichlet boundary), although it is transparent to microwaves. Therefore, the mathematical model for describing this heat conduction process consists of four time-dependent partial differential equations in spherical coordinates. Its solution shall render a temperature profile as a function of position (three dimensions) and time, for each shell and for the core. Such a mathematical model, represented by the energy balance (i.e., thermal energy), also known as the heat diffusion equation, is presented in (1), where $r_{j} \le r \le r_{j + 1}$, $j \in \left\{ {1,2,3} \right\}$, and $r_{i}$ is the radius of the $i$-th spherical element, with $i \in \left\{ {1,2,3,4} \right\}$.

$$ \frac{{\partial T_{i} \left( {r;\mu ;\phi ;t} \right)}}{\partial t}{ } = \frac{{\alpha_{i} }}{{{ }r^{2} }}\frac{\partial }{\partial r}\left( {r^{2} \frac{{\partial T_{i} \left( {r;\mu ;\phi ;t} \right)}}{\partial r}} \right) + \frac{{\alpha_{i} }}{{r^{2} }}\frac{\partial }{\partial \mu }\left( {\left( {1 - \mu^{2} } \right)\frac{{\partial T_{i} \left( {r;\mu ;\phi ;t} \right)}}{\partial \mu }} \right) $$

(1)

To solve the problem laid out in (1), initial conditions are required, such as

$$ T_{i} \left( {r;\mu ;\phi ;t = 0} \right) = f_{i} \left( {r;\mu ;\phi } \right), $$

(2)

as well as boundary conditions, which are given as follows:

Boundary conditions for the core (when $i = 1$) in the $r$-direction, i.e., for the solid sphere, are defined as

$$ \frac{{\partial T_{1} \left( {r_{1} ;\mu ;\phi ;t} \right)}}{\partial r} = 0. $$

(3)

Boundary conditions for the outer layer (when $i = 4$) in the $r$-direction, i.e., for the external spherical shell, are defined as

$$ k_{4} { }\frac{{\partial T_{4} \left( {r_{4} ;\mu ;\phi ;t} \right)}}{\partial r} = q^{\prime\prime}\left( {r = r_{4} ;\mu ;\phi } \right), $$

(4)

where $q^{\prime\prime}\left( {r = r_{4} ;\mu ;\phi } \right)$ $\left[ {{\text{W}} \cdot {\text{ m}}^{ - 2} } \right]$ is a time constant heat flux generated by the electrical resistance and delivered to the outer shell hemisphere. This flux can depend on the other two coordinates. Therefore, boundary conditions in the $\theta$-direction are given in two parts. The first one corresponds to temperature continuity within the same sphere $\left( {i \in \left\{ {1,2,3,4} \right\}} \right)$, such as

$$ T_{i} \left( {r;\mu = - 1;\phi ;t} \right) = T_{i} \left( {r;\mu = 1;\phi ;t} \right). $$

(5)

The second one refers to heat flux continuity within the same sphere, as

$$ \frac{{\partial T_{i} \left( {r;\mu = - 1;\phi ;t} \right)}}{\partial \mu } = \frac{{\partial T_{i} \left( {r;\mu = 1;\phi ;t} \right)}}{\partial \mu }. $$

(6)

Likewise, boundary conditions in the $\phi$-direction are defined in two parts. Temperature continuity,

$$ T_{i} \left( {r;\mu ;\phi = 0;t} \right) = T_{i} \left( {r;\mu ;\phi = 2\pi ;t} \right), $$

(7)

and heat flux continuity within the same sphere,

$$ \frac{{\partial T_{i} \left( {r;\mu ;\phi = 0;t} \right)}}{\partial \phi } = \frac{{\partial T_{i} \left( {r;\mu ;\phi = 2\pi ;t} \right)}}{\partial \phi }. $$

(8)

Moreover, temperature continuity at the contact interface between two shells, $j$ and $j + 1$ with $j \in \left\{ {1,2,3} \right\}$, is

$$ T_{j} \left( {r_{j} ;\mu ;\phi ;t} \right) = T_{j + 1} \left( {r_{j + 1} ;\mu ;\phi ;t} \right),{ } $$

(9)

and the heat flux continuity is defined as,

$$ k_{j} \frac{{\partial T_{j} \left( {r_{j} ;\mu ;\phi ;t} \right)}}{\partial \phi } = k_{j + 1} \frac{{\partial T_{j + 1} \left( {r_{j + 1} ;\mu ;\phi ;t} \right)}}{\partial \phi }. $$

(10)

In the previous equations, $\alpha_{i}$ [${\text{m}}^{2} \cdot {\text{s}}^{ - 1}$] is the thermal diffusivity of the sphere with material $i$. This parameter is defined as the ratio between thermal conductivity $\left( {k_{i} \left[ {{\text{W}} \cdot {\text{m}}^{{ - 1}} \cdot {\text{K}}^{{ - 1}} } \right]} \right)$ and the product of density $\left( {\rho_{i} \left[ {{\text{kg }} \cdot {\text{m}}^{ - 3} } \right]} \right)$ and specific heat capacity $\left( {c_{i} \left[ {{\text{kJ }} \cdot {\text{kg}}^{ - 1} \cdot {\text{K}}^{ - 1} } \right]} \right)$, i.e., $\alpha_{i} = k_{i} /\rho_{i} c_{i}$. $T_{i} \left( {r;\mu ;\phi ;t} \right)$ [${\text{K}}$] is the $i$-th shell temperature, which is a function of the radial position, the cosine of angle $\theta \left[ {{\text{rad}}} \right]$, the elevation angle $\phi$ $\left[ {{\text{rad}}} \right]$, and the time $t \left[ {\text{s}} \right]$. Moreover, $\mu = \cos \left( \theta \right)$ is a convenient variable change used in the literature. $g^{\prime\prime\prime}\left( {r;\mu ;\phi } \right)$ $\left[ {{\text{W }} \cdot {\text{m}}^{ - 3} } \right]$ is the internal power flux generation (volumetric heat generation), which only exists in the inner solid sphere (i.e., when $i = 1$) considered as time-invariant for this work. The energy contribution by the external electrical resistance is included in the model as a heat flux. Now, the dominant coupling between the microwaves and the core is through power absorption by the imaginary part of the electric permittivity, i.e., $\varepsilon^{\prime\prime} \left[ {{\text{F }} \cdot {\text{m}}^{ - 1} } \right]$. The core is made of a high dielectric loss material, because there are few materials which have considerable coupling to the direct magnetic field interactions, i.e., materials with high $\mu^{\prime\prime} \left[ {{\text{H }} \cdot {\text{m}}^{ - 1} } \right]$, especially at high frequencies. Therefore, in this paper, we consider that the actual field effect is heating from an electric field induced in the material by the external electromagnetic field. In this sense, the electrical conductivity effects are included within $\varepsilon^{\prime\prime}_{{{\text{eff}}}}$, and the volumetric power magnitude dissipated in the core is given by,

$$ \left| {g^{\prime\prime\prime}\left( {r;\mu ;\phi } \right)} \right| = {\text{Re}}\left\{ { - \nabla \cdot {\varvec{S}}} \right\} = \left( {\varepsilon_{0} \chi^{\prime\prime}\left( \omega \right) + \frac{\sigma }{\omega }} \right) = \omega \varepsilon_{0} \varepsilon^{\prime\prime}_{{{\text{eff}}}} E_{{{\text{rms}}}}^{2} , $$

(11)

where ${\varvec{S}}$ is the complex Poynting vector (the power flux density in the wave), $\varepsilon_{0} = 8.854 \times 10^{ - 12} {\text{F }} \cdot {\text{m}}^{ - 1} ,$ $\omega = 2\pi f$ is the angular frequency ($f = 2.45\, {\text{GHz}}$), $\chi^{\prime\prime}$ is the magnetic susceptibility as a function of frequency, $\sigma$ is the electrical conductivity $\left[ {{\text{S }} \cdot {\text{m}}^{ - 1} } \right]$ and all $E_{{{\text{rms}}}}^{2}$-field energy loss terms are comprised in $\varepsilon^{\prime\prime}_{{{\text{eff}}}}$. This internal heating shows as a contribution to the internal thermal energy of the core. Therefore, this heating term will be due entirely to core conductivity if both, permittivity and permeability, are real. It is important to remark that the proposed model can only be used by considering the thermodynamic properties for each shell as independent of position (homogeneous and isotropic media) and temperature. Boundary conditions and the internal generation are assumed time constant. Evaporation of components (i.e., latent heat) and chemical reactions are disregarded during the simulated heating process. In addition, we assumed that all surfaces are perfectly smooth, so that there is no thermal resistance at the interfaces. Finally, we use dielectric materials with low conductivities.

3 Solution Method

Recently, Singh et al. reported the analytic solution of systems composed of multiple layers, obtained via power series [21]. The main limitation of this solution is that it requires time-independent boundary conditions and heat sources. Moreover, for the case of thermodynamic properties $\left( {\alpha_{i} ;k_{i} } \right),$ these cannot depend on either temperature or position. Nevertheless, that piece of work represents a very interesting contribution to the approach for finding exact solutions in the area of heat transfer. Similarly, commercial software includes numerical alternatives for solving this type of heat conduction problems. In this work, the numerical solution of the partial differential equation system was developed using MATLAB. Numerical simulations were carried out in a DELL LATITUDE E6440 laptop, with an Intel Core i7 processor, 8 GB RAM, and Microsoft Windows 7 OS.

Figure 1 shows the experimental setup for the simulations. This scheme shows that the solid sphere with three shells hangs from the top wall of the electromagnetic multimodal applicator. This multimode microwave applicator is a closed volume, surrounded by conducting walls. It has a large cavity that allows more than one mode of the electric field. In addition, it is less sensitive to product geometry or position, adaptable to continuous or batch flow, and overall, it is suited for hybrid heating. This multimode applicator operates at 2.45 GHz to create greater field uniformity. An electrical resistance is placed at a defined distance, in such a way that it can provide uniform thermal heating in only one hemisphere. Moreover, a metallic stirrer (e.g. a low-speed multi-blade fan, 1–10 rps) is placed opposite to the resistance. This mode stirrer is a device with the sole purpose of continuously altering the field distribution. In doing so, it contributes to increasing heating uniformity. The operating conditions, which were used in the numerical simulation, are presented in Table 1. In this table, the subscript $i$ represents the number of a layer.

Fig. 1

Experimental setup of the system to be simulated (left) and sketch of the multi-layered sphere model (right)

Table 1 Parameters and operating conditions for the mathematical model and simulations

Moreover, this figure shows that the heterogeneous sphere is heated in one hemisphere, as the sphere does not rotate in this work. This physical situation is given by Eq. 4. Once microwave heating starts, temperature within the innermost sphere (the only one that absorbs/transforms microwaves) raises rapidly, as Eq. 11 describes. The temperature profile for the multi-layered system appears from the solution of Eq. 1, along with its corresponding initial and boundary conditions.

4 Results and Discussion

The Finite Element Mesh (Fig. 2) has 3 $\times$ 43 813 nodes, 10 $\times$ 30 743 elements, using a maximum element size of 1, and a minimum element size of 0.5.

Fig. 2

An example of a mesh distribution for the spherical heterogeneous system (one core, three shells)

Simulations were carried out as follows. First, the electrical resistance was turned on, while the magnetron was kept off. Figure 3 shows the resulting temperature profiles. In this heat conduction process, the shape and intensity of these temperature profiles are dependent on the thermal properties of each shell. The solution model describes quite well the temperature continuity at each material interface. As can be seen, after the first two seconds the temperature profile is quite clear: A higher temperature in the hemisphere being heated with the electrical resistance. Such a profile becomes more evident as heat flows through each layer of anisotropic material. After 97 s, it varies according to the thermal properties of each layer. By definition, the thermal diffusivity measures the heat transfer rate. So, in layers with high thermal diffusivity $($from higher to lower$:i = 1,4,2,3)$ heat transfers quickly. Nonetheless, there is a difference of about 200 °C between temperatures at the core and the outermost heated layer. Such a difference is even higher w.r.t. the hemisphere that receives no thermal radiation at all. Additionally, Fig. 4 shows how temperature profiles change as a function of time for several radial positions. Obviously, the steady state cannot be reached because of the continuous resistance heating.

Fig. 3

Temperature profiles generated using only the electrical resistance, $T\left( {r;\theta = \frac{\pi }{2}; \phi = 0;t} \right)$

Fig. 4

Time variation of the temperature profile, as a function of $r$ and $t$, when only the resistance is heating the composite structure, $T\left( {r;\theta = \frac{\pi }{2}; \phi = 0;t} \right)$

The next simulation analyzed the effect of exclusively heating through microwaves. Hence, the magnetron was set to provide energy continuously, while the resistance was turned off. Figure 5 illustrates a numerical example. As expected, if the permittivity is temperature independent, temperature at the shells increase steadily in the geometrical center (i.e., there is no thermal runaway). Because of its volumetric heating characteristics, microwaves provide a temperature profile through the entire composite structure in a short period of time. It is clear that the inner temperature profile changes drastically after a few seconds. This is due to the high thermal diffusivity of the core and to the continuous volumetric flow converted to thermal energy (an effect of the high dielectric properties). It is also evident that thermal properties of the core are significantly different from those of the layers. This leads to a strong differentiation between the inner and outer temperatures, though they are symmetric with the origin. As in the previous case, Fig. 6 shows that temperature profiles increase with time. Again, it is not possible to reach a steady state, due to the continuous heat generation within the core. As expected, the initial slopes of those curves are higher than the ones of the previous scenario (Fig. 4).

Fig. 5

Temperature profiles generated using only microwaves, $T\left( {r;\theta = \frac{\pi }{2}; \phi ;t} \right)$

Fig. 6

Time variation of the temperature profile, as a function of $r$ and $t$, when only the magnetron is heating the composite structure, $T\left( {r;\theta = \frac{\pi }{2}; \phi = 0;t} \right)$

We now move on to analyze the hybrid heating of the composite structure. Figure 7 shows there is a rapid temperature homogenization through the whole structure, due to the simultaneous effect of both heating methods. Data show that energy absorption is selective. This effect is due to the dielectric properties of the core. Additionally, microwave energy has deep penetration in the composite structure which causes bulk heating. Moreover, temperature is now more homogeneous even after a few seconds of hybrid operation. We remark that this phenomenon is evident, and it is ruled by the thermophysical properties assumed for the simulations. Nonetheless, we show that hybrid heating can be used to rapidly manipulate temperature profiles as needed. In this work we considered materials with position and temperature independent properties, as to easily illustrate the way temperature profiles of multi-layered systems can be changed. Figure 8 shows temperature profiles under different conditions. There is an evident overlap after some radiation time. Also, temperature profiles rise quickly at the beginning, as a response to microwave heating. Finally, it is evident that data come from the numerical development of differential equations that comprise the model. We highlight, once again, that analytical solutions, such as the ones found in literature [21], represent quite an advancement in analytical strategies for describing heat transfer within a heterogeneous system. But, they still lack at describing real situations.

Fig. 7

Temperature profiles generated using the hybrid heating, $T\left( {r;\theta = \frac{\pi }{2}; \phi ;t} \right)$

Fig. 8

Time variation of the temperature profile, as a function of $r$ and $t$, when using the hybrid heating, $T\left( {r;\theta = \frac{\pi }{2}; \phi = 0;t} \right)$

5 Summary and Conclusions

The main objective of this article was to simulate a method for treating a composite structure under different heating conditions. Such a structure is composed of a solid internal sphere (the core) with electrical and thermodynamic properties that allow it to be heated by microwaves. The sphere is surrounded by three microwave transparent shells. We began our work by analyzing the effect of exclusively using radiation heating, due to the energy provided by an electrical resistance that uniformly impacts one hemisphere of the outer shell. Then, we analyzed the effect of exclusively applying microwave radiation in a multimodal applicator. The reasoning behind using such an approach was to study the eventual benefits that volumetric heating may bring to the treatment of the composite structure, such as a rapid temperature increase in the core. We then simulated a combined heating system, where microwaves are used as a pre-heater.

Based on our data, we can conclude that the electrical resistance drastically changes the temperature profiles generated by microwaves, especially at the shells. Hence, this hybrid heating can be used to perform a controlled thermal treatment of composite materials. This can be particularly useful under two scenarios: when certain surface properties are required, and when fast or deep heating is desired. The first one cannot be achieved with microwaves alone, while the second one is not possible with just thermal radiation. Finally, we believe that the hybrid model presented in this work improves upon the numerical simulation of heterogeneous materials, which will positively impact the optimization of their processing.