Modeling Heat Transfer in Thin Layers

MODELING HEAT TRANSFER IN THIN LAYERS

OF GRANULAR COMPOSITE MATERIALS BY CELLULAR AUTOMATA

B.B.Khina and I.P.Samtsevich

Physico-Technical Institute, National Academy of Sciences, Minsk, Belarus

A new computer-oriented model for simulating heat transfer in thin layers of granular composite materials and calculating the effective thermal conductivity is developed using the cellular automata method. Heat transfer in gaskets for heat removal from heat-dissipating units used in Power Electronics is examined. The layer thickness accommodates only about 20 filler particles separated by the binder interlayers, and the thermal conductivity coefficients of a filler (oxide or ceramic particles) and a binder (epoxy or rubber resin) differ by two orders of magnitude, thus the structure of a composite (particle size, shape and spatial arrangement) affects substantially the overall heat conductivity factor, especially for elongated filler particles. Numerical simulation for AlN-filled silicon rubber is performed, and the effect of the spatial arrangement of the filler particles on the effective thermal conductivity is evaluated.

INTRODUCTION

Composite materials with a polymeric binder and ceramic filler can possess a unique combination of mutually contradicting properties, e.g. high thermal conductivity in combination with high insulation resistance. Thin layers (typically of about 1 mm) of such materials are used as gaskets for heat removal from heat-dissipating units (transistors, microprocessors, etc.) in Power Electronics. Elaboration of new electronic devices for special applications necessitates the development of novel composite materials with increased thermal conductivity retaining their insulating properties. Also, there are stringent demands upon the mechanical properties of such materials, viz. they must possess elasticity, plastic and adhesive properties to fit to the roughness of the contacting surfaces (e.g., the transistor base and radiator), along with durability and stability of thermal and mechanical properties at elevated temperatures (80-120 C in present-day devices, and up to 200 C in the future prospect). For producing such composite materials, an electrically insulating ceramic powder with relatively high thermal conductivity, e.g. BN, AlN, Al2O3, ZnO is used as a filler, and such substances as epoxy resin, synthetic rubber/silicon rubber are employed as a matrix. It is known that the thermal conductivity, , of the constituents differ by two orders of magnitude: for example, for silicon rubber  = (15.821.6)10-2 W/(mK) (1) while for Al2O3 30 W/(mK) (2). Besides, the thermal properties of ceramic particles reported in literature vary in a wide range depending on the production method and the crystallographic orientation of grains in polycrystalline samples used for measuring (e.g., for polycrystalline BN with wurtzite lattice  varies from 30 to 60 W/(mK) (3)).

Hence, further advance in this area demands for the development of mathematical models capable of predicting heat conduction of such composites bearing in mind the ultimate goal of optimizing the structure of the material to attain superior thermal properties. However, the specific feature of the problem is that the layer thickness (H ~ 1 mm) accommodates only about 10 to 20 filler particles with a typical diameter of 20-50 m separated by interlayers of the binder (it is obvious that the contact of several brittle ceramic particles inside the material should be avoided in order to retain elastic properties). In this situation the analytical formulas widely used for calculating thermal conduction of bulk materials (4,5 and many others), where volume-averaging gives good agreement with experimental data, are not suitable. Moreover, for elongated filler particles their spatial arrangement must affect substantially the overall thermal conductivity of the composite layer. It seems reasonable to employ stochastic methods, which acquire wide use in modeling disperse systems (6), for studying thermal properties of such materials.

In connection with the above, the objective of this research is to develop a computer-oriented method for modeling heat conduction in thin-layer composite materials, where the thermal conductivity factors of the constituents differ by two orders of magnitude, with a particular stress on simulating the effect of microstructure on the target properties. To attain the goal, we use the "cellular automata" approach (7) possessing wide potentiality for simulating heat/mass transfer in a domain with complex internal structure.

FORMULATION OF A MODEL

Master equation

A two-dimensional domain is considered: a layer of a composite material with a thickness H and width W. Conductive heat transfer through the material is described by a Fourier equation written in a dimensionless form

(1)

with the initial condition

(x, y, =0) = 0 = 0,(2)

where x=X/L and y=Y/L are dimensionless coordinates where X and Y are the corresponding dimensional coordinates, the 0X axis runs across the sample,  = t/t0 is the dimensionless time, L and t0 are the characteristic size and time scale, respectively, L = [t00/(0c0)]1/2; 0, 0 and c0 are the characteristic values of thermal conductivity, density and specific heat, correspondingly; the dimensionless thermal parameters are defined as  = /0,  = /0, c = c/c0,  = (TT0)/(T*T0), where T is temperature and T* is the characteristic maximal temperature (T*=80-120 C), T0 is the initial temperature which is equal to the ambient temperature.

At the upper surface of the flat sample (x=0) we pose the 2nd order boundary condition to equation (1):

(3)

where q|x=0 is the dimensionless heat flux per unit area dissipated by the electronic device attached to the upper surface, q = QL/[0(T*T0)], Q is the corresponding dimensional parameter. For power transistors used e.g., in TV sets, the typical value is Q = 3 to 7 W/cm2 (8). Equation (3) permits simulating heat removal not only in the steady state thermal regime (when Q=const) but also during the initial rapid heating of an electronic device, because most failures occur during transient regimes.

The lower surface (X=H, or x=h=H/L) is considered to be kept at the ambient temperature T0 (1st order boundary condition):

(x=h) = 0(4)

At the lateral surfaces y=0 and y=w=W/L the adiabatic boundary conditions (the absence of heat exchange with the surroundings) are postulated:

(5)

Implementation of cellular automata model

The cellular automata model for numerical solution of master equation (1) with initial (2) and boundary conditions (3)-(5) is built as follows. Since the temperature interval considered in the given problem is relatively narrow, about 100-150 C, the temperature dependence of all the physical parameters may be neglected, and thus the model is treated as a linear one with respect to temperature. A discrete equation describing heat exchange between adjacent cells is actually an analog of an explicit finite-difference scheme, which can be obtained by the integration-interpolation method (9). Using simple formulas (i1,j) = (1/2)[(i,j) + (i1,j)] and (i,j1) = (1/2)[(i,j) + (i,j1)] for interpolating the  value in half-points, where i and j denote the cell coordinate along the 0x and 0y axes, correspondingly, we derive the equation

(6)

where  is a temporal step and n denotes the step number, a is the cell size.

For explicit schemes, the stability criterion exists,  = /[(a2 min(c)]  0.25, which imposes limitation on the  value.

A two-color picture representing the structure of a composite material (a model image or a real microstructure entered into computer from an optical microscope) is superimposed on a field composed of square cells (not less then 100100 cells), and the thermal parameters prescribed to a particular cell, viz.c(i,j),(i,j) and(i,j) are associated with the color of the corresponding image fragment (e.g., black means matrix and white denotes filler). After that, a numerical procedure is invoked which simulates the evolution of the temperature field according to Eq.(6) with corresponding boundary conditions. Calculations are continued until the steady-state regime of heat transfer is attained.

The effective thermal conductivity factor of the composite material is calculated for the steady-state regime when the heat flux, Q, becomes constant along the 0X axis:

(7)

Local thermal conductivity in a longitudinal section (parallel to the 0Y axis), s(X), which must depend on the volume fraction, V, of the filler phase in the corresponding section, was determined in the course of simulation by the formula

(8)

RESULTS OF COMPUTER SIMULATION AND DISCUSSION

Calculations were performed for AlN-filled silicon rubber whose thermal properties are listed in Table 1. For boundary condition (3) we took the dimensional heat flux Q=5 W/cm2 (see Ref.(8)).

Table 1. Thermal parameters of the constituents

Substance / , g/cm3 / c, J/(gK) / , W/(mK) / References
AlN / 3.12 / 0.734 / 30.0 / (3)
silicon rubber / 0.975 / 1.3 / 0.17 / (1)

Figures 1 through 3 present the results of computer simulation of heat transfer in thin layers of composite materials having a close volume fraction of the filler (about 40%) and similar particle shape and size but different microstructure, i.e. spatial arrangement of the filler particles: ordered structure with elongated particles placed upright (Fig.1,a) and aflat (Fig.2,a), and disordered structure (Fig.3,a). The corresponding temperature maps (in grayscale) after the attainment of a steady-state regime are shown in Figs.1,b through 3,b, the temperature-scaling palette being placed below each figure (darker color denotes higher temperature). The lines separated differently colored areas in Figs.1,b3,b are isotherms.

From the comparison of the composite structures and the corresponding temperature maps it is seen that hot (i.e., superheated) areas in the upper part of the specimen appear where the heat-conductive particles are absent (Fig.2,b).

Fig.1. Results of modeling heat transfer in a composite layer with the filler particles placed upright: (a) microstructure, (b) temperature map /

Fig.2. Results of modeling heat transfer in a composite layer with the filler particles placed aflat: (a) microstructure, (b) temperature map /

The average temperature at the upper surface of the composite layer after the attainment of the steady-state regime and the effective thermal conductivity calculated by Eq.(7) were the following: T(X=0) = 70 C and eff = 0.69 W/(mK) for upright arrangement of particles (Fig.1,a), T(X=0) = 115 C and eff = 0.44 W/(mK) for structure with particles placed aflat (Fig.2,a), and T(X=0) = 85 C and eff = 0.59 W/(mK) for disordered microstructure (Fig.3,a). Thus, the calculated eff value varies in a relatively wide range [-25%, +17%] with respect to eff for a sample with chaotic distribution of the filler particles. So, the effect of spatial arrangement of the filler particles in thin layers of composite materials on the thermal conduction is substantial for the conditions considered in this work, viz. heat removal from power electronic devices. The optimal microstructure is that presented in Fig.1,a. Low thermal conductivity of the structure shown in Fig.2,a is connected with the presence of continuous horizontal interlayers of the low-conducting binder.

Fig.3. Results of modeling heat transfer in a composite layer with chaotically oriented filler particles: (a) microstructure, (b) temperature map /

For comparison, a volume-averaged formula (Ref.(4)) for a bulk material with cubical inclusions

eff = mmV/[(1 f /m)1 (1  V)/3] (9)

where m and f are thermal conductivity factors of the matrix and filler, correspondingly, yields eff = 0.5 W/(mK) for the same volume fraction of the filler (V=0.4). This differs by 15% from the calculated value for the disordered microstructure (Fig.3,a).

CONCLUSION

A new approach for studying conductive heat transfer through thin-layer composite materials with greatly differing thermal conductivity factors of the constituents is developed. It is demonstrated that spatial arrangement of high-conducting filler particles in a thin-layered composite material used as a heat-removing gasket has a substantial influence on the overall thermal conduction: the calculated effective thermal conductivity factor varies in a relatively wide range with changing the microstructure of the material. This is important for power electronic devices, especially for those used in special applications.

The results of cellular automata modeling lay a basis for optimizing the structure of a composite material in order to improve its thermal properties.

In this work, we implied perfect adhesion at the filler particle/binder interface, i.e. an absence of an interfacial thermal barrier (the effect of such barrier has been evaluated analytically in Ref.(5) for a volume-averaged approach, which is applicable for bulk composite materials). However, the proposed model can be generalized to take into account this effect just by including chaotically distributed thin interlayers of an additional low-conducting phase at the particle/matrix interfaces.

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