Document Type : Research Paper
Authors
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
Abstract
Keywords
Main Subjects
3D Numerical Simulation of Fibers Arrangement Effects on Thermal Conductivity of Polymer Matrix Composite
Department of Mechanical Engineering, Sari Branch, Islamic Azad University, Sari, Iran
KEYWORDS 

ABSTRACT 
Different fibers arrangement; Lattice Boltzmann method; Polymer matrix composite; Thermal conductivity. 
This study investigated the effects of different arrangements of threedimensional fibers on polymer matrix composite thermal conductivity under heat flux boundary conditions. The thermal lattice Boltzmann based on the D3Q7 (three dimensions and seven temperature vectors) method is utilized to illustrate the thermal conductivity in 7 cases of PMC with a different arrangement of 3D fibers. Nondimensional temperature fields, isothermals, nondimensional thermal conductivity coefficient, nondimensional mean, and local temperature in 7 cases of PMCs have been investigated. The nondimensional thermal conductivity coefficient in each PMC has been analyzed to predict optimal levels of factors affecting this simulation to maximize and minimize the heat transfer rate. The results signified that nondimensional temperature field in a PMC with the arrangement of a fiber, triplet, and triangular perpendicular to heat flux had a greater rate than a PMC with the arrangement of fibers along the way heat flux. Also, the Maximum and minimum of nondimensional thermal conductivity coefficient were in PMC with the arrangement of triplet fibers perpendicular to heat flux, and triangular fibers along the way heat flux, respectively. 
Polymer Matrix Composite (PMC) is a practical material made of polymer and a reinforcing phase untitled fiber. Inside the polymer, the fibers are arranged as unidirectional fibers, veil mat, chopped strands, and woven fabric. manufacturing methods of PMCs are popular is simple and low cost, as well as suitable strength. So, practical applications of PMCs are in industries such as secondary loadbearing aerospace structures, boat bodies, automotive parts. The presented paper attempted to investigate the effects of different arrangements of 3D fibers on polymer matrix composite thermal conductivity under heat flux boundary conditions with the Lattice Boltzmann Method (LBM). LBM has been evolving over the last two decades. The physics of microscopic processes is considered by simplifying kinetic models in LBM [1]. In this simulation, once streaming is done, new distribution components of Lattice nodes are calculated to obtain the updated macroscopic properties [2]. This method of calculating macroscopic values essentially differs from what is done in other traditional Computational Fluid Dynamic (CFD) methods. Algorithm simplicity, fully parallel computation, and easy implementation of complex boundary conditions are among the numerous superiorities of LBM [3]. Also, it is shown strong potential in simulating nonlinear mathematicalphysical equations [4]. It should be noted that these models always deal with stability issues, and therefore in the last few years, several researches have been done to solve this problem, such as Two Relaxation Time (TRT) models and Multi Relaxation Time (MRT) model. Peiravi et al. [5] surveyed a 3D multiphase nanofluid natural convection and radiation effect based on numerical and analytical simulation. Mal et al. [6] studied the effect of thermal conductivity and electrochemical properties of a composite solid polymerbased on polyvinyl alcohol matrix by the casting method. Torres et al. [7] analyzed the behavior of composite materials mixed with polymer and ceramic matrix. They surveyed the resistance of solid residuals against wet abrasive. Fois et al. [8] investigated heat transfer mechanisms in PMCs containing microencapsulated paraffin in latent energy storage systems. Ivosevic et al. [9] improved the resistance of thermally sprayed for the erosion and oxidation of PMCs with varying volume fractions of WC–Co. Cenna et al. [10] experimentally investigated wear mechanisms in composite materials subjected to friction by bulk solids. Roy et al. [11] analyzed four different composite materials with glass fibers against solid particle erosion. Shemelya et al. [12] studied the effect of graphite, silver, and carbon fiber geometry in an acrylonitrile butadiene styrene polymer matrix on the thermal conductivity. Han et al. [13] investigated the effect of filler incorporation and curing pressure on the thermal conductivity of carbon fiber PMC. Kim et al. [14] investigated the electrical and thermal effects of a silver flake PMC using a 3D resistor network model. Yamada et al. [15] surveyed the thermal conductivity of carbon fiber PMCs by Nanostructuring the interlaminar interface with carbon black on heat dissipation of aircraft. Alva et al. [16] presented thermal and electrical effects on boron nitride, zinc oxide, and silicon carbide ceramic nanofibers with polyvinyl butyral polymer matrix in PMC applications. Wang et al. [17] simulated the thermal reaction and erosion in glass fiber in PMCs subjected to a lightning strike. Moradi et al. [18] investigated the effect of thermal conductivity and openhole size on mechanical characteristics of PMCs. Yu et al. [19] surveyed thermal conductivity and mechanical characteristics of carbon fiber in threedimensional PMCs. Takenaka et al. [20] studied thermal characteristics of PMCs with negative thermal expansion materials in injection molding. Chen et al. [21] presented the electrical and thermal conductivity of PMC with graphene/aluminum nanofibers in electronic devices. Recently, Ouyang et al. [22] investigated the thermal conductivity of PMCs with network Al_{2}O_{3} spheres fibers. Wang et al. [23] presented the electrical effect and high thermal conductivity of PMCs with polyamide 6/carbon nanofiber composites in braze welding technology. Shigang et al. [27] numerically investigated three dimensional thermal conductivity of Woven C/C composites from 300 to 2500 K. Lu et al. [28] numerically surveyed the elastic properties of three dimensional fiber composites with a wide range of fiber aspect ratios. Klein et al. [29] used a numerical homogenization approach in order to study the principal influence of key composite descriptors of fiber and particle reinforced PEEK on the homogenized heat conductivity. Karkri et al. [30] used a threedimensional (3D) finite elements method for predicting the effective thermal conductivity (ETC) of a conductive hollow tube polymer composite. Liang et al. [31] presented a model for simulating the microscopic heat transfer processes in a woodmetal composite material. The model was developed by analyzing the microstructure of experimental samples comprising a melted alloy impregnated in a wood matrix. Karkri et al. [32] presented a numerical and experimental study of a composite material with conducting spheres embedded in a polymer matrix. Ejeh et al. [33] investigated the carbon fibers doped with nanoparticles of silicon carbide (CFSiC) and resin bonded glass fiber (RBGF). It was found that the results were distinctly different when compared with the CFRP laminate. CFSiC showed to exhibit an enhanced thermoelastic behaviour, due to the high thermal stability of SiC nanoparticles in the composite. In the presented paper, the model is validated by comparing with analytical result [24]. Then nondimensional thermal conductivity characteristics and temperature field are investigated for different arrangement of 3D fibers in PMC under heat flux boundary condition.
The effects of different arrangement of 3D fibers on the thermal conductivity of PMC under heat flux boundary condition is presented numerically. The conduction heat transfer characteristics are optimized with seven fibers and polymer matrix cases in a different arrangement. The boundary conditions for PMC under heat flux are illustrated in Fig. 1. The dimensions of each 3D enclosure are L×L×L. The present study investigated fibers effects on thermal profiles from the heat flux wall to the cold temperature wall. So, according to our research, these seven cases have the most critical effect on thermal conductivity.
Thermal conductivity coefficient in i direction for fiber and polymer matrix are and
, respectively. Where T_{1} and T_{2 }are unknown temperatures on the wall with constant heat flux for fiber and polymer matrix, respectively_{.} The conduction heat transfer equation for the problem is represented in Eq. (1) [24]:

(1) 
where, k= (k_{x}, k_{y}, k_{z}) and n= (x, y, z).
The lattice Boltzmann method is a powerful numerical method for simulating thermal conductivity. This method has many advantages in comparing conventional CFD methods of calculating energy equations.


a) A fiber perpendicular to heat flux 



b) A fiber along the way heat flux 



c) Triplet fibers perpendicular to heat flux 



d) Triplet fibers along the way heat flux 




e) Triangular fibers perpendicular to heat flux 



f) Triangular fibers along the way heat flux 



g) Triplet fibers perpendicular to the adiabatic wall 

Fig. 1. Schematic diagrams of fibers and polymer matrix in a different arrangement 
In this approach, the solid domain is made discrete in uniform Cartesian cells; each holds a fixed number of Distribution Functions (DF) that represent the number of solid particles moving in these discrete directions. Hence, depending on the dimension and number of thermal directions, different models can be used. In the LBM, integral distribution functions over boundaries must be integral and calculated. Therefore, suitable equations need for calculating distribution functions on boundaries for a given boundary condition. The present study examined 3D solid using a cubical lattice with seven temperature vectors for the temperature field.
Table 1. temperature vectors of the D3Q7 model
i 

0 
(0,0,0) 
1 

2 

3 

4 

5 

6 



Fig. 2. Schematic diagrams of 3D temperature 
The temperature vectors Of the D3Q7 model is shown in Fig. 2.
The LBM used the simplest algorithm to simulate the fluid flow and boundary conditions with good accuracy. Also, this method simultaneously analyzed data, postprocessing, and evaluated them. For each temperature vector at the D3Q7 model, a particle DF is stored that presented in Table 1. where and k is the Lattice velocity direction. The LB model used in the present work is the same as that employed in [24]. The DFs are calculated by solving the Lattice Boltzmann Equation (LBE), a special discretization of the kinetic Boltzmann equation. The thermal Lattice Boltzmann equations [25] based on a uniform Lattice with Bhatnagar–Gross–Krook (BGK) collision model is represented in Eq. (2) [24]:

(2) 
Where c_{i} is the Lattice velocity, while g_{i} is energy distribution functions. τ_{c} is the controlling factors of the rate equilibrium. The equilibrium energy distribution function for the current 3D application, based on the D3Q7 model, is expressed as [26]:

(3) 
The temperature vectors of the D3Q7 model are presented in Table 2.
Table 2. temperature vectors of the D3Q7 model [26]
i 

0 

1 

2 

3 

4 

5 

6 

The temperature vectors of the D3Q7 model are presented in Table 2 [26].

(4) 
In this section, according to Fig. 3, the results of the simulation of conduction heat transfer are compared and verified in an environment with two solids with different conductivity coefficient and [23].
According to Fig. 3, the conduction heat transfer relationship between two solid bodies are represented in Eq. (5):

(5) 
The energy equation is expressed according to the physical condition of the problem is represented in Eq. (6):

(6) 
T_{c} 
T_{h} 
Solid – Solid interface

Solid (2) 


Solid (1)

L 
Fig. 3. Schematic of solidsolid hybrid heat transfer
For the first solid body with a conductivity coefficient , the boundary conditions are represented in Eq. (7) and (8):

(7) 

(8) 
By inserting the Eq. (5) in the Eq. (8), so:

(9) 
By replacing the Eqs. (7) and (8) instead of the constant coefficients and in Eq. (6), so:

(10) 
For the second solid body with a conductivity coefficient , the boundary conditions are represented in Eq. (11) and (12):

(11) 

(12) 
Using the Eqs. (11) and (12), the constant coefficients and are represented in Eq. (13) and (14):

(13) 

(14) 
By inserting the Eq. (5) in Eq. (13), the constantcoefficient is represented in Eq. (15):

(15) 
By inserting the Eq. (5) in Eq. (14), the constantcoefficient is represented in Eq. (16):

(16) 
By inserting the Eqs. (15) and (16) in the Eq. (6), so:

(17) 
With compression of solving Eqs. (7) and (17) by the LBM and analytical solution. As shown in Fig. 4, there is good accuracy between the LBM and the analytical solution.
Fig. 4. Comparison between LBM and
analytical solution
Here, is the nondimensional temperature that is equal to .
In this research, the effects of different arrangements of 3D fibers on the thermal conductivity coefficient of PMC are presented under heat flux boundary conditions. Fig. 5 illustrates the nondimensional temperature field of PMC with the arrangement of a fiber perpendicular to heat flux and along the way heat flux at time steps 1000, 4000, 8000, 12000, 16000, and 20000. According to this Fig., nondimensional temperature field in a PMC with the arrangement of a fiber perpendicular to heat flux had a greater rate than a PMC with the arrangement of a fiber along the way heat flux. Nevertheless, Han et al. [13] increased the thermal conductivity of carbon fiber polymer matrix composite by curing pressure increase and filler incorporation. Also, Jan et al. [15] enhanced the thermal conductivity of carbon fiber polymer–matrix composites in the throughthickness direction by nano structuring the interlaminar interface with carbon black.

b) A fiber perpendicular to heat flux at 

c) A fiber perpendicular to heat flux at 

d) A fiber perpendicular to heat flux at 

e) A fiber perpendicular to heat flux at 

f) A fiber perpendicular to heat flux at 


g) A fiber along the way heat flux at 

h) A fiber along the way heat flux at 

i) A fiber along the way heat flux at

j) A fiber along the way heat flux at 

k) A fiber along the way heat flux at 

l) A fiber along the way heat flux at 
Fig. 5. Comparison between nondimensional temperature field of PMC with the arrangement of a fiber perpendicular to heat flux and along the way heat flux 
Fig. 6 illustrates the nondimensional temperature field of PMC with the arrangement of triplet fiber perpendicular to heat flux and along the way heat flux at time steps 1000, 4000, 8000, 12000, 16000, and 20000. According to this Fig., nondimensional temperature field in a PMC with the arrangement of triplet fiber perpendicular to heat flux had a greater rate than a PMC with the arrangement of triplet fiber along the way heat flux.


a) Triplet fiber perpendicular to heat flux at 

b) Triplet fiber perpendicular to heat flux at 

c) Triplet fiber perpendicular to heat flux at 

d) Triplet fiber perpendicular to heat flux at 

e) Triplet fiber perpendicular to heat flux at 

f) Triplet fiber perpendicular to heat flux at 


g) Triplet fiber along the way heat flux at 

h) Triplet fiber along the way heat flux at 

i) Triplet fiber along the way heat flux 

j) Triplet fiber along the way heat flux at 

k) Triplet fiber along the way heat flux at 

l) Triplet fiber along the way heat flux at 
Fig. 6. Comparison between nondimensional temperature field of PMC with arrangement of triplet fiber perpendicular to heat flux and along the way heat flux 
Fig. 7 illustrates the nondimensional temperature field of PMC with the arrangement of triangular fiber perpendicular to heat flux and along the way heat flux at time steps 1000, 4000, 8000, 12000, 16000 20000. According to this Fig., nondimensional temperature field in a PMC with the arrangement of triangular fiber perpendicular to heat flux had a greater rate than a PMC with the arrangement of triangular fiber along the way heat flux.


a) Triangular fiber perpendicular to heat flux at 

b) Triangular fiber perpendicular to heat flux at 

c) Triangular fiber perpendicular to heat flux at 

d) Triangular fiber perpendicular to heat flux at 

e) Triangular fiber perpendicular to heat flux at 

f) Triangular fiber perpendicular to heat flux at 


g) Triangular fiber along the way heat flux at 

h) Triangular fiber along the way heat flux at 

i) Triangular fiber perpendicular to heat flux at 

j) Triangular fiber along the way heat flux at 

k) Triangular fiber along the way heat flux at 

l) Triangular fiber along the way heat flux at 
Fig. 7. Comparison between nondimensional temperature field of PMC with the arrangement of triplet fiber perpendicular to heat flux and along the way heat flux 
Fig. 8 illustrates the nondimensional temperature field of PMC with the arrangement of triplet fiber perpendicular to the adiabatic wall at time steps 1000, 4000, 8000, 12000, 16000, and 20000. With the comparison of Figs. 8 and 6, nondimensional temperature field in a PMC with the arrangement of triplet fiber perpendicular to heat flux had a greater rate than a PMC with the arrangement of triplet fiber along the way the adiabatic wall.


a) 

b) 

c) 

d) 

e) 

f) 
Figure 8. Comparison nondimensional temperature field of PMC with the arrangement of triplet fibers perpendicular to the adiabatic wall at 
Fig. 9 illustrates the isothermal of PMC with a different arrangement of 3D fibers on thermal conductivity coefficient of PMC at time steps 1000, 4000, 8000, 12000, 16000, and 20000. According to this Fig., minimum isothermals were in PMCs with triplet and triangular fibers arrangement perpendicular to heat flux, respectively.

a) A fiber perpendicular to heat flux 

b) a fiber along the way heat flux 

c) Triplet fibers perpendicular to heat flux 

d) Triplet fibers along the way heat flux 

e) Triangular fibers perpendicular to heat flux 

f) Triangular fibers along the way heat flux 

g) Triplet fibers perpendicular to the adiabatic wall 
Fig. 9. Comparison between isothermal of PMC with the arrangement of a fiber 
Fig. 10 illustrates nondimensional local temperature on the wall with heat flux boundary condition at y direction in PMCs with a different arrangement of 3D fibers. According to this Fig., the maximum of nondimensional local temperatures was in PMCs with the arrangement of a fiber, triplet, and triangular fibers along the way heat flux, respectively. Also, the minimum of nondimensional local temperatures was in PMCs with the arrangement of triplet, triangular fibers, and a fiber perpendicular to heat flux, respectively.
Fig. 10. The nondimensional local temperature on the wall with heat flux boundary condition at y direction
Fig. 11 illustrates nondimensional local temperature on the wall with heat flux boundary conditions at z direction in PMCs with a different arrangement of 3D fibers. Similar to Fig 10., the maximum of nondimensional local temperatures was in PMCs with the arrangement of a fiber, triplet, and triangular fibers along the way heat flux, respectively. Also, the minimum of nondimensional local temperature was in a PMC with the arrangement of triplet fibers perpendicular to heat flux that is a fixed line.
Table 3 presents the nondimensional mean temperature on the wall with heat flux boundary condition of PMC with a different arrangement of 3D fibers. According to this table, maximum and minimum of nondimensional mean temperature were in PMCs with the arrangement of triangular and triplet fibers perpendicular to heat flux, respectively.
Fig. 11. The nondimensional local temperature on the wall with heat flux boundary condition at z direction
Table 3. Nondimensional mean temperature on the wall with heat flux boundary condition
Cases 
θ 
a 
0.473 
b 
0.574 
c 
0.465 
d 
0.583 
e 
0.468 
f 
0.585 
g 
0.474 
Table 4 presents the nondimensional thermal conductivity coefficient on the wall with heat flux boundary condition of a PMC with a different arrangement of 3D fibers into the thermal conductivity coefficient of the case (a). According to this table, maximum and minimum of nondimensional thermal conductivity coefficient were in PMC with the arrangement of triplet fibers perpendicular to heat flux and triangular fibers along the way heat flux, respectively.
Table 4. Nondimensional thermal conductivity coefficient on the wall with heat flux boundary condition
Cases 

a 
1 
b 
0.824 
c 
1.017 
d 
0.811 
e 
1.011 
f 
0.809 
g 
0.999 
In this paper, the effects of different arrangements of 3D fibers on the thermal conductivity coefficient of PMC were investigated under heat flux boundary conditions. The D3Q7 LB method was used to solve nondimensional temperature fields, isothermal, isothermal, nondimensional thermal conductivity coefficient, nondimensional mean, and local temperature in 7 cases of PMCs. Finally, some of the main points summarized:
Nomenclature

Boltzmann constant 
C_{i} 
lattice velocity 
k 
thermal conductivity coefficient (W/m. K) 
k_{x} 
nondimensional thermal conductivity coefficient 

lattice grade weight 
x, y, z 
coordinates (m) 
L 
dimensions of enclosure (m) 

particle energy distribution function 

equilibrium particle energy distribution function 

heat flux (W/m^{2}) 
T_{c} 
the temperature of the cold wall (K) 
α 
thermal diffusivity (m^{2}/s) 

thermal expansion (1/K) 

relaxation time relating to temperature ﬁeld 

nondimensional temperature ( 
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