Thermal Conductivity of FGM pipes
Marcelo Medeiros Jr., Ph.D.
May 15th 2022
1
Thermodynamics of FGM Composites
The general form of heat conduction equation in cylindrical coordinates for a homogeneous material is
given by:
∂T (r)
1 ∂
∂T (r)
∂
∂T (r)
∂T (r)
1 ∂
λr
+ 2
λ
+
λ
+ q = ρcp
(1)
r ∂r
∂r
r ∂θ
∂θ
∂z
∂z
∂t
where r, θ, z are cylindrical coordinates, λ, is the thermal conductivity, t time, T (r) temperature distribution,
q heat generation per unit time and per unit volume, ρ density and cp specific heat. Considering that
the cylinder is subjected to an axisymmetric temperature distribution with no variation along the length
(sufficiently long cylinder), the steady state heat equation can be reduced to:
dT (r)
1 d
λr
= −q
(2)
r dr
dr
Separating and integrating both sides yields,
Z
Z
dT (r)
dT (r)
qr 1
d
λr
dr = (−qr)dr −→ λ
= − + C1
dr
dr
dr
2
r
(3)
In an FGM the thermal conductivity varies along the radial direction, therefore λ is also a function of the
radius, λ(r). In most of the practical cases there will be no heat generated inside the cylinder so the term
related to q vanishes. If Eq. 3 is integrated once more it is possible to define the temperature at any value
of r along the radial direction as follows,
Z
1
T (r) = C1
dξ
(4)
ξλ(ξ)
In order to validate the results from the Finite Element Analysis, it is necessary to assign a functional form for
the variation of the thermal conductivity along the radial direction, λ(r) in Eq 4. If the thermal conductivity
of the FGM composite is given as λ(r) = λ0 rβ , for instance, the temperature profile as function of the radius
is given by,
C1
1
T (r) = −
+ C2
(5)
λ0 β rβ
The constants C1 and C2 are obtained considering the boundary conditions of the inside and outside walls.
When r equals the inner radius (r = a), the temperature is equal to the inner boundary temperature,
T (a) = Ta . When the radius is equal to the outer radius (r = b), the temperature is equal to the outer
boundary temperature, T (b) = Tb , therefore,
(Ta − Tb )
aβ − bβ
β
β
a Ta − b Tb
C2 =
a β − bβ
C1 = λ0 β (ab)β
1
(6)
(7)
In order to perform a heat transfer analysis of an FGM material in Abaqus it is necessary to develop
a FORTRAN user-subroutine (UMATHT) capable of addressing the variation of the thermal conductivity
along the radial direction. This subroutine is called, during the analysis, at every integration point (Gauss
Points) and returns the thermal conductivity matrix, heat flux, thermal energy among other properties based
on the x, y and z coordinates of the point. Based on that information, Abaqus assembles the global thermal
conductivity matrix and solves for the temperature and heat flow.
Figure 1 shows the comparison between the FE results from the implemented UMATHT and the analytical
solution (Eq. 5). Each curve corresponds to the temperature profile along the cylinder wall and are calculated
for three different values of β. The cylinder has an inner radius of 10cm and an outer radius of 15cm (a = 10
and b = 15).
Figura 1: Radial Temperature Profiles (λ0 = 10, Ta = 600, Tb = 300).
1.1
Thermal Conductivity of Composites
In general terms, the thermal properties of composites can be predicted by analytical micromechanical
models and finite element simulations. Micromechanical models provide closed form expressions for the
homogenized thermal conductivity as functions of the properties of the matrix and the inclusions. Hasselman
and Johnson [4] proposed the effective thermal conductivity of non-interacting, mono-disperse spherical and
cylindrical heterogeneities. The expression for the homogenized thermal conduction is given by,
h I
i
λI
λI
2λI
2 λλM − ah
+
−
1
v
+
+
2
f
M
ahc
λ
c
i
λ = λM h
(8)
λI
λI
λI
2λI
1 − λM + ahc vf + λM + ahc + 2
2
where the superscripts I and M stand for inclusion and matrix respectively, a is the radius of the spherical
inclusion and hc is the conductance of the thermal barrier between the inclusion and the matrix. This
approach is known as Effective Medium Theory (EMT) and is a variant of the Maxwell’s model for composite
thermal conductivity that takes into account the interfacial properties.
The interfacial thermal resistance plays a pivotal role on the effective thermal conductivity of the composites. It is well known that interfacial resistances between reinforcements and matrix give rise to a size effect
in the overall thermal conductivity of composites, with even highly conductive particles or fibers failing to
increase the overall conductivity when their size falls below certain limits.
EMT has been applied to Al/SiC porous composites [5]. Benveniste and Miloh [2] extended the EMT
to spheroidal particle shapes with imperfect interfaces and subsequently applied in the framework of the
Mori-Tanaka scheme [1].Torquato and Rintoul derived rigorous third-order bounds for effective conductivity
of macroscopically isotropic distribution of particles with imperfect interfaces [7]
1.2
Coefficient of Thermal Expansion
When a composite is heated, the kinetic energy of its molecules increases leading them to vibrate more
vigorously. The amplitude and the population of density of states of these vibrations (phonons) increases
with the increase in temperature, causing the material to expand. The ratio between this expansion and the
variation in temperature is called the coefficient of thermal expansion α.
A popular model to estimate the coefficient of thermal expansion of composites is the Turner model [8].
It takes into account the mechanical interaction between the phases in the heterogeneous composite material.
This model assumes homogeneous strain throughout the composite, and uses a balance of internal average
stresses to derive the effective α. It also assumes perfect bonding between the phases. According to Turner,
the coefficient of linear thermal expansion is given by:
α=
αM K M (1 − vf ) + αI K I vf
K M (1 − vf ) + K I vf
(9)
It is worthy to point out that the expression given by Turner reduces to the Rule of Mixtures (RoM)
derived by Voigt [9] if the Bulk Modulus K is replaced by the Young’s Modulus E. The Turner model is
usually employed as a lower bound for coefficient of thermal expansion of composites.
Schapery derived the upper and lower bounds for the effective coefficient of thermal expansion of isotropic
and anisotropic composite materials by using the extremum principles of thermo-elasticity [6].
M
#
M " c
I
I
K
−
K
α
−
α
vf
4G
(10a)
αupper = αI vf + αM (1 − vf ) +
Kc
4GM + 3K I
#
I " c
K − K M αI − αM (1 − vf )
4G
I
M
(10b)
αlower = α vf + α (1 − vf ) +
Kc
4GI + 3K M
where K c denotes the effective bulk modulus of the composite obtained using Hashin’s homogenization
Theory [3].
Referências
[1] Y. Benveniste. On the effective thermal conductivity of multiphase composites. ZAMP Zeitschrift für
angewandte Mathematik und Physik, 37(5):696–713, September 1986.
[2] Y. Benveniste and T. Miloh. The effective conductivity of composites with imperfect thermal contact at
constituent interfaces. International Journal of Engineering Science, 24(9):1537–1552, jan 1986.
[3] Z. Hashin and S. Shtrikman. A variational approach to the theory of the elastic behaviour of multiphase
materials. Journal of the Mechanics and Physics of Solids, 11(2):127–140, March 1963.
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[4] D.P.H. Hasselman and Lloyd F. Johnson. Effective thermal conductivity of composites with interfacial
thermal barrier resistance. Journal of Composite Materials, 21(6):508–515, jun 1987.
[5] J Molina, R Prieto, J Narciso, and E Louis. The effect of porosity on the thermal conductivity of
al–12wt.% Si/SiC composites. Scripta Materialia, 60(7):582–585, apr 2009.
[6] R.A. Schapery. Thermal expansion coefficients of composite materials based on energy principles. Journal
of Composite Materials, 2(3):380–404, July 1968.
[7] S. Torquato and M. D. Rintoul. Effect of the interface on the properties of composite media. Physical
Review Letters, 75(22):4067–4070, November 1995.
[8] P.S. Turner. Thermal-expansion stresses in reinforced plastics. Journal of Research of the National Bureau
of Standards, 37(4):239, October 1946.
[9] W. Voigt. Ueber die Beziehung zwischen den beiden Elasticitätsconstanten isotroper Körper. Annalen
der Physik, 274(12):573–587, 1889.
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