Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

49, 2, pp. 385-397, Warsaw 2011

NON-STATIONARY HEAT TRANSFER IN A HOLLOW

CYLINDER WITH FUNCTIONALLY GRADED MATERIAL

PROPERTIES

Piotr Ostrowski

Bohdan Michalak

Technical University of Lodz, Department of Structural Mechanics, Łódź, Poland

e-mail: postrow@p.lodz.pl; bmichala@p.lodz.pl

The unidirectional non-stationary heat conduction in a two-phase hol-
low cylinder is considered.The conductor ismade of two-phase stratified
composites and has smooth gradation of effective properties in the ra-
dial direction. Therefore, we deal here with a special case of functionally
gradedmaterials, FGM. The formulation of mathematical model of the
conductor is based on a tolerance averaging approach (TAA). Appli-
cation to the non-stationary heat conduction and a comparison of the
tolerancemodel with the asymptotic one is shown. The effect of geome-
try andmaterial properties of the conductor on the temperature field is
examined.

Key words: heat transfer, functionally gradedmaterial, tolerance avera-
ging approach

1. Introduction

The main aim of this paper is to consider an effect of geometry and mate-
rial properties on the temperature field in a two-phase hollow cylinder. This
consideration deals with a non-stationary heat transfer problem in a com-
posite conductor with a deterministic microstructure which is periodic along
the angular axis and has smooth and slow gradation of effective properties
in the radial direction (Fig.1). Therefore, we deal here with a special case of
functionally gradedmaterials, FGM (Suresh andMortensen, 1998).

Functionally gradedmaterials are a new class of composite materials whe-
re composition of constituents generates continuous and smooth gradation of
apparent properties of the composite. The analysis of the heat transfer in a



386 P. Ostrowski, B. Michalak

Fig. 1. Structure of the two-phase functionally graded composite in (a) micro- and
(b) macro-scale

hollow cylinder made of functionally gradedmaterials can be found in Hosse-
ini et al. (2008), Ootao and Tanigawa (2006), Sladek et al. (2003), Wang and
Mai (2005), where material properties are expressed as power or exponential
functions of the radial coordinate. The hollow cylinder presented in Hosseini
et al. (2008) has a heterogeneous microstructure and it is divided into ma-
ny subcylinders (layers) across the thickness. In the paper by Aboudi et al.
(1999), one can find applications of higher-order theory for thermal analysis
in functionally gradedmaterials.
The physical phenomenon of the heat transfer is described by the well

known Fourier equation

cΘ̇−∇· (K ·∇Θ)= 0 (1.1)

which contains (in this case) highly oscillating and discontinuous coefficients;
K – heat conduction tensor, and c – specific heat. Therefore, different avera-
ged models have been proposed. The modelling problem is how to describe a
microheterogeneous conductor by certain averaged equations. The solution to
the above problem for periodic structures based on homogenization technique
for differential equations with highly oscillating coefficients has an extensive
list. Here we can mention monographs by Jikov et al. (1994) and the paper
by Lewinski andKucharski (1992). Homogenization can be also realised using
a concept of micro-local parameters, c.f. Matysiak (1991). However, because
the formulation of averaged models by using the asymptotic homogenization
is rather complicated from the computational point of view, these asymptotic
methods are restricted to the first approximation. Hence, the averaged model
obtained by using this method neglects the effect of the microstructure size
on the heat transfer in a FGM-conductor. The formulation of themacroscopic
mathematical model for the analysis of heat transfer in the conductor under



Non-stationary heat transfer in a hollow cylinder... 387

consideration will be based on the tolerance averaging technique, c.f.Woźniak
et al. (2008), Woźniak and Wierzbicki (2000). The general description of this
technique and application to analysis of longitudinally graded stratifiedmedia
can be found inMichalak et al. (2007), Woźniak et al. (2008).

2. Model equations

The object of our considerations is a hollow conductor with microstructure
given in Fig.2. Let us introduce the orthogonal curvilinear coordinate system
Oρϕz in the physical space Ω occupied by a conductor under consideration.
The region Ω occupied by the conductor is given by Ω = Π× I, where Π
is a region in the Oρϕ plane. The time coordinate will be denoted by t.
The microstructure is determined by the unit cell ∆ with the diameter of
λ = 2π/N, where N is a number of cells in the considered composite. What
is most important, the walls width g is constant along the radial axis, which
implies smooth variation of macroscopic material properties in this direction.
Volume fractions of homogeneaus layers are denoted by ν′(ρ) = δ(ρ)/λ and
ν′′(ρ) = g/λρ. Dimensionless function ν =

√
ν′ν′′ ∈ [0,0.5] is referred to as

the distribution of heterogeneity.

Fig. 2. Fragment of a cross-section of the hollow conductor

One of the fundamental assumptions in the tolerance averaging approach
concerns the temperature field decomposition

Θ(ρ,ϕ,t) = θ(ρ,ϕ,t)+h(ρ,ϕ)ψ(ρ,ϕ,t) (2.1)

where ρ ∈ [R0,Rk], ϕ ∈ [0,2π) and t ­ 0s. Functions of averaged tempe-
rature θ and temperature fluctuation amplitude ψ are assumed to be slowly
varying, i.e. θ(ρ, ·, t), ψ(ρ, ·, t)∈ SV 1d (Ω,∆). The exact definition of the slowly



388 P. Ostrowski, B. Michalak

varying and tolerance periodic function can be found inWoźniak et al. (2008).
The expected form of the temperature oscillations, caused by discontinuity of
the coefficients in (2.1), is assured by the ”saw-type” locally periodic function,
which would be called the fluctuation shape function h.

Fig. 3. Fluctuation shape function

The second concept of the modelling technique is the averaging operation

〈f〉(ρ,ϕ) = 1
|∆|

ϕ+λ/2
∫

ϕ−λ/2

f(ρ,z) dz (2.2)

where |∆| = λ. On the grounds of this definition, we can formulate the se-
cond modelling assumption, the tolerance averaging approximation. In the
course of modelling it is assumed that terms O(d) are negligibly small, where
d is a certain tolerance parameter, c.f. Woźniak et al. (2008). For an arbi-
trary tolerance periodic function f ∈ TP1d(Ω,∆), slowly varying function
F ∈ SV 1d (Ω,∆) and fluctuation shape function h ∈ FS1d(Ω,∆), we have

〈fF〉= 〈f〉F +O(d)
(2.3)

〈f∇(hF)〉= 〈f∂h〉F + 〈fh〉∇F +O(d)

Averaging description

Bearing in mind the model assumptions, we derive from equation (1.1)
the following system of averaged equations for the unknowns θ(ρ,ϕ,t) and
ψ(ρ,ϕ,t), which can be found inWoźniak et al. (2008)

∇· (〈K〉∇θ+ 〈K∂h〉ψ)−〈c〉θ̇ =0
(2.4)

∇· (〈Khh〉∇ψ)−〈K∂h〉∇θ−〈K∂h∂h〉ψ −〈chh〉ψ̇ =0



Non-stationary heat transfer in a hollow cylinder... 389

The above equations describe two-dimensional heat conduction in the two-
phase hollow cylinder. The coefficients

〈K〉= k′ν′+k′′ν′′ 〈c〉= c′ν′+ c′′ν′′

〈Khh〉= λ2ν2〈K〉 〈chh〉= λ2ν2〈c〉
〈K∂h〉=2

√
3ν(k′−k′′) 〈K∂h∂h〉=12(k′ν′′+k′′ν′)

(2.5)

are continuous and functional. The gradient operators in the above equations
have the form

∇=(∂1,∂2) ∇=(∂1,0) ∂ =(0,∂2) (2.6)

where ∂α = ∂/∂ξα for α =1,2.
Theobtained averaged differential equations, (2.4), have smooth functional

coefficients in contrast to coefficients in equation (1.1), hence in some special
cases (stationary unidirectional conduction) analytical solution can be obta-
ined. In other cases, numerical methods have to be used. Here we shall use
the finite difference method (Cranck-Nicholson method for time integration)
to derive solutions to boundary/initial value problems formulated in the fra-
mework of the proposed tolerance model. This model takes into account the
effect of microstructure size on the overall heat transfer behaviour.

3. Examples of application

Themain aim of this section is to present the effect of someparameters on the
temperature field and relative velocity of achieving the steady state problem
– denoted in figures with the subscript st. Hence, we consider in all three
following examples the ratio of the temperature value in selected time t to the
temperature value for a steady state problem.We restrict the analysis to the
unidirectional heat transfer for a conductor with deterministic microstructure
shown in Fig.1. In general, wewrite the full anisotropic tensor of conductivity
for each component

K= k

[

1 b
b a

]

(3.1)

where a ∈ (0,1], b ∈ [0,
√
a). Fixed values of conductivity and specific heat

for both components are listed in Table 1.
Initial-boundary conditions would be given a priori. For the temperature

field given by equation (2.1), two unknown functions θ and ψ must be defined



390 P. Ostrowski, B. Michalak

Table 1.Material properties

Conductivity Specific heat
k [Wm−1K−1] c [Jm−3K−1]

phase I 58 3432000

phase II 0.045 14600

on the boundary. Let the initial conditions for θ and ψ be assumed in the
form

θ(ρ,0)= ψ(ρ,0)= 0◦C (3.2)

and the boundary conditions for every time t ­ 0s

θ(R0, t)= 100
◦C

(3.3)
θ(Rk, t)= ψ(R0, t)= ψ(Rk, t)= 0

◦C

All the above conditions and formulations will be used in the subsequent part
of this paper.

3.1. Benchmark solutions

Case 1. For verification of the postulated value of the step time parame-
ter in the Cranck-Nicholson method for time integration, we compare three
independent methods, i.e. finite difference method (FDM) for the tolerance
model, finite element method (FEM) for the microheterogeneous conductor
and analytical solution (AS) of the tolerancemodel equations. Let us consider
a uniform hollow cylinder with conductivity K = 58Wm−1K−1 and speci-
fic heat c = 3432000Jm−3K−1. Geometry as shown in Fig.1 for R0 = 1m,
Rk =3m. Analytical solution is expressed by

θ(ρ,t)= θ0+(θk−θ0)
lnρ− lnR0
lnRk− lnR0

+

(3.4)

+π
∞
∑

n=1

C0(ρ,αn)

F(αn)
J0(Rkαn)[θkJ0(R0αn)−θ0J0(Rkαn)]e−κα

2
n
t

where θ0 = θ(R0, t), θk = θ(Rk, t) and

C0(ρ,αn)= J0(R0αn)Y0(ραn)−J0(ραn)Y0(R0αn)
(3.5)

F(αn)= J
2
0(R0αn)−J20(Rkαn)



Non-stationary heat transfer in a hollow cylinder... 391

for αn, n =1,2, . . . as roots of the equation

J0(R0α)Y0(Rkα)−J0(Rkα)Y0(R0α)= 0 (3.6)
where functions J0, Y0 are well known Bessel functions. Comparison of obta-
ined results is made for t = 3600s and t = 7200s. The amplitude of tempe-
rature fluctuation in this case equals zero. All diagrams for everymethod and
at every time t are overlapped.

Fig. 4. Comparison of averaged temperature for AS – analytical solution,
FEM – finite elementmethod and FDM – finite difference method

Case2.Additionally, a comparisonof the tolerancemodelwith theasymptotic
one, which does not include the effect of microstructure size will be shown.
The governing equations of the asymptotic model are expressed by

∇·
[(

〈K〉−
〈K∂h〉2
〈K∂h∂h〉

)

∇θ
]

−〈c〉θ̇ =0 (3.7)

and the temperature fluctuation amplitude is given by the equation

ψ =−
〈K∂h〉
〈K∂h∂h〉

∇θ (3.8)

The above formulas can be found in Woźniak et al. (2008). Let us consider
the two-phase hollow cylinder (Fig.1) for R0 = 1m, Rk = 3m and material
properties as inTable 1. The number of cells is fixed at N =60 and thewidth
of the walls g = 0.5λR0. Calculations were made for a = 1 and b = 0.25 in
(3.1). Initial-boundary conditions are given by (3.2) and (3.3). Let us notice
that for the asymptotic model there is no need to impose conditions on the
temperaturefluctuationamplitude ψ.Theobtained results forbothmodelsare
covered. However, since for the asymptotic model function of ψ is expressed
by (3.8), the differences between two models occur but only nearby inner
boundary.



392 P. Ostrowski, B. Michalak

Fig. 5. Diagram of averaged temperature after t =1h for tolerance (TM) and
asymptotic (AM) model

Fig. 6. Diagram of on temperature fluctuation after t =1h for tolerance (TM) and
asymptotic (AM) model

3.2. Effect of the walls width on the temperature field

Let us consider a composite with geometry as in Fig.1 with R0 = 1m,
Rk = 3m. Initial-boundary conditions as in (3.2) and (3.3), and material
properties are as inTable 1 for a =1 and b =0 in (3.1).We denote thewidth
of the wall by

g(η) =
2πR0
N

η (3.9)

where N stands for the number of cells. In this case N =60. Diagrams of the
ratio of the averaged temperature value in a selected time t to the averaged
temperature value for a steady state problem are shown in Fig.7. Similar
diagrams for the temperature fluctuation amplitude are shown in Fig.8. The
walls width ratio η is taken as a parameter. We consider only two values of
the parameter η, i.e. η =0.25 and η =0.75.



Non-stationary heat transfer in a hollow cylinder... 393

Fig. 7. Diagram of change in time of the averaged temperature; η is taken as a
parameter

Fig. 8. Diagram of change in time of the temperature fluctuation amplitude; η is
taken as a parameter

3.3. Effect of material properties on the temperature field

Let us consider a composite with geometry as in Fig.1 with R0 = 1m,
Rk =3m. Initial-boundaryconditionsas in (3.2) and(3.3).Thenumberof cells
N =60. In this example, we consider two different values of the parameter a
in (3.1), i.e. a =0.75 and a =1, by a fixed value of b =0.5. Diagrams of the
ratio of the averaged temperature value in a selected time t to the averaged
temperature value for a steady state problem are shown in Fig.9. Similar
diagrams for the temperature fluctuation amplitude are shown in Fig.10. It
can be observed in Figs.9 and 10 that for materials with stronger anisotropic
conductivity the temperature fields achieve the steady state slower.



394 P. Ostrowski, B. Michalak

Fig. 9. Diagram of change in time of the averaged temperature; a is taken as a
conductivity parameter

Fig. 10. Diagram of change in time of the temperature fluctuation amplitude; a is
taken as a conductivity parameter

3.4. Effect of inner radius size on the temperature field

Let us consider a composite with geometry as in Fig.1 with a constant
width of the hollow cylinder Rk−R0 =1m. Initial-boundary conditions as in
(3.2) and (3.3), andmaterial properties are as in Table 1 for a =1 and b =0
in (3.1). We demand also, by various radius R0, constant effective material
properties on the inner boundary. That is why the number of cells N must be
a function of R0

N(R0)=
π

g
R0 (3.10)

where we assumed g = π/20m. We consider both cases where R0 = 5m
and R0 = 10m. From the above figures it can be observed that the inner



Non-stationary heat transfer in a hollow cylinder... 395

radius does not influence the rate of achieving the steady state for averaged
temperature.

Fig. 11. Diagram of change in time of the averaged temperature; inner radius is
taken as a parameter

Fig. 12. Diagram of change in time of the temperature fluctuation amplitude; inner
radius is taken as a parameter

4. Conclusions

The tolerance averaging approximation leads to the mathematical model of
composite conductors with functionally graded material properties. The ob-
tained model equations have continuous coefficients in opposition to discrete
models, where they are strongly oscillating. Since the proposed model equ-
ations have smooth functional coefficients then, inmost cases, solutions to the



396 P. Ostrowski, B. Michalak

specific problem, for the heat conductor under consideration, have to be ob-
tained using well known numerical methods. The tolerance model takes into
account the effect of microstructure size on the temperature field, particular-
ly on the temperature oscillation amplitude. Moreover, by changing volume
fractions or material properties of every component, we can obtain the desi-
rable temperature field inside composite. For different geometry andmaterial
properties, the temperature fields for the conductor under consideration have
a slow relative velocity of achieving the steady state:

• Formaterialswith anisotropic conductivity, the temperaturefields achie-
ve steady state slower than for materials with isotropic conductivity

• The inner radius does not influence the rate of achieving the steady
state for averaged temperature but has a low influence on the rate for
temperature fluctuation amplitude.

Acknowledgement

This contribution has been supported by the Ministry of Science and Higher

Education under grant No. NN506398535.

References

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Niestacjonarny przepływ ciepła w wydrążonym cylindrze wykonanym

z materiału o funkcyjnej gradacji własności

Streszczenie

Wpracy rozważanoniestacjonarneprzewodzenia ciepławdwu-składnikowymwy-
drążonym cylindrze. Przewodnik jest wykonany z dwuskładnikowego warstwowego
kompozytu mającego łagodną zmienność efektywnych własności w kierunku promie-
niowym. Stądmamy tutaj do czynienia ze specjalnymprzypadkiemmateriału o funk-
cyjnej gradacji własności (ang. functionally graded materiale, FGM). Zbudowanie
uśrednionego modelu matematycznego rozpatrywanego przewodnika jest oparte na
technice tolerancyjnej aproksymacji. W pracy pokazano zastosowanie otrzymanego
modelu tolerancyjnego i porównanie wyników z wynikami dla modelu asymptotycz-
negow przypadku niestacjonarnego przewodzenia ciepła. Zbadanowpływ zmienności
geometrii i własności materiałowych przewodnika na pole tempertatury.

Manuscript received June 10, 2010; accepted for print October 28, 2010