2.6 Dual-Phase-Lag Heat Conduction in Multi-dimensional Media
2.6.2 DPL Heat Conduction in Multi-dimensional Spherical
To obtain temperature field in heterogeneous spherical vessels, we should write DPL heat conduction Eq. (2.51) in the spherical coordinate systemðr;u;wÞ pre- sented in Table1.1. The thermal boundary conditions for the spherical vessel can be assumed to be: (1) Spherically symmetric one-dimensional or (2) Axisymmetric two-dimensional. As a result, the heat conduction equation is simplified in the form of the following partial differential equations in the spherical coordinate system:
1þsq @
@tþs2q
2
@2
@t2
! qcp@T
@t
¼ 1 r2
@
@r Kr2 1þsT @
@t
@T
@r
þ 1 rsinu
@
@u sinu
r K 1þsT @
@t
@T
@u
ð2:69Þ
In this section, we focus on the heat conduction in spherically axisymmetric two-dimensional problem. The material properties of the spherical vessel is assumed to vary radially according to the power law formulation, similar to those introduced in Sect.2.6.1. Except for phase lags, which are assumed constant, all other thermal properties varies according to Eq. (2.53). To simplify the solution procedure, we employ the non-dimensional parameters of Eq. (2.54).
Using the assumed material properties for the FG spherical vessel, we can rewrite Eq. (2.69) in the following non-dimensional form:
gn2þn3 1þe0 @
@fþe20
2
@2
@f2
@h
@f
¼ d0
@
@fþ1
gn1@2h
@g2 þðn1þ2Þgn11@h
@g þgn12 1 sinu
@
@u sinu@h
@u
ð2:70Þ The axisymmetric thermal boundary and initial conditions are assumed for the FG spherical vessel to enable us obtaining a semi-analytical solution for the 2D heat conduction problem in the spherical coordinate system [36]:
h g;ð u;fÞjg¼rc¼Tccosu;h g;ð u;fÞjg¼1¼cosu ð2:71aÞ h g;ð u;fÞjf¼0¼0; @
@fh g;ð u;fÞjf¼0¼0 ð2:71bÞ
Considering the above-mentioned thermal boundary conditions, temperature can be written as:
h g;ð u;fÞ ¼h1ðg;fÞcosu ð2:72Þ whereh1ðg;fÞis an unknown temperature that is needed to be determined using the initial and boundary thermal conditions. Substituting Eq. (2.72) into Eq. (2.70) and performing the Laplace transform with regard to the initial conditions (2.71b), lead to:
g2@2~h1
@g2 þðn1þ2Þg@~h1
@g Egn2þn3n1þ2þ2~h1¼0 ð2:73Þ
whereEis defined as:E¼s
e2 0 2s2þe0sþ1
d0sþ1 . The differential Eq. (2.73) can be solved as:
~h g;ð sÞ ¼gn12þ1 A1JGIgH
þA2YGIgH
for n1n2n36¼2
ð Þ
~h g;ð sÞ ¼A1gk1þA2gk2 ðfor n1n2n3¼2Þ ð2:74Þ where
G¼ 2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
n1þ1 2
2
þ2 q
n2þn3n1þ2 ;k1;2 ¼ðn1þ1Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n1þ1
ð Þ2þ4ðEþ2Þ q
2 H¼1þn2þn3n1
2 ; I¼ 2 ffiffiffiffiffiffiffi pE n2þn3n1þ2
ð2:75Þ
The integration constantsA1andA2can be also obtained in the Laplace domain as:
Forn1n2n36¼2:
A1¼ YG IrcH
þrnc12þ1TcYGð ÞI s YG IrcH
JGð Þ þI JG IrcH YGð ÞI A2¼ JG IrcH
rnc12þ1TcJGð ÞI s YG IrcH
JGð Þ þI JG IrcH YGð ÞI
ð2:76aÞ
Forn1n2n3¼2:
2.6 Dual-Phase-Lag Heat Conduction in Multi-dimensional Media 59
A1¼ Tcrck2 s rkc1rkc2 A2¼ rkc1Tc
s rkc1rkc2
ð2:76bÞ
Non-dimensional radial and polar heat fluxes in the Laplace domain are also obtained as:
Forn1n2n36¼2:
Q~rðg;u;sÞ ¼P
2gn123 A1 MJGIgH
2IHgHJGþ1IgH
þA2 MYGIgH
2IHgHYGþ1IgH
cosu Q~uðg;u;sÞ ¼ Pgn123 A1JGIgH
þA2YGIgH
sinu
ð2:77aÞ
and forn1n2n3¼2:
Q~rg;/;s
¼P A 1k1gn1þk11þA2k2gn1þk21 cosu Q~/g;/;s
¼ P A 1gn1þk11þA2gn1þk21
sinu ð2:77bÞ
whereMand Pare defined as:
M¼2GHðn1þ1Þ;P¼ TwoT1 T1
d0sþ1
e20
2s3þe0s2þs
!
ð2:78Þ
The temperature in time domain can then be obtained by implementing a numerical Laplace inversion technique. It is worth mentioning that while thermal wave can propagate in multiple directions in 2D or 3D spherical vessels, the radial thermal wave speed in all 1D, 2D, and 3D structures are the same.
To clarify the effects of each non-homogeneity indices on the thermal responses of an axisymmetric hollow sphere (2D) based on the DPL heat conduction theory, among the three different non-homogeneity indicesn1(thermal conductivity index), n2 (density index), andn3 (specific heat index), two of them are kept constant and only one varies. Figure2.21a–c show the temperature history of the mid-plane in three different cases: (a) n2¼n3¼1; (b) n1¼n3 ¼1; and (c) n1¼n2¼1.
Figure2.21a reveals that increasing the non-homogeneity index of thermal con- ductivity n1 leads to higher transient and steady-state temperature. Although increasing the non-homogeneity indices of densityn2and specific heatn3increases the amplitudes of transient temperature, the steady-state temperature does not change by tailoringn2 and n3 as shown in Fig. 2.21b, c.
Fig. 2.21 Effect of non-homogeneity indices on temperature time-history of the mid-plane of the axisymmetric (2D) hollow sphere: a Effect of thermal conductivity,b Effect of density, and cEffect of specific heat. [Reproduced from [36] with permission from World Scientific Publishing Co., Inc]
2.6 Dual-Phase-Lag Heat Conduction in Multi-dimensional Media 61
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