f xð Þ ¼;t XN
n¼0
CnPðn0;bÞ2edt1
ð2:10Þ
wheredis a real positive number andb[ 1. Furthermore,Pðna;bÞ represents the Jacobi polynomial of degreen, defined as:
Pðna;bÞð Þ ¼z ð1zÞað1þzÞbð1Þn 2nn!
dn
dznhð1zÞnþað1þzÞnþbi
ð2:11aÞ or:
Pðna;bÞð Þ ¼x Cðaþnþ1Þ n!Cðaþbþnþ1Þ
Xn
k¼0
n k
Cðaþbþnþkþ1Þ Cðaþkþ1Þ
x1 2
k
ð2:11bÞ
The unknown coefficients Cn in Eq. (2.10) are obtained by the following recurrence relation:
df xð ;ðbþ1þkÞdÞ ¼Xk
m¼0
k kð 1Þ ðkðm1ÞÞ kþbþ1
ð Þðkþbþ2Þ ðkþbþ1þmÞCm ð2:12Þ For an accurate approximation of Laplace inversion, it is recommended to choose theband das:
0:5b5 ð2:13aÞ
0:05d2 ð2:13bÞ
1þkq @
@xþsq @
@tþs2q
2
@2
@t2
!
q xð Þ ¼ k;t T 1þsT @
@t
@T xð Þ;t
@x ð2:14Þ
wherekq is the correlating length parameter for nonlocal analysis. The correlating length is equivalent, by two times, to the length parameters in the thermomass model of heat transfer in dielectric lattices. Specifically,kqandsqcould be correlated to the mean free time and the mean free path in microscale heat transport [1]. Eliminating the heatfluxq xð Þ;t between Eq. (2.14) and 1D form of the energy Eq. (1.6) leads to:
1þsT @
@t
@2T xð Þ;t
@x2 ¼ 1þkq @
@xþsq @
@tþs2q
2
@2
@t2
! 1 a
@T xð Þ;t
@t ð2:15Þ
The following non-dimensional parameters are also introduced for analysis:
h¼ TT0 TwT0;b¼ t
sq;n¼ x ffiffiffiffiffiffiffi asq
p ;Z¼sT
sq;L¼ kq
ffiffiffiffiffiffiffi asq
p ð2:16Þ
The heat conduction Eq. (2.15) and the initial and boundary conditions could be written in terms of the non-dimensional parameters. To identify different heat conduction models, two artificial coefficients A and B are included in the non-dimensional heat conduction equation as follows:
@2h n;ð bÞ
@n2 þZ@3h n;ð bÞ
@n2@b ¼@h n;ð bÞ
@b þL@2h n;ð bÞ
@n@b þA@2h n;ð bÞ
@b2 þB@3h n;ð bÞ
@b3 ð2:17aÞ h n;ð 0Þ ¼@h n;ð 0Þ
@b ¼0 ðInitial conditionsÞ ð2:17bÞ hð0;bÞ ¼1; lim
n!1h n;ð bÞ ¼0 ðBoundary conditionsÞ ð2:17cÞ As seen in Eq. (2.17), the thermal responses are characterized byZ;L;A;andB parameters. For the case ofL¼0,A¼1,B¼12andZ 6¼0, Eq. (2.17) reduces to Fig. 2.1 A semi-infinite,
homogeneous 1D medium with a suddenly raised surface temperatureTW
2.3 Non-Fourier Heat Conduction in a Semi-infinite Strip 27
hyperbolic-type DPL model, while L¼0, A¼1, B¼0, and Z6¼0 lead to parabolic-type DPL model. Furthermore, forL¼0,A¼1, andB¼0, Eq. (2.17) reduces to C-V and classical Fourier models when Z is set to 0 ðsT¼0Þand 1
sq¼sT
, respectively. The nonlocal C-V (NL C-V) model could also be derived from Eq. (2.17), by settingL6¼0,A¼1, andB¼Z¼0:
Due to the time dependency of transient thermal responses in Eq. (2.17), solu- tion for temperature is found in the Laplace transform as:
~h n;ð sÞ ¼1
sexp Ls ffiffiffiffiffiffiffiffiffiffiffi ð ÞLs2 q
þ4sð1þAsþBs2Þð1þZsÞ
2 1ð þZsÞ n
2 4
3
5 ð2:18Þ
shown by Tzou [1–3], using the partial expansion technique and the limiting the- orem in Laplace transform, Eq. (2.18) presents the thermal wave behavior for nonlocal C-V CNL CT -V
model with the following thermal wave speed:
CTNL C-V ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a
sq þ kq
2sq
2
s
þ kq
2sq¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi CCT-V
2
þ kq
2sq
2
s
þ kq
2sq ð2:19Þ which reveals that the CTNL C-V[CCT-V. Reimann sum approximation is then employed to numerically transform temperature in Laplace domain, given in Eq. (2.18), to time domain.
A code in MATLAB could be developed to numerically conduct the Laplace inversion of Eq. (2.18). Reimann sum approximation has been used here for the Laplace inversion. Figure2.2compares temperature distribution at non-dimensional timeb¼1 for classical Fourier, C-V, diffusive-like DPL, wave-like DPL, and NL C-V heat conduction models. As shown in Fig.2.2, C-V, wave-like DPL, and NL C-V models result in a wave-like behavior for temperature and reveals a sharp wavefront in thermal wave which divides the thermal response domain into the heat affected and unaffected zones. In accordance with Eq. (2.19), NL C-V model predicts a higher thermal wave speed compared to the C-V model. Furthermore, as derived in Eq. (1.22 ), thermal wave speed of the wave-like DPL model is related to the C-V model as:
CTDPL¼CTC-V
ffiffiffiffiffiffi p2Z
. As a result:
CTDPL\CTC-V for Z\1
2 ð2:20aÞ
CDPLT CTC-V for Z1
2 ð2:20bÞ
For the assumedZ ¼10 in Fig.2.2, the wave front of the wave-like DPL model is ahead of the C-V and NL C-V models. As observed in Figs.2.2 and 2.3, the mixed derivative termZ@3h n;bð Þ
@n2@b in Eq. (2.17a) removes the singularity at the thermal wavefront, compared to the wave-like DPL model. As Z increases in the diffusive-like DPL model the thermal wavefront is completely destroyed and
temperature responses show a diffusive behavior same as classical Fourier heat conduction, Fig.2.3a. Although both of the diffusive-like DPL and classical Fourier heat conduction models do not show a finite thermal wave speed, the thermally affected zones are not the same for these two models. While for Z[1, the diffusive-like DPL model reveals a wider affected zone compared to the Fourier heat conduction; the affected zone is narrower for Z\1. For diffusive-like DPL model, thermal wave with discontinue temperature distribution around the wave- front is detected in Fig.2.3b. Thermal wave speed increases by increasing Z. As opposed to diffusive-like DPL, decreasing Zcould result in a diverged and noisy temperature distribution for very low values ofðZ0Þ.
The effect of non-dimensional correlation length L on temperature distribution is illustrated in Fig.2.4, at non-dimensional time b¼1 using the NL C-V heat conduction model. As seen in thisfigure, the NL C-V model reduces to the C-V model for = 0kq¼0
. AsLincreases, the wavefront of the NL C-V thermal wave advances and temperature in the heat affected zone is raised. The NL model for L\2 has been shown by Tzou to be identical to the thermomass heat transfer model in phonon gas.
Fig. 2.2 Non-dimensional temperature distribution predicted by Fourier, C-V, diffusive-like DPL, hyperbolic-type DPL, and NL C-V heat conduction models at non-dimensional timeb¼1 2.3 Non-Fourier Heat Conduction in a Semi-infinite Strip 29
Fig. 2.3 Effect of temperature and heatflux phase lag ratioð ÞZ on non-dimensional temperature distribution predicted bya diffusive-like DPL and b wave-like DPL at non-dimensional time b¼1
Fig. 2.4 Effect of non-dimensional correlation length ð ÞL on non-dimensional temperature distribution at non-dimensional timeb¼1 using NL C-V heat conduction model. [Reproduced from [1] with permission from Elsevier Masson SAS]