Vladimir V. Kulish
Mem. ASME
e-mail: mvvkulish@ntu.edu.sg
School of Mechanical & Production
Engineering,
Nanyang Technological University,
50 Nanyang Ave., Singapore 639798
Vasily B. Novozhilov
Assoc. Mem. ASME
Faculty of Engineering,
University of Ulster,
Shore Rd., Newtownabbey,
Co. Antrim, BT37 0QB, United Kingdom
An Integral Equation for the
Dual-Lag Model of Heat Transfer
An integral relation is obtained between local temperature and local temperature gradient
for the dual-lag model of heat transport, which substitutes classical Fourier law at short
time scales. Both the heat flux lag and the temperature gradient lag are considered,
however, it is shown that only difference between the two affects temperature profile.
Being applied at exposed surface of material, the integral equation predicts surface temperature variation for any form of imposed heat flux. The solution is tested considering
solid heating by a picosecond laser impulse. Results are compared with the classical
solution of the parabolic heat transfer equation and available experimental data.
关DOI: 10.1115/1.1797034兴
Keywords: Analytical, Heat Transfer, Laser, Thin Films
Introduction
Heat transport at small scales receives increasing attention due
to application in modern electronics 关1兴, 关2兴, and 关3兴. Many commonly used devices, such as personal computers or cellular
phones, operate already on nanosecond time scale, and the spatial
scales for the energy transport are of the order of a single atom.
It has become established that the heat transport equation is
different from the classical one at such time scales 关4兴. In particular, one has to modify the Fourier law, which relates heat flux and
the temperature gradient. According to Fourier law, heat flux adjusts immediately to the imposed temperature gradient, i.e., there
is no relaxation time for the heat flux. At those scales where the
Fourier hypothesis fails, one has to take into account the lag between the heat flux and the temperature gradient. In the most
consistent way, such model for micro-scale heat transport has
been proposed by Tzou 关5兴, who introduced the two lags, namely
the heat flux time lag and the temperature gradient time lag. Such
a model represents a new type of constitutive relation between the
heat flux and the temperature gradient, which supercedes the Fourier law at small scales. Being combined with the energy conservation law, such relationship leads to the energy transport equation, which is different from the classical parabolic one.
Numerical solution of the modified heat transport equation has
been attempted in 关5兴, using phase lags as small parameters and
obtaining Taylor expansion of the constitutive relationship.
The present study generalizes analysis of the one-dimensional
homogeneous energy transport equation with the dual phase lag.
The integral form of the solution is obtained for the most general
form of the constitutive relationship. It is shown that the general
formulation, which does not involve approximate Taylor series, in
fact leads to quite simple heat transfer equation and integral relationship for the solution.
Analysis is restricted to a one-dimensional semi-infinite domain
with constant material thermal properties. Note that materials behave as thermally thick in most microheating applications. Further, in many typical cases 共such as the one considered below兲 the
temperature rise is very small; therefore the constant material
properties assumption is an excellent approximation.
The energy transport equation is reduced to the form, closely
reminiscent of the classical heat transfer equation, but with the
time lag. Integral equation, which relates local values of the temperature and its gradient at any point inside the domain, is
Contributed by the Heat Transfer Division for publication in the JOURNAL OF
HEAT TRANSFER. Manuscript received by the Heat Transfer Division October 22,
2002; revision received June 3, 2004. Associate Editor: G. S. Dulikravich.
Journal of Heat Transfer
obtained. This equation is then applied for prediction of the surface temperature under heating conditions, and the results are
compared with the available experimental data.
Mathematical Model and Preliminary Analysis
Consider an infinite solid, ⫺⬁⬍x⬍⫹⬁. According to the
dual-phase-lag model proposed by Tzou 关5兴, for any ⫺⬁⬍x
⬍⫹⬁, ⫺⬁⬍t⬍⫹⬁ there is a lag between the local temperature
gradient, ⳵ T/ ⳵ x, and the local heat flux, q ⬙ ,
q ⬙ 共 x,t⫹ ␶ q 兲 ⫽⫺k•
⳵T
共 x,t⫹ ␶ T 兲 ;
⳵x
⫺⬁⬍x⬍⫹⬁, ⫺⬁⬍t⬍⫹⬁
(1)
Here ␶ q is the phase lag of the heat flux, ␶ T is the temperature
gradient phase lag, and k is the thermal conductivity of the material. This is in contrast to Fourier’s law, where heat flux adjusts
immediately to the imposed temperature gradient
q ⬙ 共 x,t 兲 ⫽⫺k•
⳵T
共 x,t 兲 ;
⳵x
⫺⬁⬍x⬍⫹⬁, ⫺⬁⬍t⬍⫹⬁ (2)
Constitutive relation 共1兲 results in the new form of energy conservation equation, which in general form is written as
⳵T
1 ⳵ q ⬙ 共 x,t 兲
;
•
共 x,t 兲 ⫽⫺
⳵t
␳cp
⳵x
⫺⬁⬍x⬍⫹⬁, ⫺⬁⬍t⬍⫹⬁
(3)
The nature of the modified equation can be established if one
expands the left and right hand sides of Eq. 共1兲 to get
q ⬙⫹ ␶ q
冋
册
⳵q⬙
⳵T
⳵ 2T
⫹o 共 ␶ q 兲 ⫽⫺k
⫹␶T
⫹o 共 ␶ T 兲 ;
⳵t
⳵x
⳵x⳵t
⫺⬁⬍x⬍⫹⬁, ⫺⬁⬍t⬍⫹⬁
(4)
Upon substitution into the conservation law 共3兲, the following
heat transport equation is obtained 关2兴
␶ q ⳵ 2T 1 ⳵ T ⳵ 2T
⳵ 3T
⫹o 共 ␶ q 兲 ⫹o 共 ␶ T 兲 ;
⫽
⫹
⫹
␶
T
␣ ⳵t2 ␣ ⳵t ⳵x2
⳵ x 2⳵ t
⫺⬁⬍x⬍⫹⬁, ⫺⬁⬍t⬍⫹⬁
(5)
for the temperature field, T(x,t), in the domain.
It is clear from Eq. 共5兲 that the lagging behavior should be
taken into account for the processes whose characteristic time
scales are comparable to ␶ q , ␶ T .
Copyright © 2004 by ASME
OCTOBER 2004, Vol. 126 Õ 805
Different approaches may be taken to solve the heat transport
equation with the lags numerically. Tzou 关5兴, for instance, considers the truncated Eq. 共5兲 and then obtains a solution of this equation in the Laplace space. The solution thus obtained cannot be
inverted. Therefore, approximate methods, such as partial expansions and the Riemann–Sum approximation, have to be applied.
Although the numerical results agree quite satisfactory with most
of the experimental results, the exact solution is believed hard—if
at all possible—to obtain.
In fact, truncation of the constitutive relation 共1兲 is not necessary. Even involving conventional finite volume methods, one can
easily solve the conservation equation 共3兲 with the general constitutive relation 共1兲.
Moreover, in the next section we demonstrate that a simple
integral equation can be obtained involving the constitutive relation 共1兲. Therefore, the problem is considered here in the most
general form, involving the both lags and untruncated constitutive
relation.
Note that Eq. 共1兲 can be written in equivalent form as
q ⬙ 共 x,t 兲 ⫽⫺k•
It is obvious now that the temperature and its gradient are related locally. Taking the derivative of Eq. 共12兲
d⌰ 共 x,s 兲
⫽⫺C 1 共 s 兲 •
dx
⫺⬁⬍x⬍⫹⬁, ⫺⬁⬍t⬍⫹⬁
(13)
and comparing to Eq. 共12兲 itself,
⌰ 共 x,s 兲 ⫽⫺
L ⫺1
⫺
⳵T
⳵ 2T
共 x,t 兲 ⫽ ␣ 2 共 x,t⫺ 共 ␶ q ⫺ ␶ T 兲兲 ;
⳵t
⳵x
冉冑 冉
⌬␶
␣
•exp ⫺
•s
s
2
冑 冕
␣
•
␲
t
0
冉
u t *⫺
冑
T 共 x,t 兲 ⫽T 0 ⫺
T 共 x,t 兲 ⫽T 0 ⫹
Consider now the problem for semi-infinite solid
t苸 关 ⌬ ␶ ;⬁ 关 ;x苸 关 0;⬁ 关
(8)
This domain is initially in thermal equilibrium. Therefore, the
initial and boundary conditions are written as
␽ 共 ⬁,t 兲 ⫽0 (9)
The condition at x⫽0 depends on particular specified rate of
solid heating given by the flux q ⬙ (0,t), and is a direct consequence of constitutive relation 共6兲.
Here the excess temperature is introduced as ␽ ⫽T⫺T 0 , and
⌬ ␶ ⫽ ␶ q ⫺ ␶ T is the difference between the lags.
Differential equations with lag are readily handled by Laplace
transform.
Taking transform of Eq. 共8兲 with respect to t, L( ␽ (x,t))
⫽⌰(x,s), and involving initial condition, one gets the ODE
dx
s
•exp共 ⌬ ␶ •s 兲 •⌰⫽0
␣
(10)
with the general solution
⌰ 共 x,s 兲 ⫽C 1 共 s 兲 e ⫺ 冑s/ ␣ exp共 ⌬ ␶ •s 兲 •x ⫹C 2 共 s 兲 e 冑s/ ␣ exp共 ⌬ ␶ •s 兲 •x
The latter must be bounded at x→⬁, therefore, C 2 (s)⬅0, and
the solution is written as
806 Õ Vol. 126, OCTOBER 2004
冊
•
冉 冊
冊冊 冑 冑
␣
•
␲
⫽
u t⫺
⌬␶
2
t⫺
⌬␶
2
⳵␽
共 x,t⫺t * 兲 dt * ⫽ ␽ 共 x,t 兲
⳵x
冑 冕
␣
•
␲
(12)
⳵T
共 x, ␨ 兲
⳵x
t⫺⌬ ␶ /2
冑*
0
t ⫺
1
冑␲ k ␳ c p
•
冕
t⫺⌬ ␶ /2
⌬␶
⫺␨
2
•d ␨
q ⬙ 共 x, ␨ ⫹⌬ ␶ 兲
冑
0
(15)
(16)
(17)
⌬␶
t *⫺
⫺␨
2
•d ␨
(18)
Being applied at the surface (x⫽0), Eq. 共18兲 provides the
means to predict behavior of the surface temperature of the material, T s (t)⫽T(0,t), based on the imposed flux q s⬙ (t)⫽q ⬙ (t,0),
T s 共 t 兲 ⫽T 0 ⫹
1
冑␲ k ␳ c p
•
冕
t⫺⌬ ␶ /2
0
q s⬙ 共 ␨ ⫹⌬ ␶ 兲
冑
t *⫺
⌬␶
⫺␨
2
•d ␨
(19)
Surface temperature is the most important parameter in microscale heat transfer applications.
For the case ␶ q ⫽ ␶ T ⫽0, the above equation reduces to integral
relation obtained for the classical heat transfer equation by Kulish
and Lage 关7兴.
Model Validation
A test solution is obtained by integrating Eq. 共19兲, in order to
predict surface temperature of the material, subjected to prescribed heat flux. The heat flux applied at the boundary imitates
laser impulse according to Gaussian distribution in time, namely
冋 冉 冊册
⬙ exp ⫺
q s⬙ 共 t 兲 ⫽q s,max
(11)
⌰ 共 x,s 兲 ⫽C 1 共 s 兲 e ⫺ 冑s/ ␣ exp共 ⌬ ␶ •s 兲 •x
(14)
Integral equation 共17兲 relates values of temperature and its gradient at the same point, and holds everywhere inside the domain.
In terms of the heat flux, Eq. 共17兲 can be written as
Integral Equation
⫺
2
⌬␶
2
⌬␶
t *⫺
2
(7)
Equation 共7兲 has the same form as the classical heat transfer
equation, but with the lag ␶ q ⫺ ␶ T in the diffusion term.
An immediate and important conclusion from Eq. 共7兲 is that
solution in the dual-lag model does not depend on ␶ q and ␶ T
separately, but on the difference ␶ q ⫺ ␶ T only.
d 2⌰
冊
⌬␶
d⌰ 共 x,s 兲
␣
•exp ⫺
•s •
s
2
dx
or, after simple re-arrangement, t⫺t * ⫽ ␨ ,
⫺⬁⬍x⬍⫹⬁, ⫺⬁⬍t⬍⫹⬁
1
⳵␽
共 0,t⫺⌬ ␶ 兲 ⫽⫺ q ⬙ 共 0,t 兲 ,
⳵x
k
冉
冑
The inverse Laplace transform of Eq. 共14兲 can be found involving the table transform 关6兴
(6)
Substitution of 共6兲 into the conservation equation 共3兲 yields
␽ 共 x,0兲 ⫽0,
s
exp共 ⌬ ␶ •s 兲 •e ⫺ 冑s/ ␣ exp共 ⌬ ␶ •s 兲 •x
␣
where u(t) is the unit step-function, and applying the convolution
theorem 关6兴. This results in the following integral equation:
⳵T
共 x,t⫺ 共 ␶ q ⫺ ␶ T 兲兲 ;
⳵x
⳵␽
⳵ 2␽
共 x,t 兲 ⫽ ␣ 2 共 x,t⫺⌬ ␶ 兲 ;
⳵t
⳵x
冑
t⫺b
␴
2
;
t⭓0
(20)
with b⫽10 ps and ␴⫽5.0 ps 共Fig. 1兲. Characteristic time of the
considered heating process is, therefore, of the order of picoseconds. In the course of computations, the maximum value of the
⬙ , was 103 W/m2 .
surface heat flux, q s,max
Transactions of the ASME
Fig. 1 Laser impulse for the test case
Material properties are taken as those of gold 共Au兲, i.e., k
⫽315 W/共m K), ␣ ⫽1.2495⫻10⫺4 m2 /s, ␶ q ⫽0.7438 ps and ␶ T
⫽89.286 ps 关5兴. The time lags cited in 关5兴 have been determined
experimentally.
For a given heat flux history, integration of Eq. 共19兲 is straightforward, and is carried out by explicit time advancement using
trapezoidal rule for integration. In order to handle singularity, analytical integration is performed in the vicinity of the upper limit.
The time step is chosen in such a way that the relative error of the
solution does not exceed ␧⫽10⫺6 . Upon achieving this criterion,
the solution also becomes independent of further reduction in time
step.
The evolution of the normalized surface temperature, ␪ ⫽(T s
⫺T 0 )/(T m ⫺T 0 , is shown in Fig. 2. Also shown in this plot is the
profile for the classical heat transfer equation ( ␶ q ⫽ ␶ T ⫽0).
It is obvious from Fig. 2 that the classical solution deviates
significantly from solution of Eq. 共19兲 at time scales comparable
with ␶ q , ␶ T , and, therefore, cannot be used at such scales. In
particular, the maximum surface temperature is predicted at later
time by the classical equation, and the shape of the temperature
profile is quite different from the solution of Eq. 共19兲.
An interesting mathematical exception is the case ␶ q ⫽ ␶ T ,
where the solution of Eq. 共19兲 coincides with that of the classical
equation. However, such a case is unlikely for real materials, since
the lags seem to differ significantly 共see above example of gold兲.
Note also that for real materials ␶ T ⬎ ␶ q so that the upper limit
in the integral 共Eqs. 共17兲–共19兲兲 is always positive.
The lags ␶ q , ␶ T are material properties, which are known, however, with limited accuracy. It is worthwhile, therefore, to investigate sensitivity of the solution to these parameters. Plotted in
Fig. 3 is a family of solutions that are obtained for different values
of ⌬␶, which deviate ⫾10% from the base values given in 关7兴. The
sensitivity is small, compared with difference between the classical and dual-lag solutions.
From Fig. 3, one can see that the less value of the lag is, the
closer the solution is to the classical model 共no lag兲.
Fig. 2 Normalized surface temperature obtained for a bulk
sample of gold „Au… in the case of the dual „B… phase lag, and
classical energy equation „D…
Journal of Heat Transfer
Fig. 3 Time evolution of the surface temperature for different
values of the phase lags: „B… ⌬␶ÄÀ88.5422 ps „ ␶ q Ä0.7438 ps,
␶ T Ä89.286 ps…—base case; „F… ⌬␶ÄÀ97.3964 ps; „D… ⌬␶
ÄÀ79.6880 ps
Fig. 4 Comparison between the numerical solution and experimental data
Finally, Fig. 4 shows the numerical solution of Eq. 共19兲 compared with the available experimental data by Brorson et al. 关8兴
and Qiu et al. 关9兴. Direct comparison of the maximum temperature with the measurements was not possible, since experimental
data 关8兴 and 关9兴 are presented in normalized way. The maximum
excess temperature obtained in the present calculations was
⌬T max⬇0.06 K.
Conclusions
The heat transport equation with the dual-phase lag constitutive
relation is considered. In contrast to previous studies, general formulation of the constitutive relationship is retained in the analysis.
It is shown that such a general formulation results, in fact, in
much simpler solution of the problem. Heat transfer equation with
the dual lag is reduced to the form, similar to the classical parabolic equation. The solution of such equation depends only on the
difference between the two lags. Therefore, the lagging behavior
of the particular material is, in fact, described by the single parameter.
The method of Laplace transform has been used to obtain relationship between the local temperature and the local heat flux for
the considered model. Such a relationship is written in the form of
a simple integral equation.
Being applied at the surface, the integral equation provides a
convenient way to predict maximum 共surface兲 temperature during
heating. The integral equation has been tested for the case of
material heating by a picosecond laser impulse. The result has
been compared with the solution of the classical 共parabolic兲 heat
transfer equation and with the experimental data available in literature.
In case of zero lags 共or for time scales much longer than time
lags兲, the proposed integral equation reduces to the previously
established integral relation 关3兴 and 关7兴, applicable to problems,
described by classical Fourier law.
OCTOBER 2004, Vol. 126 Õ 807
Nomenclature
b
cp
k
L
q⬙
q s⬙
s
t
t*
T
T0
Ts
Tm
u(t)
x
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
⫽
mean of normal distribution, s
specific heat of solid material
thermal conductivity, W m⫺1 K⫺1
Laplace transform operator
local heat flux, W m⫺2
surface heat flux 共at x⫽0), W m⫺2
Laplace transform variable
time, s
dummy time variable, s
absolute temperature, K
initial temperature 共at t⭐0), K
surface temperature 共at x⫽0), K
maximum temperature during heating, K
unit step-function
co-ordinate normal to material surface, m
Greek Symbols
thermal diffusivity, m2 s⫺1
excess temperature, ⫽T⫺T 0 , K
Laplace transform of ␪
normalized non-dimensional temperature,
⫽(T s ⫺T 0 )/(T m ⫺T 0 )
␴ ⫽ variance of normal distribution, s
␳ ⫽ density, kg m⫺3
␶ q ⫽ phase lag of heat flux, s
␣
␽
⌰
␪
⫽
⫽
⫽
⫽
808 Õ Vol. 126, OCTOBER 2004
␶ T ⫽ temperature gradient phase lag, s
⌬␶ ⫽ difference between lags, ⫽ ␶ q ⫺ ␶ T , s
References
关1兴 Brorson, S. D., Kazeroonian, A., Moodera, J. S., Face, D. W., Cheng, T. K.,
Ippen, E. P., Dresselhaus, M. S., and Dresselhaus, G., 1990, ‘‘Femtosecond
Room Temperature Measurement of the Electron-Proton Coupling Constant in
Metallic Superconductors,’’ Phys. Rev. Lett., 64, pp. 2172–2175.
关2兴 Qiu, T. Q., and Tien, C. L., 1994, ‘‘Femtosecond Laser Heating of Multilayered Metals—I. Analysis,’’ ASME J. Heat Transfer, 37, pp. 2789–2797.
关3兴 Kulish, V. V., Lage, J. L., Komarov, P. L., and Raad, P. E., 2001, ‘‘FractionalDiffusion Theory for Calculating Thermal Properties of Thin Films From Surface Transient Thermoreflectance Measurements,’’ ASME J. Heat Transfer,
123共6兲, pp. 1133–1138.
关4兴 Cattaneo, C., 1958, ‘‘A Form of Heat Conduction Equation Which Eliminates
the Paradox of Instantaneous Propagation,’’ Compte Rendus, 247, pp. 431–
433.
关5兴 Tzou, D. Y., 1997, Macro to Microscale Heat Transfer: The Lagging Behavior,
Taylor & Francis.
关6兴 Abramowicz, M., and Stegun, I. A., 1964, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, Dover, New York.
关7兴 Kulish, V. V., and Lage, J. L., 2000, ‘‘Fractional-Diffusion Solutions for Transient Local Temperature and Heat Flux,’’ ASME J. Heat Transfer, 122, pp.
372–376.
关8兴 Brorson, S. D., Fujimoto, J. G., and Ippen, E. P., 1987, ‘‘Femtosecond Electron
Heat-Transport Dynamics in Thin Gold Film,’’ Phys. Rev. Lett., 59, pp. 1962–
1965.
关9兴 Qiu, T. Q., Juhasz, T., Suarez, C., Bron, W. E., and Tien, C. L., 1994, ‘‘Femtosecond Laser Heating of Multilayered Metals—II. Experiments,’’ Int. J. Heat
Mass Transfer, 37, pp. 2799–2808.
Transactions of the ASME