An analytical solution of the hyperbolic heat conduction equation for the case of a finite
medium symmetrically heated on both sides
M. Lewandowskaa, L. Malinowskib*
a
Institute of Physics, Szczecin University of Technology, 70-311 Szczecin, Al. Piastów 48,
Poland
b
Faculty of Maritime Technology, Szczecin University of Technology, 71-065 Szczecin, Al.
Piastów 41, Poland
*
Corresponding author E-mail address: lmal@ps.pl (L. Malinowski); tel.: +48 91 4494827; fax:
+48 91 4494488
Abstract
This paper presents an analytical solution of the hyperbolic heat conduction equation for the
case of a thin slab symmetrically heated on both sides. In the mathematical model adopted, the
heating is treated as an internal heat source with capacity dependent on coordinate and time,
while walls of the slab are assumed to be insulated. The solution is obtained by Laplace
transforms method taking advantage of the method of superposition. The analytical solution is
validated by comparison with the results from a numerical model.
Nomenclature
a
thermal diffusivity  k /  c p 
cp
specific heat at constant pressure
g
capacity of internal heat source
I
laser incident intensity
Ir
arbitrary reference laser intensity
I0
modified Bessel function, 0th order
k
thermal conductivity
l
thickness of the slab
L
dimensionless thickness of the slab  wl / 2a 
L-1
inverse Laplace operator
1
R
surface reflectance
s
Laplace variable
t
time
ti
duration of laser pulse
tk
relaxation time of heat flux
T
temperature
Tm, T0
arbitrary reference temperatures
u  
unity step function
w
speed of heat propagation  a / tk 1 / 2
x
Cartesian coordinate
X
dimensionless Cartesian coordinate

dimensionless absorption coefficient
  
Dirac delta function
  
dimensionless rate of energy absorbed in the medium

absorption coefficient

dimensionless temperature

density

dimensionless time
i
dimensionless duration of laser pulse

dimensionless capacity of internal heat source
0
constant coefficients related to the dimensionless capacity of internal heat
source

frequency of a periodic heat source
Superscript
transformed variable
2
Keywords: Hyperbolic equation of heat conduction; Finite medium; Analytical solution;
Laplace transforms method; Superposition method
1. Introduction
In highly unsteady situations, the parabolic heat conduction equation based on the Fourier law
fails, so the need for more adequate model of heat conduction, which permits the finite speed of
heat flux, has arisen. There have been numerous attempts to formulate a new model in the
literature [1-4], but it seems that, at present, the most frequently used is the hyperbolic model of
heat conduction introduced by Cattaneo [5]. This model owns its popularity to simplicity and
effectiveness. Various cases of hyperbolic heat conduction in a finite medium were studied
analytically [6-12] and numerically [10, 13-15]. Recently, Torii et al. [16] solved numerically
the case of a thin film subjected to a symmetrical heating on both sides. In this paper, we solve
the same problem analytically by the method of Laplace transforms.
2. Model
We consider a thin slab of thickness L, initially at temperature T  x,0  T0 , with constant
thermophysical properties and insulated walls. At time t  0 , laser heat generation starts at both
walls of the slab, giving rise to two thermal waves travelling in opposite directions. The
temperature field in the slab can be described by the following hyperbolic equation of heat
conduction
tk
 2T T
 2T
1  g



a

 g
 tk
2
2
t
c p  t
t
x

(1)
where tk is the relaxation time which represents a delay of the heat flow after a temperature
gradient has been imposed. The relaxation time is related to the speed of propagation of thermal
wave in the medium, w, by
tk  a / w 2
(2)
The heat source term in Eq. (1) is modelled as
g  x, t   g l  x, t   g r  x, t 
(3a)
3
where
g l x, t   I t 1  R  exp  X 
(3b)
g r  x, t   I t 1  R  exp   L  X 
(3c)
g l  x, t  and g r x, t  are the capacities of the internal heat sources acting at the left-hand side
wall and at the right-hand side wall of the slab, respectively. Eq. (3b), used by Blackwell [17]
and Zubair et al. [18], describes internal absorption of laser radiation. For convenience of
subsequent analysis, we introduce the following dimensionless quantities
X  wx / 2a 
(4a)
  t / 2tk 
(4b)
  T  T0  / Tm  T0 
(4c)
  gtk /c p Tm  T0 
(4d)
Eq. (1) is expressed in terms of the dimensionless variables (4a)-(4d) as
 2
  2


2

2
 4
2
2
 X


(5)
The dimensionless forms of Eqs. (3a)-(3c) are
  X ,    l  X ,    r  X , 
(6a)
 l  X ,    0  exp  X 
(6b)
 r  X ,    0   exp   L  X 
(6c)
where
 0  I r 1  Rtk / c p Tm  T0 
(6d)
 ( )  I 2tk  / I r
(6e)
  2 wt k 
(6f)
The dimensionless initial conditions for the present problem are
  X ,0  0
(7a)

 X ,0  2  X ,0

(7b)
4
The dimensionless boundary conditions are

0,   0
X
(8a)

L,   0
X
(8b)
Eq. (7b) is derived from the energy conservation equation on the assumption that there is no
heat flow in the body at the initial moment.
3. Analytical solution
The boundary value problem of Eqs. (5), (6a) - (6f), (7a), (7b), (8a), and (8b) is solved by the
method of Laplace transforms. At first, we solve Eq. (5) for
  X ,    l  X , 
(9a)
to obtain the solution  l  X ,  . Next, we solve Eq. (5) for
  X ,    r  X , 
(9b)
to obtain the solution  r ( X , ) . Finally, as the problem is linear, we superimpose the two
solutions
  X ,    l  X ,    r  X ,    l  X ,    l L  X , 
(10)
We substitute Eq. (6b) for  in Eqs. (5) and (7b) to obtain, respectively
 2 l
 l  2 l
 


2

 2 0 
 2  exp X 
2
2




X


(11)
l
 X ,0  2 0 0exp X 

(12)
Taking the Laplace transform of Eq. (11) and using the initial conditions given by Eqs. (7a) and
(12), yields
d 2l  X , s 
 ss  2l  X , s   2 0 s  2 s exp X 
dX 2
(13)
Transforming the boundary conditions given by Eqs. (8a) and (8b) gives
dl
0, s   0
dX
(14a)
5
dl
L, s   0
dX
(14b)
The solution of Eq. (13) satisfying boundary conditions (14a) and (14b) is the function
l  X , s   A0 s exp X   A1 s exp Bs X   A2 s expBs X 
(15a)
where
A0 s  
A1 s  
A2 s  
2 0 s  2 s 
ss  2   2
(15b)
A0 s 

expBs L  exp L
exp Bs L  expBs L
(15c)
A0 s 

exp Bs L  exp L
exp Bs L  expBs L
(15d)
Bs 
Bs 
Bs   ss  2
1/ 2
(15e)
To invert Eq. (15a) and find the time solution, we expand the terms A1 ( s ) exp  B( s ) X  and
A2 ( s ) exp B ( s ) X  in binominal series
  exp B( s )2nL  X 

 

B( s )
 n 0

A1 ( s ) exp B( s ) X   A0 ( s ) 


 exp(  L) exp B( s )( 2n  1) L  X 



B( s )
n 0
(16a)
  exp B( s )2nL  X 

 

B( s )
 n 1

A2 ( s ) expB( s ) X   A0 ( s ) 


 exp(  L) exp B( s )( 2n  1) L  X 



B( s )
n 0
(16b)
Moreover, we use the following pair of transforms
L1

exp  as s  b 
ss  b1 / 2
1/ 2

0

2
2
exp 0.5b I 0 0.5b   a


The inverse Laplace transform of solution (15a) is
6

1/ 2

for
for
a   0
 a
(17)

 


h
(
2
nL

X
,

)

 hi (2nL  X , )
  i
n 1
 n 0




f Hi ( ) exp(  X )    exp(  L)  hi ( 2n  1) L  X , 

n 0





 exp(  L)  hi ( 2n  1) L  X , 



n 0
 l ( X , )   0
(18a)
where
0

hi ( p, )  
exp( u ) I 0 u 2  p 2

p

f Hi ( ) 
1


  1  2

1/ 2
f
for
p   0
  u )du for   p
Hi (
(18b)

 (u) p exp m (  u)   m exp  p (  u)du
(18c)
0

1/ 2
(18d)
m   1
(18e)
 p   1
(18f)
4. Solutions for special cases of heat source capacity
The RHS of Eq. (18c) can be considerably simplified for some particular forms of    [20].
Below, there are presented expressions for f Hi ( ) for the cases examined in this paper.
4.1. Instantaneous source,  ( )   ( )
f H 1   
1


p

exp m    m exp  p 
(19)
4.2. Source of time independent strength,     u  
f H 2   
 2p exp m    m2 exp  p   4
 p m
(20)
4.3. Rectangular pulse source,     u    u    i 
for  i    0
 f  
fH 3   H 2
 f H 2    f H 2    i  for    i
(21)
4.4. Periodic source,     sin    1
7
C


f H 4    2  A exp  p   B exp m   sin    D cos   f H 2  



(22a)
where
A
B
C
  3 2   m 2  4  4



2  4  2 2 2   2   4

(22b)
  3 2   p 2  4  4



2  4  2 2 2   2   4

(22c)
 2 2
 4  2 2 2   2   4
D


(22d)
 2  2  4
 4  2 2 2   2   4


(22e)
5. Validation of the solution
The analytical solution given by Eqs. (10), (18a) - (22e) was validated by a numerical solution
of the problem given by Eqs. (5), (6a) – (6c), (7a) - (8b). The MacCormack algorithm [19] was
used. Agreement between the analytical and numerical solutions was very good. We
additionally confirmed the correctness of our results by using the energy balance equation for
the whole slab
L
   X , dX 
0
4 0


1  exp L  d
(23)
0
We also compared our results with those reported by Torii et al. [16] and found some
discrepancies between them. Moreover, we found some errors in the paper by Torii et al. [16].
It seems that in the caption to Fig. 2, the equation   1 should read   1 . In the caption to
Fig. 3, the equation   1 should read   1 . In the captions to Figs. 4 - 9, it should be
c0 x0 /( 2 ) , not c0 x0 /  . In Figs. 4a and 4b,  should be equal to 0.01, not 0.1. In the caption
to Fig. 6, it should be  0  1 ,  ( )  1  sin(  )/ 2 , not  ( )  1 ,  0  1  sin(  )/ 2 .
6. Sample calculations and discussion
Using our analytical solution, we performed sample calculations of temperature profiles in the
slab for all considered types of heat source and chosen values of: L, ,  , 0 , and other
8
parameters concerning a particular type of heat source. The results of calculation are presented
in Figs. 1(a) - 3(b). The heat waves excited at the vicinity of both side surfaces of the slab travel
in the opposite directions, superimpose, and reflect back and forth between the surfaces. It is
worth noting that the heat wave travels the distance of x in time of t  x / w . Taking into
account Eqs. (2), (4a), and (4b) we obtain X   . So, the heat wave covers the distance of L in
time equal to L.
Figs. 1(a) and 1(b) displays the time-dependent temperature distribution in the slab for the
instantaneous heat sources for which  ( )   ( ) (par. 4.1). It is difficult to model numerically,
with good accuracy, such a source, but in the analytical solution, the function f H 1   takes
quite a simple form. Torii et al. did not consider the instantaneous heat sources in their work
[16]. It is seen in. Fig. 1(a) that the fronts of both waves meet in the middle of the slab for
  2.5 . They meet for all times calculated from the equation   nL / 2 for n  1, 2,  .
Temperature profiles for the sources of time independent strength,     u   (par. 4.2), are
presented in Figs. 2(a) - (2c). For L  1 and   1 (Fig. 2(a)), the heat produced by both
sources is nearly evenly distributed through the slab which explains the flat temperature
profiles for particular times. For large values of  and/or L, the strength of heat generation
varies a lot across the slab thickness that is manifested by substantial variation of temperature
profiles (Figs. 2(b) and 2(c)). There are significant quantitative differences, growing with time,
between the majority of results presented in Fig. 2(a) - 2(c) and the results for the same data
sets reported by Torii et al [16]. The temperature profiles calculated from our model lie higher.
For L  10 and   10 , our results overlap with those presented by Torii et al.
Figs. 3(a) and 3(b) shows the temperature profiles in the slab for the rectangular pulse heat
sources,     u    u    i  , (par. 4.3). In the limiting cases of very small and very large
values of  i , the time characteristic of the considered source becomes close to the time
characteristic of the instantaneous source, or of the source of time independent strength,
9
respectively. In Fig. 3(a) there is seen an overshooting of temperature for   0.8 and   1 ,
specific for the hyperbolic model of heat conduction.
For the periodic heat sources,     sin    1 , (par. 4.4), our results are the same as those
reported by Tori et al [16]. For the case of pulsed heat sources considered by Torii et al. [16],
we obtained higher values of temperatures; only for duration of the pulse equal to 5 the results
are similar. In general, we observe better agreement between our and the Torii et al. results for
large values of L and  .
6. Conclusions
The problem of thermal response of a thin slab to a symmetric rapid heating in the vicinity of
its both side surfaces is solved analytically. The results are validated by numerical calculations.
The agreement is very good. The results are also compared with those obtained from a
numerical model by Torii at al. [16]. In great part (especially for smaller values of L and  )
our results are higher than those reported by Torii at al., probably because of inaccuracy in the
Torii at al. calculations.
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Figure captions
Fig. 1. Temperature profiles in the slab for the instantaneous heat sources,  ( )   ( ) .  0  1 ,
  5 . (a) L  1 , (b) L  10 .
Fig. 2. Temperature profiles in the slab for the heat sources of time independent strength,
    u( ) .  0  1 . (a) L  1 ,   1 , (b) L  1 ,   10 , (c) L  10 ,   1 .
Fig. 3. Temperature profiles in the slab for the rectangular pulse heat sources,
    u    u    i  .  0  1 ,  i  0.4 . (a) L  1 ,   10 , (b) L  10 ,   1 .
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