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Two-temperature discrete model for nonlocal heat conduction
S. Sobolev
To cite this version:
S. Sobolev. Two-temperature discrete model for nonlocal heat conduction. Journal de Physique III, EDP Sciences, 1993, 3 (12), pp.2261-2269. �10.1051/jp3:1993273�. �jpa-00249081�
Classification Physics Abstracts
05.60
Two-temperature discrete model for nonlocal heat conduction
s. L. Sobolev
Institute of Chemical Physics Academy of Sciences of Russia, 142432 Chernogolovka, Moscow Region, Russia
(Received12 January 1993, revised 2 August 1993, accepted 23 September 1993)
Abstract. The two-temperature discrete model for heat conduction in heterogeneous media is
proposed. It is shown that the discrete model contains as limiting cases both hyperbolic and
parabolic heat conduction equations for propagative and diffusive regimes, respectively. To obtain these limiting cases two different laws of continuum limit have been introduced. The evolution of the two-temperature system comprises three stages with distinct time scales : fast relaxation of each subsystem to local equilibrium, energy exchange between the subsystems and classical
hydrodynamics.
1. Introduction.
The classical heat conduction theory rests on the assumption that the system is local in time and space. In other words the system is in a state of local thermodynamic equilibrium and the
thermodynamic equations are valid in each small element of the system. The latter implies the absence of any inner structure in the medium and so it can be considered as a continuous one.
Strictly speaking, transfer processes are inherently nonlocal since excitations arriving at a
point have brought their energy from surrounding points where the excitations were at the previous time moment. It is clear that the assumption of lpcal equilibrium is not satisfactory if the time scale of the transfer process is comparable to the relaxation time to local equilibrium [1-3, 9, 13, 25-27]. These processes may be treated as nonlocal in time. If the space scale of the transfer processes is comparable to the characteristic scale of internal structure of the system, it cannot be considered as a continuous medium. Hence, when one is interested in
sufficiently rapid phenomena, or when heat conduction process occurs in the system with heterogeneous internal structure, the nonlocal heat conduction theory should be used. Such a
nonlocal theory may be useful to study heat transfer processes in solids subjecting to ultrashort laser irradiation [4-6], in laser-produced plasmas [7], in capillary-porous media and cellular systems [8], in polymers, polyelectrolyte solutions, liquid crystals or more general materials
involving an internal structure [I], and in other extremes.
Other causes of nonlocal effects are found when the molecules have intemal degrees of
freedom or when a chemical reaction takes place in the gas [I]. The nonlocal theory can be
2262 JOURNAL DE PHYSIQUE III N° 12
applied to the analysis of some problems in low temperature physics [1-3, 10]. Mass transfer
processes under extreme conditions, for instance, diffusion in porous media or neutron
diffusion in nuclear reactors also exhibit the nonlocal behaviour [1-3, 11, 12]. For the analysis
of heat transfer processes in heterogeneous media the two-temperature model may be used [3, 4, 6, 9, 13-17]. If the time to establish equilibrium between subsystems of the heterogeneous
media is greater than the relaxation time to local equilibrium of each subsystem taken
individually, then one can attribute to each subsystem its own temperature. Introduction of the two-temperature model is necessary if the time to establish equilibrium (I.e. the time of heat
exchange between-the subsystems) is comparable to (or greater than) the characteristic time of
heat transfer process as a whole. Otherwise ordinary one-temperature model may be used.
Many systems of great practical interest can be treated as two-temperature ones. Complex systems consisting of solid, liquid, and gas, e-g- porous media, alloys, catalyst pellets, suspensions, pastes, etc. provide examples of two-temperature heterogeneous media [3, 8].
Other fields where two-temperature model is directly relevant are the mixtures of particles of different kinds, e.g. electrons and ions in plasma [7], electrons and lattice in metals [2-4, 6, 14, 15], or the molecules with internal degrees of freedom [1, 9]. The two-temperature models
are normally formulated within the space-time continuum using differential calculus [3, 4, 6, 9, 13-15]. In recent years, however, there has been a substantial growth of interest in discrete
approach, where in contrast it is assumed that time and length may be discrete variables [3, 12, 13, 16-24]. Perhaps the most familiar of such discrete models are cellular automata [18, 12, 21]. In these models both space and time are discrete ; the allowed values of a dynamical
variable are also discrete. For coupled-map models [19] and cell dynamical systems [20] space and time are again discrete but the dynamical variables are allowed to take on a continfium of
values. The idea of discrete modeling of transfer processes is based on a random walk theory
[3, 13, 16, 22-24] and has also some in common with coupled-map and cell dynamical approaches. It seems natural to represent the transfer processes in terms of a random walk on a lattice, since energy is transported by many different kinds of particles or excitations, which
can be treated as random walkers. The discrete model does not adopt the local equilibrium and the spatial locality assumptions and, hence, it can be used to study both classical local systems and local-non-equilibrium ones. The discrete approach provides heat (or mass) transfer
equations in discrete form, hence, it can be particularly useful for numerical simulation of cellular systems and various kinds of deterministic fractals, etc.
Our purpose is to propose the discrete two-temperature model to the problem of simulating
nonlocal heat conduction in heterogeneous systems. We will establish relations between the discrete model and continuous ones. This provides a « molecular level » connection between the discrete microscopic parameters and the thermodynamic coefficients. It can be done by
introducing a different law of continuous limit for the diffusive (dissipative) and the wave
(propagative) regimes [3, 13, 16].
2. The two-temperature discrete model.
One can derive the discrete heat-mass transfer model taking as a starting point a random walk
on a lattice [3, 12, 13, 16, 17, 22-24]. Usually, random walk theory describes a universe consisting of a homogeneous array of cells each cell evolves in discrete time step and all the cells compute their new state simultaneously. To constructe two-temperature model, let us consider a 2D heterogineous array consisting of two coupled subsystems (sublattices) [17].
For simplicity we shall assume that each subsystem contains square cells of size h and the cells
are arranged as on a chess-board. A characteristic length of the lattice h is of the order of a
mean free path of the molecules. For solids and liquids this length can be understood as a
representative mean free path of the thermal excitation. In contrast to ordinary homogeneous
discrete models where all the cells evolve in the same time step, in our heterogeneous model each subsystem (sublattice) has its own time step. The cells forming subsystem I evolve in the
discrete time step vi and the cells forming subsystem 2 in the discrete time step
r~. It means, that the heterogeneous system has two internal time scales a state of such a
system at time t is attained by n = t/rj steps of subsystem I and k
= t/r~ steps of subsystem 2.
Let r~/rj
= m > I. In this case the cells of subsystem 2 interact with their neighbours from both subsystem I and 2 at each step k. If n = km, the cells of subsystem I interact with their
neighbours from both subsystem I and 2 too. But if n # km, they interact with their neighbours
from subsystem I only. Let us denote by D the probability of interaction between cells of the
same type at step k and by D * the probability of interaction between cells of different type at
step k, with D + D *
=
I. At steps n # km the cells of subsystem 2 do not evolve and, hence, they do not interact with the cells of subsystem I. It means that D(n)
= I and D*(n)
= 0 at
n # km. In view of the above descRption, the probabilities of interaction at step n depend on n D(n)= ~~' ~ ~~ ~~ ~'~'~"
l ~ #kR~
(1)
~~~~~~ ~~~~ ~#~~.
According to theory of random walk on a homogeneous lattice [3, 13, 16, 22-24] the temperature of a cell at a particular time step depends on the sum of temperatures of its nearest
neighbors at previous time step. Such an uniform rule has also been used by coupled-map and cell dynamical approaches II 9, 20]. Taking into account the interaction between the jublattices
in our heterogeneous system we obtain the recursion relations for the temperatures oj(I, j, n) and o~(a, fl, k) of the sublattices I and 2, respectively [17]
oj(I, j, n + 1)
=
~~~~ lo
j(I + I, j + I, n) + oj(I I, j + I, n) + oj (I + I, j I, n) +
4
+ 0j(I I, j I, n)1+ ~(~~~10~(I + I, j, n) + 0~(I I, j, n)
+ 02(1, J + I, n) + 02(1, J I, n)I + Fj(0j(1, J, n)) (2) o~(a, p, k + 1)
=
~ [o~(a + I, p + I, k) + o~(a I, p + I, k) + o~(a + I, p I, k) +
+ o~(« i, p i, k)i + ( ioj(«
+ i, p, k) + oj(« i, p, k)
+ oj(«, p + i, k) + oj(«, p i, k)i + F~(o~(«, p, k)). (3) The single cell dynamic functions Fj(oj(I, j, n)) and F~(o~(a, p, k)) describe the
dynamic of an isolated cell and play the role of discrete energy sources. The pair of coupled equations (2) and (3) can be used to study both local-equilibrium and local-nonequilibrium
processes directly in the discrete form. Computer simulation and detailed analysis of the discrete model would be an interesting topic for following work.
Now let us consider relation between our discrete two-temperature model and partial
differential ones. It can be done by passing from the lattice picture to a continuum limit. The
probabilities of interaction I are highly fluctuating functions of step n resulting in the need for temporal averaging to obtain accurate continuum temperature fields.
In view of the above description, it seems natural to take the average values of
Dj(n) and D*(n) in (I) over m step
(D(n))
=
~ ~ ~
; (D *(n))
=
~ (4)
m ~n
2264 JOURNAL DE PHYSIQUE III N° 12
Moreover, to proceed to the continuum limit it is necessary to connect the discrete source
functions Fj and F~ in the discrete model (2) and (3) with source functions Wj and
W~ in partial differential models. It has been shown that the discrete source function
F, is the integral of the continuum source function W~ taken from t to t + r, [3, 13, 16, 17]
F, (o, )
t
~ ~ ~'
w, (t) dt
= r, w, (t + y, r, (5)
where 0~ y~~ I is a numerical coefficient, which depends on the form of function
W, (t).
The continuum limit of the discrete equations (2) and (3) can be obtained by setting
x = ih, y
= jh, t = nrj for sublattice I ; x = ah, y
=
Ph, t
= kr~ for sublattice 2 and by letting h and r, both tend to zero. The Taylor's series expansions of the discrete equations
contain an infinite number of terms with two small parameters and h. To obtain partial
differential equations with a finite number of terms one must specify the limiting behavior or,
in other words, the law of continuous limit, I-e- the relation between r~ and h as
r, - 0 and h
- 0 [3, 13, 16, 24]. This limiting behavior is determined by the nature of the system and the processes occurring in it. The two most typical cases will be considered [3, 13, 16].
2.I DIFFUSION FORM OF CONTINUUM LIMIT. The diffusion form of continuum limit requires
that thermal diffusivity a
=
h~/2 r~ remains finite as r, and h tend to zero [3, 13, 16, 17]
lim h~/2 r,
= a~ = Cte. (6)
~,, h -0
It should be pointed out at once that the diffusion form of continuous limit determines infinite velocity v, of propagation of thermal perturbations (thermal wave), since
v, = h/r,
=
2 a~/h
- co as h
- 0.
In the first order approximation in
r for the diffusion form of continuum limit (6) the two- temperature discrete model (1)-(3) with allowance for (4) and (5) results
3o vi 3~o
j aj h~
~ h2 ~ awj
+ -j = aj hoi + g(o~ oj + A oj + Ao~ + Wj + yj vi (7)
at 2 at 12 2 At
302 r2 3~o2 a~ h~
~ h2 ~ aW~
$ ~ 2 at~ ~~ ~~~ ~ ~~~~ ~~~ ~ 12 ~ ~~ ~ 2 ~~~ ~ ~~ ~ ~~ ~~ at ~~~
g =
D */r2 is the coupling constant, aj = (m I + D h~/2
vi m is the thermal diffusivity for subsystem I and a2 "
Dh~/2
r2 for subsystem 2. The terms of equations (7) and (8) containing
r, reflect time-nonlocality of the system, the terms containing h-space-nonlocality, terms
containing g reflect coupling between the subsystems. In the zero order approximation, I-e- in the local limit, the two-temperature model (7) and (8) reduces to a pair of coupled diffusion
equations
30j
= aj hoi + g(o~ 6
j) + Wj (9)
at
36~
= a~ A6~ + g(6j 6~) + W~. (lo)
at
This two-temperature local model will be considered in more detail below.
2.2 WAVE FORM OF coNTINuous LIMIT. Let us now consider the case where the velocity of thermal wave (thermal perturbations),
v~ = h/r,, remains finite when h and r, tend to zero [3, 13, 16, 17, 24]
lim h/r,
= v~ = Cte (I1)
~,, h -0
This form of limiting behavior will be called the « wave » form.
In the first approximation in r~ for the wave form of continuous limit (I I) we obtain from the discrete model (2) and (3)
'
~
~i ~ j 2
a~
~' ~° + g(o~ o + w
aW~
' ~ ~i ~i ~ ~~~
36~ T~ 3~o~ 3W~
$ ~ 2' at~ ~~ ~~~ ~ ~~~~ ~~~ ~ ~~ ~ ~~ ~~ at' ~~~~
Equations (12) and (13) are partial differential equations of hyperbolic type with a finite speed
of propagation of thermal waves vi = (2 aj/rj)~/~ and v~ = (2 a~/r~)'/~, corresponding to
subsystem I and 2, respectively [3, 13]. Some features of the hyperbolic heat conduction
equation (HHCE) will be discussed below and complete account of references may be found in [2, 3, 10, 25-27].
The two-temperature HHCE model may be applied to heat exchange between the electron gas and the lattice as a metal surface is irradiated with ultrashort laser pulses [3, 4, 6, 14, 15].
For this situation it is usually assumed that the energy of the laser pulse is absorbed only by the electron gas and heat conduction through the lattice can be neglected [15, 4, 6]. With these
assumption equations (12) and (13) take the form
3Tj vi 3~Tj 3Wj
~
' at
~ ~
' 2 at~ ~ ~~~ ~ ~~~~ ~~ ~ ~~ ~ ~~ ~~
at
~~~~
C~3T~
=
~e(Tj T~) (15)
where the subscripts I and 2 refer to electron gas and lattice, respectively ~e is electron-
phonon coupling factor (W/cm~ K) ; C, is volumetric heat capacity ; Wj is the laser heat
source. The two-temperature model (14) and (15) describes the space-time evolution of the
electron and the lattice temperatures on a time scale (vi is momentum relaxation time of electrons). The HHCE (14) implies that heat pulse propagates through the electron gas with finite velocity v = (Aj/C
j vi )'/~. Choosing reasonable values for the heat transfer parameters (Aj = 3 W/cm.K, Cl " 3 x 10~~ J/cm~ K and
vi =
10~ '~ s) one can find that v
-
10~ cm/s, I-e- heat pulse moves at a velocity close to the Fermi velocity of electrons V~. This result is in
agreement with the fact that heat carrying electrons move at V~, since those electrons which lie close to the Fermi surface are the principal contributor to heat transport. Brorson, Fujimoto and Ippen [4] have observed wave heat pulse transport in thin gold films with velocity
v = V~. These experiments can be described by two-temperature hyperbolic model (14) and
(15) rather than diffusive one.
Moreover, the model predicts that on a vi time scale the temperature of electron gas can be much greater than it is predicted by classical diffusion model. This overheating or localization
of heat near the heat source has a local-nonequilibrium nature [3, 25-27].
2266 JOURNAL DE PHYSIQUE III N° 12 3. Two-temperature diffusion model.
Let us now consider a pair of coupled diffusion equations (9) and (lo) in more detail. For
simplicity in the above description we assumed that the volumetric heat capacities of sublattices C~ = I. In general case equations (9) and (lo) may be rearranged as
3TjCj-=AjATj+~e(T~-Tj)+Wj (16)
at
C~ 3T~ = A
~AT~ + ~e(Tj T~) + W~ (17)
at
where C~ is the volumetric heat capacity, J/cm~ K ; ~e is the coupling factor, W/cm~ K. From
(16) and (17) we obtain the heat conduction equation governing the temperatures of the
subsystems Tj and T~ taken separetely [13]
~~'
+ r *
~~~' i( ~ ATj
=
a* ATj + W + r *
~~'
a~ I(A~Tj I( A
~j (18)
at at2 at at Hi
~~
+ r* ~~j~ I( ~ AT~
=
a* AT~ + W + r *
~~~
a~ I(A~T~ I( A
~j (19)
dt jt dt dt H~
Here r*
=
Cj C~/~e(Cj +C~) is the characteristic time of heat transfer (heat exchange)
between the subsystems, i~ = (H( + H()'/~ and i~ = (Hi H~)'/~ are the characteristic space scales of the heterogeneous two-temperature system, H)
= r * a~ is the characteristic heat scale of subsystem I, a~
= A~/C~ is thermal diffusivity of subsystem I, W
= (Wj + W~)/(C + C~)
W~ = W,/C,, a*
= (Aj + A~)/(Cj + C~) is thermal diffusivity of the heterogeneous two-
temperature system, and a(
= a a~. It should be reminded that C, and A are the parameters of the subsystem I per unit volume of the heterogeneous system. The temperature and the
thermodynamic parameter determined experimentally, however, are usually average values.
Hence, it is important to consider a heat conduction equation for average temperature
I, which we define as I
= (C Tj + C~ T~)/(C
j + C~). After some algebraic manipulation the
heat condiction equation goveming the average temperature T takes the form [13]
~~~~~~ ~~~t~~ ~~~~~~~~~~ ~~~~~~~ Cj~C2~~~j~~j~ ~~~~
Equations (18-20) clearly demonstrate that heat conduction processes in heterogeneous two- temperature system (9) and lo) may be treated as nonlocal both in time and space. The terms of equations (18-20) containing r* reflect time-nonlocality of the transfer processes, so r * is a scale of time-nonlocality. The terms containing i~ and i~ reflect space-nonlocality, so
i~ and i~ are the scales of space-nonlocality. It is natural that these equations reduce to a
classical PHCE (diffusion equation) in the local limit, when r*«t* and I «L, here
t * and L are the characteristic time and space scales of a heat conduction process (macroscopic scales). Note that this local limit does not coincide with the local limit in section 2. II because
the latter stands for inequalities r~ « t*, h ML and, as a rule, r* » r,, I » h.
For more detailed analysis it is necessary to estimate all the terms of equations (18-20). Let 3T/3t l, then
~
a2 r *
~2 ad 12
r )t2 t~, ~ at L2
