The principle of minimum entropy production for diffusion and heat

The principle of minimum entropy production for diffusion and heat

RENATA BUJAKIEWICZ-KOROŃSKA∗, JAN KOROŃSKI**
THE PRINCIPLE OF MINIMUM ENTROPY
PRODUCTION FOR DIFFUSION AND HEAT
TRANSFER IN OPEN SYSTEMS1
ZASADA MINIMUM PRODUKCJI ENTROPII
DLA DYFUZJI I PRZEWODNICTWA CIEPLNEGO
W UKŁADACH OTWARTYCH
Abstract
Equilibrium thermodynamics describes a closed system that tends towards an equilibrium characterized by minimum energy and
maximum entropy. Non-equilibrium thermodynamics describes an open systems that tends towards a steady state characterized
by minimum production of entropy. The open systems are systems which are not isolated systems, i. e. systems which exchange
as well as the mass (molecules) and any other form of energy with surroundings. The energy delivered to the system from its
surroundings is dissipated into new structures. These structures are called dissipative structures or shortly: patterns. In particularly
soil-plant systems exchange both energy and matter with their surroundings and are consequently examples of open biological
systems. They should therefore tend towards a steady state described by non-equilibrium thermodynamics and characterized
by minimum production entropy. In this paper we study the principle of minimum entropy production for diffusion and heat
transfer in open systems. The general formulae and exemplary values of entropy production for diffusion and heat transfer for any
substances involving particular boundary conditions are obtained.
Keyword: open systems, dissipative structures, principle of minimum entropy production, diffusion, heat transfer
Streszczenie
Równowaga termodynamiczna cechuje układy zamknięte (izolowane) w takim sensie, że tendencja do równowagi charakteryzuje
się stanem minimum energii i maksimum entropii w układzie. Nierównowaga termodynamiczna cechuje układy otwarte dążące do
stanu minimum produkcji entropii. Poprzez układy otwarte rozumiemy układy, które nie są układami izolowanymi, tzn. układy,
które wymieniają z otoczeniem masę (cząstki) i inne formy energii. Energia wymieniana z otoczeniem jest dyssypowana na nową
strukturę. Takie struktury powstałe w układach otwartych nazywa się strukturami dyssypatywnymi lub krótko patterns.
W szczególności rośliny wymieniają z otoczeniem energię i masę i są w konsekwencji przykładami otwartych układów
biologicznych. Zatem wykazują tendencję do nierównowagi termodynamicznej i dążą do stanu minimum produkcji entropii.
W niniejszym artykule badamy zasadę minimum produkcji entropii dla procesu dyfuzji i przewodnictwa cieplnego w układach
otwartych. Uzyskujemy ogólne formuły i przykładowe wartości produkcji entropii w układach otwartych dla procesu dyfuzji
i przewodnictwa cieplnego dla dowolnych substancji przy spełnieniu pewnych szczególnych warunków brzegowych.
Słowa kluczowe: układy otwarte, struktury dyssypatywne, zasada minimum produkcji entropii, dyfuzja, przewodnictwo cieplne
∗
PhD. Renata Bujakiewicz-Korońska, Faculty of Mathematics, Physics and Technical Sciences,
Institute of Physics, Cracow Pedagogical University.
**
PhD. Jan Koroński, Faculty of Physics, Mathematic and Informatics, Institute of Mathematics,
Cracow University of Technology.
1
The paper is extended version of the lecture at the XIII National Conference Application
of Mathematics in Biology and Medicine – Serpelice nad Bugiem, 18-22 September 2007 [4].
24
1. Introduction
This paper deals with the principle of minimal entropy production for diffusion and heat
transfer in dissipative systems [1], [2]. We will discuss here the following issues:
– deriving formulae for entropy production by mass diffusion and heat transfer,
– problem of minimization of functional expressions for entropy production in component
diffusion and heat transfer,
– effective analytical solutions (in the case of one-dimensional space variable) for which
functional expressions for entropy production take minimal values,
– graphic presentations of the obtained solutions,
– general formulae and exemplary values of entropy production for diffusion and heat
transfer for any substances involving particular boundary conditions.
The systems which are not isolated systems are called open systems. The open systems
exchange mass (molecules) as well as the any other form of energy with their surroundings.
The mass and energy delivered to the system from its surroundings are dissipated into new
structures. These structures are called dissipative structures [1], [2], [3] or shortly: patterns.
Let us consider of thermodynamics aspects of life in open biological and medical
systems. In [3], [4] and [5] the thermodynamic concepts and methods for the macroscopic
description of biological systems are considered.
Living beings are natural ordered and information-processing macroscopic systems
originating from processes of self-organization and natural evolution [5]. The basic
thermodynamic properties of living systems are openness, heterogeneity and large distance
from equilibrium. It seems to us that the self-organization of information-processing by
evolution is the central aspect of the evolution of life.
Entropy belongs to the central concepts of natural sciences including the life sciences
[5]. Life is bound to nonequilibrium situations; it is surely impossible in thermodynamic
equilibrium. All processes in living systems originate from physical processes. Living
beings are open thermodynamic systems which permanently exchange matter, energy,
entropy and information with the surrounding. All these exchange processes are closely
connected. The exchange of matter as well as the process of information transfer between
systems are always connected with the transfer of energy and entropy. However, matter and
information are not identical with energy and entropy. From the physical point of view,
information denotes a quantity which is exchanged between a source and a receiver and
which reduces the uncertainty about the state of the system. Information is a binary relation
between two systems: source – receiver and it is always connected with the flow of energy
and entropy between these systems. In [2] the example of dissipative structure in biology
describing the life cycle of the amoebae Dictyostelium discoideum is considered.
Equilibrium thermodynamics describes a closed systems that tends towards an
equilibrium characterized by minimum energy and maximum entropy. Non-equilibrium
thermodynamics describes an open systems that tends towards a steady state characterized
by minimum production entropy [3].
In this paper we will emphasize those features which can be understood from a general
point of view from principle of minimum entropy production. This principle is assisted with
understanding of self-organization processes in open systems in particularly of open
biological and medical systems. We deal with the principle of minimum entropy production
for diffusion and heat transfer in dissipative systems.
25
2. Derivation of formulae for entropy production for mass diffusion
and heat transfer
As it has been already shown in [1] chapter 1, the requirement of stationary state makes
one of the coupled flows zero. Thus there arises the problem of finding relation expressing
entropy production for mass diffusion and heat transfer.
In the case of continuous distribution of thermodynamic forces, entropy production
is given by the formula:
di S
= σdV ,
dt
∫
V
where σ is density of entropy production within a region of volume V, which can be
expressed as the sum of products of local flows Jk and thermodynamic forces Xk:
N
σ=
∑J
k Xk ,
k =1
where index k denotes the respective flow types.
In the case of unary flow (k = 1), entropy production for continuous distribution of
thermodynamic forces is given by the formula:
di S
= JX dV .
dt
∫
V
For mass diffusion there hold relations:
 µ  −1
X m = ∇   = ∇µ,
T  T
Jm = −
L
∇µ,
T
µ = RT ln C ,
where R denotes the gas constant, C – component concentration, µ – chemical potential,
and L is a phenomenological coefficient. Taking the above relations into consideration, we
obtain
di S
=
dt
∫
V
LR 2 (∇C ) 2
dV .
C
C
Taking into account Fick's law:
Jm = –D C
and
∇µ = RT
∇C
C
26
and on the other hand:
J m = − LR
∇C
,
C
we can get:
LR
= D.
C
In effect, the following formula for entropy production in mass diffusion is obtained:
di S
(∇C ) 2
= RD
dV .
dt
C
∫
V
Repeating analogical considerations for heat transfer and taking into account that:
1
1
X th = −∇   = − 2 ∇T ,
T
T
 
J th = −k ∇T ,
we get the following formula for entropy production:
di S
=k
dt
∫
V
(∇ T ) 2
T2
dV ,
where k is the coefficient of heat conductivity, and the other symbols have meaning as
defined previously.
3. Problem of minimization of functional expressions for entropy production
in diffusion and heat transfer
Now we are going to determine for what temperature and mass distributions the above
expression for entropy production in heat and mass flows reaches a minimum. By
calculating variations of those functional and equaling them to zero one gets the condition
for their minimal values.
For diffusion we have:
δ
di S
1
1 

= DR  δ((∇C ) 2 ) + (∇C ) 2 δ( )  dV =
dt
C
C 

V
∫
1 
2
= DR  ∇C∇ 2 C − (∇C )3 2  δxdV =
C
C 

V
∫
= DR
∫
V
∇C  2
1
2∇ C − (∇C ) 2  δxdV = 0.

C 
C
27
Hence we obtain the equation:
2∇ 2C =
(∇C ) 2
.
C
So
∇2C −
(∇C ) 2
= 0.
2C
Similarly for heat transfer we have:
1
1
d S


δ  i  = k  δ((T )2 ) 2 + δ( 2 )(∇T ) 2  dV =
dt
T
T




V
∫
=k

(∇T )2 
∇T ∇ 2T −
 δxdV = 0.
T 
T

V
∫
2
2
Hence we get the equation:
∇ 2T −
(∇ T ) 2
= 0, or
T
∆T −
(∇T ) 2
= 0.
T
Note that the two non-linear differential equations describe concentration and
temperature field in a stationary state, while in the classical description concentration
and temperature field in a stationary state are described by the linear Laplace equations:
∆C = 0
and
∆T = 0.
In this situation, there arises an interesting problem: How much do the solutions
of the nonlinear equations derived before, describing concentration and temperature field
within open systems, differ from solutions of the above Laplace equations describing
concentration and temperature field in the classical sense satisfying the relevant boundary
conditions?
4. Effective solutions on linear and nonlinear equations
in one-dimensional case
The condition for minimization of the value of the functional entropy production
in the diffusion, derived in 3.
∆C −
(∇C )2
=0
2C
(1a)
in one-dimensional case (in respect to the independent space variable) takes the form
of the following nonlinear differential equation
28
d 2C
dx 2
2
=
1  dC 

 .
2C  dx 
(1b)
The condition for minimization of the value of the functional entropy production in the
heat transfer, derived in 3.
(∇ T ) 2
=0
T
in one-dimensional case takes the form of the following nonlinear equation
∆T −
d 2T
dx 2
(2a)
2
=
1  dT 

 .
T  dx 
(2b)
The respective differential equations corresponding to equations (1b) and (2b) in
classical description have the form (in one-dimensional case) of linear differential equations
d 2C
dx 2
=0
(1c)
and
d 2T
(2c)
= 0.
dx 2
Equations (1c) and (2c) can be immediately solved by twice integrating. Thus the
general solution of the equation (1c) is a two-parameter family of straight lines
C ( x) = k 1 x + k 2 ,
(1d)
where k 1 and k 2 are arbitrary constants resulting from integration. Analogically, the
general solution of the equation (2c) is a two-parameter family of straight lines
T ( x) = k 1 x + k 2 ,
(2d)
where k 1 and k 2 are also integration constants.
In the further, we are going to compare how much the solutions of equations (1b) and
(1c), and equations (2b) and (2c) differ from each other. In order to do this we solve
effectively the nonlinear equations (1b) and (2b). Using proper substitutions of dependent
variables, the considered equations can be reduced to nonlinear equations of the first order,
which we will solve by doing relevant integrations. Eventually we go back to the sought
solutions, making use of the previous substitutions and integrating them. In short, we solve
effectively the above nonlinear equations by reducing the considered second-order
equations to equivalent systems of first-order equations.
General solution of the equation (1b)
In order to find effectively the form of a solution of the equation (1b), we make
the following substitution:
dC
= u (C ),
dx
29
Hence we get:
d 2C
dx
2
=
du dC du
=
u.
dC dx dC
Taking this into account in the equation, we obtain the following differential equation in
respect to the unknown function u = u(C):
du
1 2
u=
u ,
dC
2C
provided u ≠ 0
hence after dividing the latter equation by u and separating of variables we obtain:
du
1
=
dC.
u 2C
By integrating the latest equation we have
ln u = (1/ 2) ln C + ln k1 ,
or
u = k1C1/2 .
Making the previous substitution one gets:
dC
= k1C1/2 .
dx
By integrating both sides of this equality one obtains the general solution of the
equation (1b) in the form
C ( x) = (k1 x + k2 ) 2 ,
where k 1 and k 2 are respectively arbitrary constants resulting from the integrations.
General solution of the equation (2b)
By analogical argument like that used in solving equation (1b) one can gets the general
solution for the equation (2b) in the form
T(x) = k2 exp(k1x),
where k1 and k2 are arbitrary constants resulting from the integrations.
5. Graphic presentation of solutions obtained in 4
Graphic presentation of solutions of the equations (1b) and (1c)
In order to provide a graphic presentation of the difference between the nonlinear
equation (1b) and the linear equation (1c), we will consider them within a some interval
(a, b), where a, b ∈ R, along with the given boundary conditions, for instance C(a) = 0,10
and C(b) = 1,00 (understood as the limiting values when x → a, or x → b, respectively).
30
Thus we have a typical boundary problem – finding a solution satisfying the following
conditions:
d 2C
dx 2
2
=
1  dC 

 , x ∈ (a, b),
2C  dx 
C(a) = 0.1,
C(b) = 1
In the previous section 4., we obtained the general solution of the considered nonlinear
equation in the following form:
C(x) = (k1x + k2)2.
Taking into account the boundary conditions we determine the values of constants
k1 = 0.00683 and k2 = 0.317. Consequently, the particular solution of the considered
boundary problem has the form:
C(x) = (0.00683x + 0.317)2.
while the solution of the boundary problem:
d 2C
dx 2
= 0, x ∈ (a, b),
C(a) = 0.1,
C(b) = 1
is of the form:
C(x) = 0.009x + 0.1.
The difference between the solutions of both the boundary problems considered is
presented in the Fig. 1.
Fig. 1. Solutions of boundary problems for equations (1b) and (1c)
Rys. 1. Rozwiązania problemu brzegowego dla równań (1b) i (1c)
31
In the Fig. 1, the dashed line denotes the solution of the respective linear boundary
problem, and the continuous line denotes the solution of the respective nonlinear boundary
problem. As it can be seen from the diagram the respective values of solutions for the linear
and nonlinear boundary problems are different. In the Fig. 2, it is shown that difference is
larger for larger values of the difference between the boundary values, and when the
difference between the boundary values goes to zero, the difference between the solutions
vanishes too, due to the negligible nonlinear component in the respective nonlinear equation.
In the Fig. 2 the same conventions for the meaning of dashed and continuous lines are used.
Table 1
The solutions of (1b) and (1c) equations for different boundary conditions
Lp.
1.
2.
3.
4.
5.
C (0) [%]
0.1
0.1
0.1
0.1
0.1
C (100) [%]
0.5
1
1.5
2
2.5
Solutions of (1b) equation
C(x) = 0.004x + 0.1
C(x) = 0.009x + 0.1
C(x) = 0.014x + 0.1
C(x) = 0.019x + 0.1
C(x) = 0.024x + 0.1
Solutions of (1c) equation
C(x) = (0.004x + 0.317)2
C(x) = (0.007x + 0.317)2
C(x) = (0.009x + 0.317)2
C(x) = (0.011x + 0.317)2
C(x) = (0.012x + 0.317)2
Fig. 2. Graphic representation of increasing difference between the solutions of the considered linear and
nonlinear problems with increasing difference of boundary values of the concentration C
Rys. 2. Graficzne przedstawienie rosnących różnic pomiędzy rozwiązaniami liniowych i nieliniowych
zagadnień ze wzrastającą różnicą wartości brzegowych koncentracji C
We have shown that after limiting our considerations, for the sake of effective solving
of equations, to the one-dimensional case, the nonlinear and linear partial differential
equations are reduced to the respective, nonlinear or linear, ordinary differential equations,
which could be solved by using relevant substitutions. When comparing the solutions
of those equations, we can find that the difference between the nonlinear equations and the
corresponding linear equations increases with increasing difference between the boundary
conditions set at both ends of the interval within which the equations are considered in the
one-dimensional case. For the same values of boundary conditions the solutions of linear
equations and respective nonlinear equations are equivalent, due to vanishing of the
nonlinear part.
32
Graphic presentation of solutions of the equations (2b) and (2c)
The similar effects as for diffusion, shown above, can be demonstrated for heat transfer.
The function of temperature field, which is the solution of the relevant linear boundary
problem, also differs from the solution of the corresponding nonlinear boundary problem,
and the difference increases with increasing difference of boundary values of the function
T. The effect is showed graphically below in the Figs. 3 and 4.
Fig. 3. Solutions of boundary problems for equations (2b) and (2c)
Rys. 3. Rozwiązania zagadnień brzegowych dla równań (2b) i (2c)
Fig. 4. Graphic representation of increasing difference between the solutions of the considered linear
and nonlinear problems with increasing difference of boundary values of the temperature T
Rys. 4.Graficzne przedstawienie wzrastających różnic pomiędzy rozwiązaniami
rozważanych problemów liniowego i nieliniowego związanych ze wzrostem
różnicy wartości brzegowych tempartury T
33
Table 2
The solutions of (2b) and (2c) equations for different boundary conditions
Lp.
1.
2.
3.
4.
5.
T (0)
0.1
0.1
0.1
0.1
0.1
T (100)
0.5
1.0
1.5
2.0
2.5
Solutions of (2b) equation
T(x) = 0.004x + 0.1
T(x) = 0.009x + 0.1
T(x) = 0.014x + 0.1
T(x) = 0.019x + 0.1
T(x) = 0.024x + 0.1
Solutions of (2c) equation
T(x) = 0.1exp(0.016x)
T(x) = 0.1exp(0.023x)
T(x) = 0.1exp(0.027x)
T(x) = 0.1exp(0.029x)
T(x) = 0.1exp(0.032x)
It is worth noticing here that, from a formal point of view, differential equations (1b)
and (2b) differed just by a constant value. And yet that difference did lead to essentially
different forms of general solutions of the considered equations. For equation (1b)
the solution is the superposition of a linear function with a quadratic one, while for the
equation (2b) – superposition of a linear function with an exponential one.
Basing on the potential theory, the similar results can be obtained for the multidimensional case, i.e. when the partial equations are considered in two- or three-dimensional spaces.
6. General formulae and exemplary values of entropy production
(in one-dimensional case) for arbitrary materials
and boundary conditions for diffusion and heat transfer
In one-dimensional case, entropy production can be described by the following
functional:
b
2
di S
1  dC 
= DR 
 dx.
dt
C  dx 
∫
a
After substituting into this equality, first the general solution of the linear equation, i.e.
function C ( x) = k 1 x + k 2 , we get the formula for entropy production for diffusion
described by a linear equation, and then the solution C ( x) = (k1 x + k2 x)2 of the nonlinear
equation, the formula for entropy production for diffusion described by a nonlinear
equation is obtained.
In the second case we have:
k1 x + k2 = z
b
b′
b
di S
1
1 dz
2
= DR
k1 dx =
= DR k12
= DRk1 ln k1 x + k2 .
1
a
dx = dz
dt
k x + k2
k1 z
a 1
a′
k1
∫
∫
After taking into account the boundary conditions the entropy production for diffusion
described by a linear equation is given by the formula:
di S
k 1b + k 2
= DRk1 ln
.
dt
k1a + k2
34
And in the latter case:
b
b
di S
k12
b
= DR
(2(k1 x + k2 ))2 dx = 4 DR k12 dx = 4 DRk12 x a .
2
dt
(
k
x
+
k
)
2
a 1
a
∫
∫
Thus the entropy production for diffusion described by a nonlinear equation is given
by the formula:
di S
= 4 DRk12 (b − a).
dt
Using analogical argument, the formula for entropy production in heat transfer can be
obtained. And thus when heat transfer is described by a linear equation, entropy production
is given by the formula:
b
di S 1
= k k 1 ( k 1 x + k 2 ) −3 .
a
dt
3
and when heat transfer is described by a nonlinear equation, entropy production is given by
the formula:
di S
= kk12 (b − a).
dt
The derived formulas allow for effective calculating of entropy production for diffusion
and for heat transfer in the considered situations.
In Table 3 we provided values of entropy production for diffusion, along with
difference between the respective values of entropy production as derived from relevant
functional for linear and nonlinear case, for materials of diffusion coefficient D.
We consider the particular interval (a, b) = (0, 100), i.e. a = 0, b=100, along with
several particular boundary values of concentration C.
Table 3
The difference between values of entropy production in the linear
and nonlinear cases with different boundary conditions
C(x) = 0
C(x) = 100
0.1
0.1
0.1
0.1
0.1
0.1
0.10
0.15
0.25
0.50
0.75
1.00
di S –2
10
dt
in linear case
0
0.020RD
0.137RD
0.644RD
1.310RD
2.072RD
di S –2
10
dt
in nonlinear case
0
0.019RD
0.130RD
0.608RD
1.210RD
1.850RD
Percentage difference
of entropy production between
linear and nonlinear cases
0
3.36
5.11
5.28
7.63
10.71
The last column of the Table 3 gives the percentage difference between entropy
production in the linear and nonlinear cases, regardless of values of the coefficients D
and R for any materials.
35
An analogical can be made up for heat transfer, where the percentage difference in
entropy production between the linear and nonlinear cases is much greater, but otherwise
the table is quite similar, so we do not provide it here.
References
[1] B u j a k i e w i c z - K o r o ń s k a R., O przemianach energii i informacji w strukturach dyssypatywnych (On the Transformations of Energy and Information in Dissipative Structures),
Monograph, Wydawnictwo AP, Kraków 2000.
[2] B u j a k i e w i c z - K o r o ń s k a R., K o r o ń s k i J., Dissipative structures and biology,
Proceedings of the IX National Conference Application of Mathematics in Biology and Medicine,
Piwniczna, 9–13 September 2003, Institute of Mathematics Jagiellonian University, Cracow
2003, 19-26.
[3] B u j a k i e w i c z - K o r o ń s k a R., K o r o ń s k i J., Synergetic efficiency and biology,
Proceedings of the IX National Conference Application of Mathematics in Biology and Medicine,
Piwniczna, 9–13 September 2003, Institute of Mathematics Jagiellonian University, Cracow
2003, 27-32.
[4] B u j a k i e w i c z - K o r o ń s k a R., K o r o ń s k i J., On the principle of minimum production
entropy in open systems, Proceedings of the XIII National Conference Application of
Mathematics in Biology and Medicine, Serpelice nad Bugiem, 18–22 September 2007, Institute of
Mathematics Jagiellonian University, Cracow 2007, 25-30.
[5] E b e l i n g W., Entropy and the evolution of biological information, Physica A 163, 1990,
398-402.