Asymmetrical heat conduction in inhomogeneous materials

Physica 131A (1985) 449-464 North-Holland, Amsterdam

ASYMMETRICAL

HEAT CONDUCTION MATERIALS Heinrich

Sektion Kalorimetrie,

Universitaet

IN INHOMOGENEOUS

HOFF

Urn, Oberer Eselsberg,

Received

22 October

D-7900

Ulm, Fed. Rep. Germany

1984

Fourier’s law of heat conduction is able to describe nonlinear global properties, if the dependence of the heat conductivity on the temperature is taken into account. Even asymmetrical heat conduction, where the transport is different in opposite directions, can be predicted under certain conditions.

1. Introduction Transport

of heat by conduction

is described

by Fourier’s

law’)

j = -AVT. Combining

(1.1)

it with the local balance

of heat

(1.2)

V-j=-cpg, where no heat sources are assumed equation of heat conduction

in the medium,

yields

the well

known

aT cp at = AV2T.

This equation, however, holds only for a constant it must be written as (see ref. 1)

v.(AVT)=

heat conductivity,

c$-$

0378-4371/85/$03.30 @I Elsevier Science Publishers (North-Holland Physics Publishing Division)

otherwise

(1.4)

B.V.

H. HOFF

4.50

on

In this paper space-either

we shall deal with those cases, where A is assumed to depend explicitly, because the heat conducting medium is in-

homogeneous, temperature. depends

or implicitly, We

shall

on temperature

discontinuous,

treat

because even

and

A depends the

on space.

which corresponds

on

combination The

the

local

of both

dependence

to a composite

medium

value

cases,

on space

of the where may

A be

- e.g. a heat conduc-

ting rod consisting of pieces of different material-or it may be continuous. Examples for the latter case are an alloy or an imperfect solid, the composition or imperfection of which continuously vary in space. The reason why these cases are not investigated in earlier treatments is easy to understand: taking into account the dependence of A on the local temperature yields a nonlinear equation of the heat conduction, which produces more complicated mathematical problems than the linear case treated in most textbooks. The inhomogeneous medium with temperature dependent heat conductivity, which is the most general case, requires the most difficult mathematical framework, of course. Within this paper we shall investigate the global properties of those systems, where external sources drive a one dimensional steady state transport by producing a constant input and output flow. These global properties are described by the functions J(T, AT), where system, ?i=the average of both temperatures

J denotes the heat flow across the at the left and the right boundary.

and AT their difference. The asymmetrical interesting result of this paper, corresponds odd function of AT:

heat conduction, which is the most to a case, where J(T, AT) is not an

J
-AT),

in one direction

the results

differs

from

that

hold also for the transport

in the opposite of matter

related problems because of the mathematical analogy. theoretical framework originally has been developed

of heat conducting

by diffusion

and

Thus a great deal of the for the transport across

membranes’). The results have been presented in that paper, a detailed discussion and deduction in a preliminary form.

2. Global properties

one for a

however,

without

systems at steady state

Before treating the above-mentioned cases, we have to find the language, how to express the global properties of a steady state transport system. The global properties are described by those quantities which can be measured at

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4.51

the surface of the system. Thus it consists of the relation between the flow densities, j(r,, t) and the temperatures T(rO, t) at the surface 0. In the case of one dimensional heat transport at steady state it is given by J(T’, T”), where T’, T” denote the temperatures at the right and the left surface, respectively, and J the heat flow, which is equal at both surfaces. As the flow J vanishes in equilibrium, the average and the difference of T’, T”:

T = ;(T’+

AT

T”),

= T’-

(2.1)

T”

is preferred. AT is considered as a “global force” and T as a “reference state” 3*7 ). Thus the global properties near equilibrium are described by the ansatz J(?; AT) = -L(T)AT,

(2.2)

where L(T) is the “coefficient of heat conductivity”. Eq. (2.2) is the near equilibrium approach of the relation between J and AT, which holds if AT is sufficiently small. It can be extended to the nonlinear range by either assuming that L depends also on the force AT, or by considering (2.2) as the first term of an expansion of J(T, AT) around the equilibrium state AT = 0. It can be demonstrated (to be published), that the first concept is very disadvantageous, if coupled processes are taken into account. So we prefer to extend (2.2) to the nonlinear range by the ansatz

J(T, AT) = -

2

Lk(T)ATk

.

(2.3)

k=l...

There are certain restrictions with respect to the L,(T), such as the second law of thermodynamics, the consequences of spatial symmetry and the condition of stability. The latter is necessary for keeping the state steady. Thus the stability condition defines the range of validity of the global description (2.3). 2.1. Consequences of the second law and of spatial symmetry The well known density of the production heat conduction is given by

a=j*V

Consequently

(

+

>

of entropy4.‘) with respect to the

>O.

the weighted density of entropy productior#)

(2.4) is written as

452

H. HOFF

8=

j.VT
(2.5)

In the case of one dimensional example

-integration

steady

over the volume

state yields

transport

-occurring

that the weighted

in a rod for production

of

entropy 8= J(T,AT)AT
(2.6)

for any value

of T, AT. It should

be mentioned

that the production

of entropy or the weighted production of entropy can be determined completely by the data which can be measured at the surface. Thus information about the inner structure of the system is not needed. This is a consequence of the assumption of steady state. Now we discuss the consequences of spatial symmetry. The response of any given system is related to its inner structure by the theorem of Neumann’): it has a higher or the same degree of symmetry as its inner structure. As we are dealing here with one dimensional transport, we can only discuss the two cases of an asymmetrical and of a symmetrical structure. In the latter case the transport has to be symmetrical, i.e. J is an odd function of AT, or L,(T)

= 0

(k = 2,4,.

. . even,

all T).

(2.7)

This has to be due to the special choice of T as a reference state, which is invariant with respect to interchanging T’, T’17). Asymmetrical transport (1.5) where L,(T) (k even) does not vanish, cannot be due to an asymmetrical structure alone, as Neumann’s theorem expresses only a necessary condition. In other words: there are systems of an asymmetrical structure, which reveal a symmetrical transport behaviour. 2.2. Consequences of the stability condition Before

investigating

the stability

we want

to compare

the nonlinear

global

description (2.3) with the more familiar one of electrotechnics, where J, AJ are analogous to the electric current and to the potential difference, respectively. The dependence of the heat flow on the “reference state” T, however, has no analogy in electrotechnics. Thus we cannot apply the well known condition of stability that the electric current is a monotonous increasing function of the potential difference by translating it into the analogies of the heat conduction. This kind of stability condition would be incomplete, because of the dependence of the heat flow on 5: Thus we have to derive the stability condition of heat conduction. This leads to a special problem, which concerns the global

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453

description of steady state transport J(T, AT): the discussion of stability is achieved by comparing the steady with nonsteady states. The latter cases cannot be represented by J(F, AT), however, as they are beyond the possibilities of such a simple concept of the global description. Thus we have to restrict the comparison to the cases of quasisteady transport described by J(T(r), AT(t)), where the boundary conditions L?(t), AT(t) are weak functions in time. The steady and the quasisteady states are performed by an arrangement, which is illustrated schematically in fig. 1. The system consists of two phases (‘), (“) and of a nonequilibrium region (“‘) in between, the transport properties of which are discussed. A realization of these conditions requires large coefficients of conductivity L’, L”, which can be performed by the choice of appropriate materials and by a large cross sectional area of the phases of (‘), (“). The steady state is achieved by external sources, which produce a constant input flow J into the phase (“) and the same output flow from (‘), In the case where the steady state J = J” is not reached, the changes of T’(t), T”(t) are assumed to be so small that no heat is stored in the nonequilibrium region (“‘), which requires large capacities C’, C” compared with C”‘. These assumptions are just the same as those of the so-called discontinuous system (cf. the textbooks of irreversible thermodynamics4’5)). According to fig. 1 we have the following balance equations:

SJ = J -

J”

=

;(&f_

C,,j-f,).

(2.9)

Inserting (2.1) and c’=

&+AC,

C”=

c-+C

(2.10)

yields &-

AC ?A?, 4C

(2.11)

Fig. 1. Schematical illustration of a transport system at steady or quasisteady state. The capacities C’, C” and the coefficients of conductivity L.‘, L” are large compared to those of (“‘).

H. HOF’F

(2.12)

The steady

state (superscript

well known

conditions

00) is an “attractor”

of asymptotic

of the quasisteady

stabihty6)

are fulfilled,

ones, if the

i.e.

;{AT(r)_aT;)‘
(2.13)

6 AT(t) -$““*-

T”jkO.

(2.14)

dT(t) From

(2.12) and (2.13) we conclude

immediately

6J6AT
(2.15) 6J is carried

out,

which

yields

with

the aid of the constraint

a

-J(~“;AT~)-~~~J(1:~,AT~)
(2.16)

(2.16) is understood as a necessary and sufficient condition: the necessary condition is given by the case of equal capacities, i.e. AC = 0 of the phases (‘), (“):

&

J(T”,

AT”) < 0

i.e. J(T, AT) decreases instability

occurs

a -J(T”,AT”) aAT

for any T, AT,

monotonously

(2.17)

in AT. The sufficient

condition

is that no

for any /AC’/ < 2C, which yields ~J(T=,AT-)~

,

(2.18)

i.e. the dependence of J(T, AT) on T is not allowed to be too strong. The latter condition does not exist for electric transport, because the electric current depends only on the force- the potential differencebut does not depend on any kind of reference state. The second condition (2.14) is already included by (2.13) and the constraint (2.11).

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455

It should be stated that (2.17) is more restrictive than (2.6) which is an “integrated formulation” of the second law. Before violating (2.6) the system must become unstable.

3. Comparison

with Fourier’s law

In order to investigate the relation between the inner structure and the nonlinear global properties, we shall consider the following three cases: a) h(z): the conductivity depends on space explicitly because of inhomogeneity, i.e. the medium is composite - A(z) is discontinuous in this case -or its properties vary as a continuous function of space because of the reasons already mentioned. The dependence of A on temperature is neglected. b) A(T(z)): the conductivity depends on space implicitly because it depends on temperature. This corresponds to a homogeneous medium. c) A(T(z), z): the conductivity A depends on space implicitly and explicitly. This corresponds to an inhomogeneous medium (see case (a)), where the dependence on temperature is not neglected. According to the case of one dimensional transport the system is assumed to consist of a rod of length 1 and of a cross sectional area A. Its left and right boundaries are situated at z = - l/2 and z = + l/2, respectively. 3.1. Inhomogeneous

system

with temperature

independent

conductivity

This case corresponds to the above mentioned kind of system, where the dependence of A on T is neglected, because it is too weak or because the system is close to equilibrium where T’ = T”. According to (1.4) the equation of heat conduction is written as cpz=/\VZT+VA*VT=O

(steadystate).

(3.1)

Because of the linearity of (3.1) the solution of the one dimensional case is easy to obtain for the steady state heat transport. Nevertheless we want to mention it, because the results are fairly instructive with respect to the global properties.

g

=

-r(z)j, l/Z,

T(t)

= T(0) - j [ r(r’)dz’

.

(3.2)

0

denotes the local “thermal resistivity”, which exists always because of (2.4). The global properties J(T AT) are given by r(z)

H. HOFF

456

A

J=-

I

(3.3)

AT,

112

r(z)dz

L i.e. J does not depend

on T and is linear

in AT. The same result

would

have

been obtained for a homogeneous medium with a constant thermal conductivity A. Thus the coefficient of conductivity L does not reveal any information about

the inner

structure

of the system.

Eq. (3.3) excludes the case of asymmetrical transport structure is permitted to be asymmetrical, where r(z) # r(-z) This simple

6

Z s

result demonstrates

a necessary sufficient

(-l/2

condition

enough

of

the inner

U) .

(3.4)

clearly

that Neumann’s

asymmetrical

to cause an asymmetrical

3.2. Homogeneous

(1 S), although

transport. transport

theorem

expresses

Obviously

(3.4)

only is

not

behaviour.

system with temperature dependent conductivity

The dependence on F must be taken into account in those cases, where the difference of temperature is large. According to (1.4) the equation of heat conduction is extended as

cpg=AV2T+$CT*VT=0

The solution of (3.5) is more difficult z(T) is obtained by integration:

(steady

because

state).

of the nonlinearity.

(3.5)

The inverse

T

z(T)=

-;

A (r)dT I T(O)

(3.6)

with the aid of the boundary conditions z(T’) = l/2, z(T”) = --l/2, which are used to eliminate T(O), j. The r.h.s. integral is called “heat potential” ‘,6), which can be introduced to linearize (1.1) and (3.9, respectively. Determining T(z) from (3.6) requires inserting a certain A(T) in order to carry out the integration and to calculate the inverse function of z(T). The latter must exist by

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457

consequence of the second law (2.4). This law implies that for any given j the function T(z) is monotonous. Thus the problem of calculating the inverse of z(T) has a solution for any arbitrary dependence of the heat conductivity on temperature. We renounce to treat a particular example because of brevity. The global properties described by J(T, AT) are yielded as i=+AT/Z

J=_4

1

h(T)

dT.

(3.7)

?-AT/Z

In order to compare this representation with (2.3) the integral on the r.h.s. is expanded around AT = 0, which requires calculating the partial derivatives of J with respect to AT. The differentiation cancels the integration, which yields

(3.8) Higher order derivatives are obtained as

which yields

&J(T,O)= -$h(T)=-L,(F),

(3.10)

as --I(T,O)=--&-&L,(T) aAT” &J(tO)=O

(s=1,3,...,odd),

(s=2,4

,...,

even).

(3.11)

(3.12)

So the Taylor series is written as

J(T,AT)=

-

c Ll&k,(T)ATk. k=lI.3 _,(2k-’ k! dTk-’

(3.13)

L,(T)

According

to Neumann’s theorem

(3.13) includes only odd powers of AT In

458

H. HOFF

contrast

to the previous

?‘. Nonlinear coefficient

terms

L(T)

This

be

system

is the most

the transport

predicted

on the reference

3.3. Inhomogeneous

cording

subsection can

from

general

on the reference

dependence

case and-of

dependent course-

of heat conduction

aT c~dt=hV2T+~VT.VT+V,,,h.VT=0

linear

conductivity

the most is extended

difficult

one.

Ac-

as

(steadystate).

In the case of a composite medium, number of homogeneous parts, where

(3.14)

i.e. a medium consisting A is discontinuous in z:

of a certain

A,(T)N2s.z sP,, ; A,(T)P,_, s z s l/2.

A(T,z)=

state

of the

state.

with temperature

to (1.4) the equation

depends the

(3. IS)

T(z) is yielded as a solution of n problems example, that T”, j are given as boundary immediately

of the kind (3.6): suppose for conditions. From (3.6) follows

(3.16)

UPI)

p,=-l

hi(r)dr

- ; ,

(3.17)

i T” T

zZ(T) = P, - + 1 J

A*(r)dr

(P, s z2 s P2),

(3.18)

W’I)

T(h)

P2 = P, - 1 i

I W’I)

(3.19)

&(T)dT

and so on. The whole system of equations

function

T(z)

is obtained

by calculating

successively

the

HEAT CONDUCTION

IN INHOMOGENEOUS

T

z,(T)=

l--1-,!1

Ai(T)dT,

p,=Pi_,-i 1 j

Ai(

W’-I)

Pi_lGz;-(zJi).

,...,II;

459

W’i)

r(P)-I) (i=

MATERIALS

(3.20)

In the case where the boundary conditions are expressed by T’, T”, we have to replace j by W’I) j=-l

T’

&(r)dr + . . . + 1 if T”

j-

WW)

(3.21)

3

TV-I)

which requires resolving the whole system of coupled equations (3.20) simultaneously. Anyway the solution T(z) exists because of the same reason (2.5) as in the previous subsection. This method, however, will fail in the case of a continuous inhomogeneous medium described by a thermal conductivity h(T, z), which is continuous in z. Thus another method is preferred yielding more elucidating results with respect to the global behaviour. From (1.1) follows immediately the integral equation

T(z)= T(0)- f j-r(T(z'), z’)dz’ ,

which is resolved by the application of perturbation expanded around T:

(3.22)

theory. Thus r(T(z), z) is

r(T(~),z)=r(%r)t$r(T,r)+~$r(T,z)(T(z)-f)’+.-..

Inserting the first term of the r.h.s. of (3.23) into the integral equation yields the first order approaches

(3.23)

(3.22)

I

T”‘(z)

= T(0) - $ / r( F, z’)dz’ ,

(3.24)

l/2

r( F, z)dt

.

(3.25)

460

H. HOFF

In order

to determine

the second

term

the second

of the series

have to calculate

order

phenomena,

which

(3.23) into the integral

the first order

approach

equation

requires (3.22)

inserting we first

of (3.26)

f/2)) .

T”‘(2) - T(‘) = T”‘(2) - ; (T”‘( l/2) + P’(-

Thus the second order approach is determined and so on. We do not write out the tedious calculations because of brevity. Anyway the total procedure defines an algorithm which permits calculating successively the higher order approaches. J= 0:

Summing

AT=

-

2

up these

approaches

yields

an expansion

of AT

R,(T)Jk,

around

(3.27)

k=l...

which is inverse

to (2.3). The R-coefficients

are given by

R,@)= ; j- r(T, z)dz,

(3.28)

R,(T)= + j- ; r(T;z)B(z)dz,

(3.29)

-//2

f/2

R,(T)=;

where

B(z),

j-

C(z)

are defined

2

C(Z)= - j 0

as l/2

I

B(z) = - j-

(3.30)

(~r(%z).C(z)+f~r(~z).B(l)‘)dz,

r(T, z’)dz’+

; 1 (4% Y) - r(z

-y))dy,

(3.31)

i/2

s r(z z’kJz’+ i 1 (2 r(%

y)B(y)

-

2

r(T, -y)B(-y))dy

.

0

(3.32)

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461

3.4. Discussion and comparison with the numerical solution of the pseudolinear equation of heat conduction

First we want to demonstrate

the stability: from (1.1) we conclude

f+ATIZ

(3.33) t-AT/2

aJ -=--_

A

aAT

21

aJ -=

a7

--

A

(3.34)

(3.35)

1

The conditions of stability (2.17) and (2.18) are obviously fulfilled, as could be expected. Thus the condition of stability cannot be considered as being a restriction of h(T, z) or L,(F) except that these quantities have to be positive. The local formulation of the second law (2.4) and the pseudolinear ansatz (1.1) do not permit that pseudolinear systems ever can reach a limit of stability. Additionally (3.34) expresses that J is monotonous in AT. Hence the polynomials (3.26) and (2.3) are equivalent to each other, which permits comparing their coefficients. This can be applied as a first control, whether the calculations yielding the R-coefficients are correct. As the homogeneous system is a special case of an inhomogeneous one L,(T), L3(F) should agree with R,(F), &(F), if r(T, z) is assumed to be constant with respect to the explicit dependence on t. In this case the integrals of (3.28) and (3.30) can easily be evaluated. In order to avoid boring the reader with dull calculations, we only say that this criterion is fulfilled. As a second criterion, we apply Neumann’s theorem, which implies that there is no asymmetrical transport, if the inner structure is assumed to be symmetrical. Indeed B(z) turns out to be odd in z as well as the total integrand of (3.29), if r(T, z) = r(T, -2) is inserted. Thus R2(F) vanishes, as it should be. Additionally R,(T) (3.29) permits understanding even the “sufficient” condition of asymmetrical transport, which cannot be concluded from Neumann’s theorem as the latter expresses only is a necessary condition. This “sufficient” condition, however, is plausible indeed: imagine a heat conducting rod consisting of two different materials, the left conductivity of which increases with increasing temperature, whereas the right one decreases. This composite rod will show a larger transport of heat, if the reservoir of the higher temperature is

462

H. HOFF

connected

with the left side of the rod, than in the opposite

independent

conductivities

demonstrated

cannot

produce

asymmetrical

case. Temperature

transport

of heat as is

by (3.3).

Finally calculated

it should be mentioned from the linear global

reference

state

T has been

This had to be expected, A(T),

which

really

new information.

that the nonlinear coefficient L,(T),

measured,

however,

is proportional

provided

since

to L,(T).

The relationship

the system

the total

Thus

L,(T)

between

global properties if its dependence

is homogeneous.

information

is included

(k = 3, . . .) cannot the linear

can be on the in

contain

and the nonlinear

global properties is lost in the case of an inhomogeneous system: the Z?,(T)coefficients (k > 1) cannot be calculated from R,(T). Thus the nonlinear global properties give some information about the inner structure or r(T, z), which could be important for those problems, which really require a black box description as for diffusion across membranes. Because of the integrations (3.28),

(3.29),

(3.30) the total

r(T, z) is hidden,

function

hence

it cannot

be

calculated from the global properties. The global properties R,(T), R,(T), R,(F) only could be used as a check, whether the system is homogeneous or whether any assumptions about the inner structure hold. Of course the nonlinearities caused by local pseudolinearity are small. In order to demonstrate their magnitude, we have calculated numerically two examples (Al,O, and the combination of Al,O,-BaO) by the well known relaxation method”‘). The values of A(T) have been taken from ref. 12 (the data of BaO have been extrapolated to the room temperature range). The results are plotted in figs. 2-5. The phenomenon of asymmetrical heat transport is found to amount to 20% (see fig. 5)-a small value but nevertheless measurable.

70

LO

60

80

K

-AT Fig. 2. Heat flow across a conducting rod of .&lzO~(length: 10 cm, cross sectional area 1 cm*) versus the difference of temperature AT at several average temperatures.

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Fig. 3. Temperature versus position curved in spite of the steady state.

IN INHOMOGENEOUS

z of a conducting

LO

20

MATERIALS

rod of AlrOs.

60

The shape

463

of temperature

is

K

80

Fig. 4. Heat flow across a combination of two conducting rods consisting of AlrOs and BaO (cross sectional area: 1 cm’, length of each rod: 5 cm) versus the difference of temperature at several average temperatures. The heat flow is larger if the piece of A1203 is connected with the bath of lower temperature than in the opposite orientation.

2

L

6

8

cm

--------c.Z Fig. 5. Temperature versus position I of the same arrangement as in fig. 4 for both orientations. Upper curve: the BaO rod is connected with the bath of the lower temperature.

H. HOFF

464

4. Final remarks The

results

modynamics instability It should

demonstrate is able

that

to describe

must be excluded, be interesting

the

so-called

certain

linear

nonlinear

nonequilibrium

properties.

Phenomena

therof

however. to ask for the consequences

of pseudolinearity

with

respect to the phenomena of coupling of various transport processes. An important question, which is associated with this problem, is, whether the reciprocity relations can be extended to the range of global nonlinearity. This problem has been investigated in ref. 3, but we do not agree with the result of that article that this extension would have no physical meaning (to be published).

Acknowledgements I want to thank Prof. Dr. G. Dickel, helpful discussions.

Dr. G. Hoehne

and Dipl. phys. P. Jung for

References 1) H.S. Carslaw and J.C. Jaeger, Conduction of Heat in Solids (Oxford Univ. Press, Oxford, 1959). 2) H. Hoff, Reactive Polymers 2 (1984) 143. 3) F. Sauer, Handbook of Physiology-Renal Physiology (American Society of Physiology, Washington, D.C., 1972). 4) R. Haase, Thermodynamics of Irreversible Processes (Addison-Wesley, Reading, Mass., 1969). 5) S.R. de Groot and P. Mazur, Non-equilibrium Thermodynamics (North-Holland, Amsterdam. 1962). 6) P. GlansdortI and I. Prigogine, Thermodynamic Theory of Structure, Stability and Fluctuations (Wiley-Interscience, London, 1971). 7) H. Hoff, J. Chem. Sot., Faraday Trans. 1, 77 (1981) 2325. 8) S. Bhagavantam, Crystal Symmetry and Physical Properties (Academic Press, London, 1966). 9) PI. Schneider, Conduction Heat Transfer (Addison-Wesley, Reading, Mass., 1957). 10) U. Grigull and H. Sandner, Waermeleitung (Springer, Berlin, 1979). 11) H. Hoff, Thermochimica Acta 69 (1983) 45. 12) Y. S. Touloukian, Thermophysical Properties of Matter (Plenum, New York, 1972).