Hierarchies of irreversible energy conve

J. theor. Biol. (1985) 115, 319-335

Hierarchies of Irreversible Energy Conversion Systems.
II. Network Derivation of Linear Transport Equations
LEONARDO PEUSNER

P.O. Box 380, York, Maine 03909, U.S.A.
(Received 2 April 1984)
A previously introduced network approach (Peusner, 1970, 1983) is shown
to yield known linear transport equations without using Onsager's theory.
Although these equations are well known, the technique demonstrates how
practical descriptions can be readily made to agree with Onsager's formalism and how the hybrid descriptions (mixed tensors) introduced in section
l lead to an explicit formulation in which each phenomenological matrix
(tensor) entry represents a measurable quantity such as mobility, electrical
resistance, etc.
1. Introduction

Onsager's theory postulates that there is a bilinear form, the dissipation
function qb
• =E x~,

(1)

which, in the steady state, yields the individual forces when partially
differentiated with respect to the individual conjugate flows,
(aO)

x~; 7

(2)

Therefore, the dissipation function can be looked upon as defining a
potential "surface" in flow space--although the argument can be turned
around so that the flows are obtained from the forces when the independent
variables are the forces.
Network Thermodynamics (NT) expanded the Onsager view by formally
showing that the macroscopic structure of non-equilibrium thermodynamics
was homologous to a family of networks which obey Kirchoff's laws
(Peusner, 1970; Oster, Perelson & Katchalsky, 1971). Moreover, it also
demonstrated that a reciprocal system such as Onsager's could be represented either by a connected system consisting of linear resistances or, alternatively, by resistances inter-connected by ideal, lossless energy conversion
devices (Peusner, 1970, 1983; Oster et al., 1971).

319
0022-5193/85/150319+ 17 $03.00/0

(~) 1985 Academic Press Inc. (London) Ltd

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PEUSNER

A previous communication (Peusner 1983, henceforward referred to as
(I)), presented in detail the behavior o f four different two force-two flow
transducers which are reproduced again in Fig. I, for the purpose o f
reference.

-

(o)

J1 H,,

-

J2
~4 ~/~

(c)

-

(b)

~'~ J~-

!_'
Xl

4

P'~J2

(d)

FIG. 1. The four basic energy transducers used in the text (from Peusner 1970, 1983). (a)
R representation of phenomenological two port equations using resistances and series controlled force sources; (b) L representation of the phenomenological equations in which force
controlled flow sources are placed in parallel with conductances; (c) hybrid H representation

in which the independent variables are Jl and Xz; (d) hybrid P network has independent
variables X~ and J2.
The four transducers were labelled R, L, H, P - - i n order to agree with
nomenclature previously used by engineers--and each of these consists of
two dissipative elements (conductances or resistances) and two controlled
sources (forces, flows or mixed).
To avoid unnecessary complications, (I) did not elaborate on the rigorous
mathematical foundation which allowed the use of these boxes as a complete
set for the two port description. For the record, we point out that R and
L are the fundamental covariant and contravariant metric tensors (Peusner,
1985); we shall defer discussing the tensorial characteristics of these boxes
until non-linear systems are considered in a subsequent paper. We shall

distinguish, however, the R and L boxes from the H and P boxes by
contrasting the topological connectivity properties of the first set with the
(partial) energy reversibility properties of the second set.
We indicated, in (I), that the reciprocity R~2 = R2j of the R boxes implies
that a connected resistive network ( T network) can be given which is
properly described in terms o f independent flows at the ports. Similarly,
the reciprocity L~2 = L21 of the L sets leads to a conductance (Tr) network
which is properly described in terms o f the port forces. Moreover, by
differentiating the dissipation function in the R network with respect to the
flows, we obtain the conjugate forces and by differentiating the dissipation

DERIVATION

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321

function in the L network with respect to the forces we obtain the conjugate
flows. These and similar considerations readily demonstrate that these two
networks are at the heart of the connected Onsager description: the dissipation function serves both as a potential and glue to relate (or messenger to
compare) interactive processes.
Onsager's theory stops there, because it does not consider mixed tensors.
The antisymmetry of the H and P boxes (H,2 = - H 2 , and P12 = - P 1 2 ) are
very different in nature, however; they do not lead to connectivity, but to
a certain degree of reversibility, as the controlled sources act as ideal
dissipation transformers of the type considered by Thomson in his treatment
of the Peltier effect (Thomson, 1874). While Onsagerian theorists discount
Thomson's efforts as being pseudo-irreversible, the mathematical basis of
both theories is the same when placed in terms of any procedure which
changes the frame of reference or keeps a bilinear form invariant (Jacobian
techniques, network transformations, tensorial equations, etc.).
The practical difference is the following: while the Onsager--or function
of state--technique gives the elegant, potential theory view, the description
based on the hybrid, or energy conversion boxes, yields the parameters

most accessible to physical measurement. Thus, if the basic nature of the
process is understood in descriptive terms, and the forces and flows
measured at the outputs agree with actual phenomenology, the "proper"
forces and flows can be obtained easily by looking for antisymmetries--i.e.,
the P and H boxes.
This apparently trivial result has far reaching implications, because if the
practical equations of a linear system are known, it is not necessary to find
the dissipation function. The system is first represented by a transducing
machine and network substitutions are utilized to find a reciprocal or
antireciprocal engine. The network paradigm then assures us that the representation found is considered with Onsager thermodynamics. Network
theory then allows the definition of "proper" forces and flows without going
through the procedure of finding basic thermodynamic potentials and the
dissipation function whenever a new system is being analyzed.
These concepts will now be applied to derive several known transport
equations connected with the Onsager formalism from practical equations.
In particular, we shall consider Kedem-Katchalsky equations, electrical
phenomena and the steady state properties of such systems when two or
more membranes are attached.
A point of clarification is here in place. The validity of the Onsager
formalism and the linear assumptions have been questioned repeatedly (see,

for example, Mikulecky, 1977). The topological basis of the physical
assumptions involved will be rigorously dealt with elsewhere and will not

322

L. PEUSNER

be considered here. We are not attempting to either validate or disprove
Onsager's model, but simply to show how network transformations yield
the same formalism from practical observations. Of course, the final decision
regarding the validity of such a model will depend on each specific situation.

2. Kedem-Katchalsky Equations
The well known set of transport equations proposed by Kedem &
Katachalsky (1963a, b) which describe the interaction of water and solute
flows in a membrane are derived following the rules of Onsager thermodynamics:
(i) Find the dissipation function.
(ii) Transform the dissipation function to include proper practical forces
and flows.
(iii) Use the proper forces and flows to give macroscopic phenomenological equations. (These should be reciprocal, according to the

Onsager model.)
In general terms, the derivation proceeds as follows (Curran & Katchalsky, 1965). First, the dissipation function is written as

q~ = J,.AI~,,. + LAg,.

(3)

Introducing the explicit potential differences in dilute solutions, it is possible
to show that the dissipation function transforms to

dP = ( J~, Vw + J~ f/~) A P + ( J-~ - ff/.~,Jw) A'rr

(4)

in which t~. is the average concentration of solute in the membrane; A~r is
the ideal osmotic pressure, RTAc,, (,',. and 19~are the partial molar volumes
of water and solute, respectively; and Ap is the hydrostatic pressure difference across the membrane.
According to Onsager, the new dissipation function is also the product
of conjugate flows and forces. If we rewrite equation (2) in a "shorthand"
notation it can be abbreviated to

= .Iv AP + Jo AT"r,

(5)

in which
L = (JwVw+ L'~,),

(6)

and
jD = Js

c~

~"wJw.

(7)

DERIVATION OF LINEAR TRANSPORT EQUATIONS

323

Non-equilibrium thermodynamics says that Jo and JD a r e the flows conjugate
to the forces Ap and Art respectively, and the phenomenological equations
associated with the new dissipation, equation (3), are
J r = G A P + G D ATr

(8)

JD = LDp A P + LD ATr,

in which Lp (hydraulic coefficient), LD (dittusional coefficient), Lpo (ultrafiltration coefficient) and Lop (osmotic flow coefficient) are numbers that
depend on the given system and, according to Onsager's theory, L o p = Lpo.
Onsager's theory excludes--or, rather, ignores--the possibility of having
mixed sets of flows and forces like those of the H and P converters.
Moreover, the resultant equations involve Jo, which is an indicator of the
relative velocities of solute and solvent, but the actual measurable flow is
J , the solute flow. It is therefore convenient to put the Kedem-Katchalsky
equations in a form consistent with the formalism of irreversible thermodynamics which includes measurable quantities
Ap_Azr_gs(l-cr)2Lv+tojv
o:,Lp

Azr~

1 - tr

Cs

to

~-

1

(1 - o') j~
to

(9)

Jv +--2- Js,
toCs

in which ~r = Staverman's reflection coefficient, to = solute mobility in the
membrane, gs = average concentration of solute inside the membrane.
While it is simple to go from the Kedem-Katchalsky equations to Staverman's (by "collapsing" the equations), there is no way, within the formalism
of non-equilibrium thermodynamics, to go in the reverse direction. Using
Network Thermodynamics--in its basic idea that Onsager's theory is
equivalent to connected processes--it is possible to accomplish this step.
Moreover, a hybrid description which provides a more elegant physical
interpretation than those which can be extracted from the KedemKatchalsky equations is obtained in the process.

3. Derivation of Kedem-Katchalsky Equations from Practical Equations
Using Network Transformations
We shall now derive equation (5) from practical equations and the rules
of network thermodynamics. By simple algebraic manipulation the KedemKatchalsky equations (9) "collapse" to the practical set, known from

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L. P E U S N E R

experiment,

Jv = Lp( A P - crATr)
(10)
Js = es(l -cr)Jv+w ATr

in which the reciprocity of equations (8) has disappeared.
Equations (10) give the required practical information, as well as good
intuitive meaning for the physical processes involved. These equations are
from experiment, known independently of the thermodynamic formalism;
at first sight, they seem to have destroyed the reciprocity of equations (8)
and the information of "'proper" forces and flows--as given by Onsager's
dissipation. Surprisingly; network theory reveals this information is only buried
and can be retrieved.
To this end, we first represent the practical equations (9) by means o f
two disjoint networks N, and N: as shown in Fig. 2. Network NI expresses
the flow Jv by means of a "battery" (Ap--crATr) applied across the end
nodes of a resistance 1/L,, while network N2 expresses the total solute flow
Js as a sum of two flow terms ~s(1-cr)Jv and toA1rs. The first term in N2
is a controlled flow proportional to Jr, while the second term is a conductance of value w with an applied force Aus across its nodes.

+

( )es c1-,,I,,,,, E3

Z)Trs

FIG. 2. Practical representation o f the Staverman's equations.

We now proceed to show that the connected network derivable from the
two networks above lead to Kedem-Katchalsky equations in any of their
equivalent forms. The central idea consists in obtaining a reciprocal network
by using network transformations.
We can first split the force element or battery X = A p - ~ r A~r into two
force terms. This leads to the network shown in Fig. 3, which, by simple
deformation of the drawing can be changed into the network shown in
Fig. 4. The complete network then, can be given as shown in Fig. 5; this is

FIG. 3. Equivalent transformation of the input, in which the force sources have been split.

aAr

FIG. 4. Further input transformation in which the two force sources are separated.

FIG. 5. Equivalent input-output representation of the Staverman's equations. Note that the
network is not consistent with Onsager thermodynamics because the controlled sources are
not reciprocal.

326

L. P E U S N E R

clearly an H hybrid model having Jv and A~r as independent variables.
Note that the new network is neither reciprocal nor antireciprocal; and is,
therefore, not homologous with the Onsager model. To be consistent with
the Onsager formalism the matrix describing the H network should be
anti-symmetric (Peusner, 1970), which insures that a connected resistive
representation may be given.
We can achieve this in a m a n n e r consistent with network transformations
as follows:
1. Add the null force X = 0 to network NI (Fig. 6)--i.e. a battery which
is, for all practical purposes, not there.

v

x=O

FIG. 6. Adding a null force source at the lower branch does not change the network.

2. Express the null force element as a sum of a force ATr and its inverse
-Azr, as shown in Fig. 7. Transformations 1 and 2 leave the network
unaltered.

Z~

z3rr

FIG. 7. Representation of the null force by two opposing forces Art and -A~r.

3. We can now express the difference in force sources (trATr-ATr) as a
single force source

( ~ A ~ - A~') = (o'- I)A~
and thc forces A P and - A ~ by the single sourcc (AP-A~-). The resultant
network is now that shown in Fig. 8.

DERIVATION

OF

LINEAR

TRANSPORT

p-ATr

327

EQUATIONS

+
( a ' - I),~rr

FIG. 8. Application of the associative property to the forces.
4. The force (or- 1)A~r is now multiplied and divided by cs; this clearly
leaves the source invariant. The complete H network is that shown in
Fig. 9(a).
5. To conform to Onsager's formalism, the resultant H network must
have antisymmetric cross coefficients and identical independent variables
on both halves of the network. Given the consideration o f the previous
paper, these variables must be, for the hybrid system, .Iv and A~/cs, while
the dependent variables will be .Is and ( A P - A ~ r ) J v . If the output force

- - --.---~-o')Jv
(a)

I/Lp
+
4-

+

+

(b)

FIG. 9. Final steps in transforming the equations to a network reciprocal form---completely
equivalent to Kedem-Katchalsky's equations. Note that only network transformations were
used, Onsager's theory, while consistent with the network formalism, was not required.

328

L. P E U S N E R

source is divided by cs and the conductance to is multiplied by E's,the value
of the output flow Js will remain invariant.
The equivalent network now becomes that shown in Fig. 9(b) and is given
by the hybrid H matrix

r

l/Lp

- (1 - t r ) ~ ]
oJ~,
j"

H = L(l-o.)~s

(ll)

These equations are antisymmetric and therefore conform to the Onsager
model. Note that the resultant matrix (l l), derived directly from practical
equations, has as entries the simplest coefficients of the membrane description

as it separates the individual contributions of mobility, filtration, and average
concentrations. This result cannot be obtained, incidentally, using Onsager
thermodynamics.
The H matrix (ll) leads to the following practical definitions for the
membrane coefficients:

\

1":/~,:o

Jv

Js

es(1-o-)= ~vv a=/e,=o

\ A"-~/'---~/A,=o"

Although matrix H represents the hybrid equation:

L

J

it can readily be transformed to L or R coefficients using the results of the
conversion table (Table 1).
The corresponding L and R matrices are
L=

Lp
Lp(l-cr)e,

(1- o.)6,Lp ]
we~+(l-o.)2e2LeJ

(13)

o~+ ( l - o.)2esLp (o-__2-1
R=

Lpw
(o-- 1)

w
1

tO

O)C s

"

(14)

Clearly, the L and R matrices above are more involved than the simpler
hybrid matrix obtained by rearrangement of the practical equations, but
are in agreement with the Kedem-Katchalsky formalism.

DERIVATION

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329

EQUATIONS

TABLE 1

C o n v e r s i o n s a m o n g two port coefficients

To

[R]
R;t

From

[L]
L22
L

R12

det

[g]
R2)

R22

R22
det g

- RI2
det R

-RE!

R~I

det R

det R

[P]

- L~2
det L

det H
H22

- -

HI2
/-/22

__

1
Ptl

- Plz
Ptl

Lit

-H21
H22

1
H22

P21
PI I

detP
PI t

-L21
det L

[H]

det

L

1
L,~

L,2

~

- HI2
Hl l

det P
P22

Pt2
P2--2

L21

L22

--

H2~

detH

-Pzl

I

Htt

Hl z

P22

P22

H,l

Hi2

P2~
det P

-- PI2
det P

H21

Hz2

-P21
det P

Pll
det P

PR J

Pt 2

[L]

det R

[H]

--

RI2
R22

--

1
L I~

- LI2
L,i

-R21
R22

I
g2--~

Lzl
L~--~

detL
Li,

1
R~ i

-- RI2
R ii

det L
L22

--

LI2
L22

det

/-/22
H

- HI2
det H

R21
R~I

detR
R~

-L2i
L22

1
L22

-,/'/21
det H

Hll
det H

R22

--

[P]

4. Liquid Junction Potentials

In the case of motion of charged molecules in the presence of osmotic
and hydrostatic gradients, the dissipation function must include the additional electrical terms (Curran & Katchalsky, 1965)
q)=J~ A~'+ J~, A P + I E .
(15)
In the next two sections we utilize networks to derive proper equations for
junction potentials and electrokinetic phenomena without invoking the
dissipation function.
The practical equations describing linear liquid junction potentials for
the motion of a uni-univalent ionic compound may be given by
Js = coA~ + r i l / F

Kr Air
I . . . .

F ~,

(16)
~ KE

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L. P E U S N E R

in which J~ = flow of salt through the m e m b r a n e , I = ( J + + J - ) electrical
current density, r~ = transference number, K = m e m b r a n e conductivity, F =
Faraday constant, and ~o, Avr, ~ are the same as before.
The practical equations can be conveniently represented by the network
shown in Fig. 10, which has two flow sources controlled by Azr~ and /,

.

.L

us

FIG. 10. Practical representation of the phenomenological
equations.

-

liquid junction potential

respectively. In order to agree with one of the four standard network
representations, we replace the conductance/parallel flow source combination at the input by a series resistance/force source, as shown in Fig. II.

'

K~_

q\ F ] Cs

I

o~$

z~Tr$

-6

FIG. I I. Transformations leading to a reciprocal--i,e, Onsager consistent--network.

Note that this hybrid network is already antisymmetric, so that no further
transformations are required to agree with Onsager thermodynamics. The
resultant ( H ) matrix is

Y,

=

rL T,,'K
JF

]

~C-, JLAT'r/~', "

(17)

This representation has obvious advantages over those used in standard
treatments (e.g. Curran & Katchalsky, 1965) which always lead to L and

DERIVATION

OF

LINEAR

TRANSPORT

EQUATIONS

331

R matrices. The main advantage resides in the possibility of spelling out
what electrokinetic measurements should be performed to describe the basic
physical characteristics of the system with minimal effort
IlK =(EIl)a,~,/~,=O,

)

r,/F=(JJl)a.,/e,=O=-(~_--~E

(18)

k l . a W l C s / 1=0

and
Js

We can also express these coefficients, if desired, in terms of R coefficients
using the conversion table (Table 1)
R l l = ( H i IH22 - H ~ z H ~ I ) / H2z !

K

(.,e,/K ) + (~-~/F-')
COt

2

Ti

wF-G '

R,2 = R2, = H12/ H22 = - H 2 t / H22 = - r , / FO~w,

and
R22 = 1/ H22 -- 1/ tocs.
These transformations lead to the resistive formulations

A~T

=

"

I

ILLJ

(19)

which are in agreement with results obtained using Onsager's theory.
5. Electrokinetic Phenomena

Given the observation that the passage of electrical current leads to a
volume flow when two compartments are separated by a charged, porous
membrane, we can postulate the fo!!owing practical linear equations relating
volume flow to the hydrostatic pressure across the membrane and the cur ent
t before making any Onsager type assumptions
J~ = L p A P + ~ I

(20a)

332

L. P E U S N E R

Jv

I

FIG. 12. Practical representation of electro-osmosis.

in which fl is defined as the electro-osmotic permeability (Fig. 12)
fl=(JdI)AP=O,

and Lp is the hydraulic conductivity at zero current flow
L, = ( J J h P ) , = o .

If we accept the premise that Onsager's theory is valid, it follows that the
corresponding hybrid equation
E = -flAP+RI

(205)

will also hold--i.e, the presence of a hydrostatic pressure difference in the
absence of an electric current will lead to a potential difference across the
membrane. The main diagonal term, R, in (20b) is the resistance that will
be measured in the absence of a hydrostatic pressure by simply applying
potential difference and measuring the resultant electrical current. The
parameters described by (20a) and (20b) are in the P form. We can also
write a corresponding set o f H equations, of which we consider here only
the partial set
I = H2,J~, + H22E,

(21)

in which
H22

K

is defined as the conductivity in the absence of volume flow.
From equation (20a) and conversion Table 1 of (I) we obtain immediately
the R coefficients R i b Rl2 and R22
RI1 = I/ PIj = ILL,,,

R , . = - e, d P,, = - ~ / t . .

DERIVATION

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EQUATIONS

333

and
R22 = 1/H22 = IlK.
Moreover, because
R12 = Rzl = - H21/Hz2 = - H21/K
it follows that
H2, = ,SK / L v

and
Hlz-- - flK I Lp
(by the assumption of antisymmetry).
Given the complete R equations

=L-~I%

IlK

JLIJ

(22)

the remaining H coefficient, H,, follows from the conversion table (Table
l)
H,, = det R / R22 = ( 1 / % ) - ( f 1 2 K / % )
and the complete H matrix is given by
(23)

/3K/ Lp
An additional change in variables leads back to the P coefficients

(24)
The newly found P2z must be equal to R. This result can be put in a more
familiar term by defining the conductance K' measured at zero current flow
by
K

t

1
--

1
--

R

(tlK)-(~2IL.)

"

Since

~2t L. = R,2R2,1 R , ,

=

R2,UR, ,.

it follows that
1

....
K

1
#
K

R2d Rll,

(25)

334

L. PEUSNER

+

_

,

~p

_l

+

,/j.( + ~ a v

E

Lp

Y

FIG. 13. Resultant reciprocal network obtained by requiring reciprocity. This is consistent
with the presentation of Curran & Katchalsky (1965).

a well known result of Onsager thermodynamics (Fig. 13) (Curran &
Katchalsky, 1965).
One of the results of the analysis of energy conversion by means of
transducers (I) was that, for our Onsager system, the maximum value of
energy conversion efficiency for forward and backward processes is the
same. This leads to the result
P21H2! = PI2HI2,

which can be easily read from equations (23) and (24).
As a final check of the procedure, we can also write, using the conversion
formulaes o f Table l, the L equations for the system

t

_

p 1/.

ntL,]pap]

(26)

In the limit in which I = O,
E = -flAP,
so that

3v- (IIK)-(fl2/Lp)
( l l L p K ) - ( ~ 2 / L 2)

AP - A P
Lp '

in agreement with the most basic linear phenomenology for volume flow
in the presence of a hydrostatic pressure.
Discussion

It can be argued that the basic results obtained above are not new.
However, we have gained a large amount of information in this process.

DERIVATION

OF LINEAR TRANSPORT

EQUATIONS

335

First, the resultant hybrid equations--which cannot be obtained from NET
without introducing tensor algebra--contain the most relevant physical
quantities in each case. In addition, the resultant hybrid transducing steps
conserve dissipation, in the manner originally conceived by Thompson in
his theory of pseudoirreversible processes. Second, and foremost, the present
train of thought indicates how little information is necessary in order to
build Onsager consistent theories. Clearly, a breakdown in reciprocity will
lead to the breakdown of antisymmetry of H and P, as pointed out in (I).
l would like to thank my colleagues D. C. Mickulecky and B. Bunow for constructive criticism, as well as for the suggestion that these examples should precede more
advanced material which is to follow in later papers of this series.

Note Added in Proof

The flow source R2~Jl in Fig. l(a), and in Fig. 7 of Peusner (1983), should
be a force source.
REFERENCES
CURRAN, P. F. & KATCHALSKY, A. (1965). Non-Equilibrium Thermodynamics in Biophysics.
Cambridge, Massachusetts: Harvard University Press.
KEDEM, O. & KATCHALSKY, A. (1963a). Trans. Faraday Soc. 59, 1931.
KEDEM, O. & KATCHALSKY, A. (1983b). Trans. Faraday Soc. 59, 1941.
MIKULECKY, D. C. (1977). J. theor. Biol. 69, 511.
ONSAGER, L. (1931). Phys. Rev. 38, 2665.
ONSAGER, L. (1931). Phys. Rev. 37, 405.
OSTER, G. F., PERELSON, A. & KATCHALSKY, A. (1971). Nature 234, 393.
PEUSNER, L. (1970). PhD. Thesis: The Principles of Network Thermodynamics. Cambridge,
Massachusetts: Harvard University.
PEUSNER, L. (1983). J. theor. Biol. 102, 7.
PEUSNER, L. (1985). Z Chem. Phys. (in press).
THOMSON, W. (1874). F~'oc. R. Soc. Edin. 8, 325.