by W.D. Bauer ©, released 7.12.99, corrected 16.10.01, 06.03.02
The essence of physics consists in the
mapping of real processes to theoretical mathematical models which should
show quantitatively the same features as found in reality. By defining
a physical measurement the variables in the theory are connected to the
physical reality. If the concerning variables of the theory coincide quantitatively
with the real measurements then we say the theory is correct, comp. fig.1.
Acc. to Newton physics is obliged to the principle of "hypotheses non fingo", i.e. the implementation of physically (or psychologically) motivated principles as mathematical constraints of a theory should be avoided because this entrains a loss of generality and the danger of arising inconsistencies. Nevermind, sometimes it is done for tentative purposes, and sometimes this strategy even seems to yield correct predictions. However, if such heuristically motivated constraints cannot not be founded in a more general framework with time they survive sometimes as "natural laws" for centuries because their preliminary heuristic character is forgotten. They are vulgarized, overgeneralized, are combined with economic interests and exert a considerable suggestive blocking pressure on learning people.
"Natural laws" are not made by god but by a man fitting the mapping of a mathematical theory to the experimental reality using the term "natural law" as power amplifier for selling his products and opinions . In reality any physical description is a compromise between what we can see and what we want to see. And what we want to see is determined by our social, economic, moral, religious and instinct motivated prepolarisation which sets up our projections on that what we regard to be the reality.
Due to such psychologically motivated reasons misinterpretations oftenly apply especially to the fundamental laws of physics like the first and second law. Therefore, we start our article with very simple and banal mathematical ideas.
2. The generalized first law
2.1 Definition of a machine
We define a machine as a coherent volume of space which exchanges one or more different qualities of quantities with its environment (for instance mass, energy, angular momentum). The volume of the machine (or the system border) can change as well. We demand that the volume change and all exchange processes with environment can be done periodically. We may or may not know what happens in the inner of the machine. A simple scetch of a machine can be found in fig. 2. It is clear that this definition extends not only to conventional machines but to many physical processes as well, comp.[1,2]
2.2 The generalized first law
Physical processes are described normally by differential equations. These can have relaxing, exploding or periodic solutions. Otherwise, strange attractors or complete chaotic solutions are possible. It is clear that a machine has to be able to work periodically or -if we use a physical term- reproducible which means that the machine process can be done periodically.
The coordinate of the exchanging qualities has to be localized uniquely, i.e. in- or outside of the machine. This means: the qualities are deterministic, i.e. uniquely defined in space und time. The quantities can cross the border between the machine and environment. With these concepts in mind we can formulate "our" generalized first law
1) a machine, which works reproducible
2) an exchange of qualities Qi with the machine, which are additive and deterministic
then the generalized first law means:
After a cycle of the machine the in- and effluxes kQi of the qualities Qi are zero
It is clear that a first law balance holds
generally only after a cycle is performed completely. Sometimes special
cases exist where the balance is fulfilled even from point to point of
a cycle but it does not hold generally if only a part of the cycle is proceeded.
An example for this is liquid water under 4 degree Celsius. If we take
off heat from it by cooling it down it expands and performs work. In sum
we get heat dQ and work dW by doing this process from A to B. The energy
balance dQ +dW =0 is fulfilled only after a complete cycle is proceeded
and the substance comes back to the initial starting point of the cycle.
After many cycles the impression appears that the quantities are conserved because the mean of the total balance is conserved in the mean if time goes to infinity.
In order to apply the first law correctly we have to find out the range of validity of the above assumptions for each qualitity Qi. It is clear that the proposition can not cover quantum mechanically defined quantities because these are deterministic only for limit cases. Furthermore, it is critical to apply it to irreversible non-reproducible processes, because it is hardly possible to prove anything under irreproducible conditions.
If concreticized the qualities Qi can be energy, charge, mass, momentum and angular momentum. The applicability and the consequences of the above proposition will be discussed for each of these qualities in the next sections.
2.3. The first law applied for the quality
In classical physics the quality energy E is defined generally by the equation
1) Conservative systems
If we make concrete x and f with
All these examples seem to look quite trivial,
however if we change the meaning of the variables in examples of figures
4 we can easily find analog systems which appear to be perpetuum mobiles
of first kind, comp. tab.2 showing these analogies. For instance, if we
postulate a "strange" magnetic material coupling to a conservative magnetic
field then we can get running a magnetizable bowl on a circular track in
a magnetic field as shown in . Here, acc. to the theory field energy
tapped is changed into work even if no "influx of magnetic energy" can
be detected from outside. Similarly, if we have (electric or magnetic)
capacitive material properties, which change parametrically in time, analogous
"overunity" effects are principally possible as well.
In sum, we can conclude: Regardless, whether energy in- or effluxes of a machine can be found experimentally or not, a balance hold for every system under description. The "first law" is used as a mathematical tool to calculate and to describe cycles: This means, all machines including perpetuum mobiles of first kind fulfill the first law per definition !
2.4 The first law applied for the quality
It is quite trivial that mass is an additive quantity. It should be mentioned that -similarly for energy- this does not hold for relativistic cases because then mass and energy cannot considered separately. In this article such problems are not followed.
2.5 The first law applied for the quality
As well as mass charge is an additive quantity and this hold even for the theory of relativity.
2.6 The first law applied for the quality
Momentum is regarded as a conserved quantity in the common sense of any physicist. However, critical examination shows that it is appropriate to be more careful with this assumption. Generally, momentum is not conserved for any system, whose Lagrangian is dependent not only from a velocity coordinate but from a space coordinate as well. Momentum conservation means always a special case of the equation of motion, i.e.
2.7 The first law applied for the quality
of angular momentum
It is clear that analogous considerations like that of the last section should be possible for angular momentum. Consequently, it should exist an angular moment balance, an angular moment potential for cyclic processes for periodic machines. However, the author did not find balance equations of angular momentum in literature at first sight. The cause of that may be that the standard definition of angular momentum is not appropriate for a many particle system. The standard definition of angular momentum is
3. A cycle with fluids in a field
3.1 Previous works
The idea that cycles binary mixtures in fields may violate the second law was due to B. van Platen who became known as the inventor of adsorption refrigerator. A. Serogodsky made measurement on a similar cycle and claimed to have measured such violations. Calculations with a modern equation of state without fields did not confirm any claim of both [6,7]. These claims were tested out using two possible axiomatizations of thermodynamics. The first axiomatization was based on the conventional Sears-Kestin version of the second law which forbids per dogma any equation of state which give such "strange"solutions. The other "axiomatization" was based on the extremum principle of the potentials. The second possibility is only apparent but not really an axiomatization of thermodynamics because it can be motivated by a derivation of thermodynamics from mechanics . It has the advantage that it reduces the number of hypotheses. No additional "physical axiom" like second law based on experience or overgeneralization (dependent from your personal point of view) is necessary. The alternative axiomatization derives the second law as a consequence of the mathematics of the system. It allows the second law to reverse if fields are present as shown in . Acc. to [6,7], both axiomatizations yield no second law violation for calculation of a van Platen or Serogodsky cycle using a equation of state (EOS).
Therefore, in order to test out the van Platen claim completely it is necessary to include the dependence from the field whose influence can be significantly in the neighborhood of critical points. The idea is near to use space-dependent thermodynamics including fields to test out the equation of state behavior in fields as proposed indirectly already in van Platen's patent.
A formal similar problem (but more simple 2-variable system independent from any space coordinate) has been discussed recently by the author. This model of a polymer solution system describes phase transitions in an electric field. It is confirmed qualitatively in parts by experiments. Consequent application of this model predicts overunity behavior due to "irreversibilites in the other direction" which are possible if one accepts the our interpretation of second law.
By applying analogous considerations to other potential fields it is possible to enlarge the number of possible candidates for second law violations. In tab. 2 we give an overview of some possible fields which allow such cycles. In appendix 3 the second variation of the least action functional is derived for gravitation and centrifugal fields indicating the possibility 2nd law violating cycles.
3.2 General formalism of space dependent equilibrium thermodynamics including potential fields
3.2.1 General thermodynamic theory
It is well known that any static thermodynamic material property can be described generally by a potential like inner energy U(S,V,ni) or G(P,T,ni) chosen acc. to the problem under question.
If fields are taken into account, the thermodynamic equations become dependent from the (here one-dimensional) space coordinate r additionally . Then, the ansatz of inner energy U* in field can be written 
3.2.2 Phase equilibrium with field
Acc. to the standard Gibbs the variation the inner energy at a interface of two phases
In general, if a field is present, pressure gradients are generated which induce as well concentration gradients in the volume. Therefore, phase equilibria differ from point to point along the field direction. At each point r of the volume, however, the local equilibrium state and the rate constants can be determined acc. to usual thermodynamics without field, if P(r),T, and xi(r) are known. Due to the global coupling in the volume, the equilibria shift in sum relatively to the uniform state without field, especially if
1) the molecular weights of the components are high
2) high pressure differences arise in the fluid due to a high field which induces high differences in the activities of the components as well, or
3) the local different barometric pressures in the system are in the neighbourhood of a critical point and even slight variation of parameters of P generated by the field can produce big variations in activities and densities of the fluid.
Furthermore, phase equilibria can depend
from the shape of the vessel due to global coupling of the particle number
over the whole volume. The dependence of the pressure profile depends from
the form of the vessel i.e. the hydrostatic paradoxon fails to exist generally
for multicomponent systems.
3.3 The computation of thermodynamic equilibrium of a mixture in a field with a equation of state
In the following we will describe how the general theory of the last section can be made concrete by a numerical solution of a phase equilibrium. We will describe subsequently the different program modules (or numeric functions) which - if built together - allow to calculate the phase equilibrium in a field.
3.3.1 The Bender equation of state-
the subroutine EQOFSTATE
In order to get a good accuracy in our calculation we take a modern generalized Bender equation of state which allows to calculate analytically for all thermodynamical properties and which can be programmed economically using the Horner sceme[14,15]. The equation has the form
3.3.2 Calculation of phase equilibrium
- the subroutine NEWTONGAUSS
In order to calculate the phase equilibrium the equations of the Gibbs fundamental system have to be solved. The first equation equals pressure in vapor and liquid
3.3.3 The inverted equation of state
V(P,T,xi) - The subroutine V_PTX
The Bender equation of state is formulated analytically in the form P(v,T,xi) with v, T and xi as independent variables. For some purposes however (as we will see below) it is appropriate to have the same equation of state (EOS) in the form V(P,T,xi). If we want to know the thermodynamical state in dependence from P0, T and xi, the inversion is done numerically by solving the equation
3.3.4 The thermodynamic state dependent
from P and fi - the subroutine VX_PF
Because the inverted EOS is needed for the least square algorithm a subroutine has been to evaluate the EOS values dependent from P,T and xi. With T constant and skipped here such a set consists of equations for
3.3.5 The thermodynamic statedependent
from P and µi - the subroutine VTX_PMu
In order to calculate the compartment array of a fluid under a field it is appropriate to have the set of constitutive equations dependent from P and µi. Such a set consists of equations for
3.3.6 The volume array calculation in
an arbitrary potential field- the subroutine VOLUME (Footnote)
If a field is applied pressures and chemical potentials differ from point to point. The subroutine VOLUME calculates the profiles of pressure, concentration and chemical potentials along the space coordinate of the known field if in one point of the volume P and xi are known. Our program calculates only one-dimensional profiles in a tubular housing,
Generalization to more dimensions and other forms of housings would be easily possible but is not done here. For calculation the whole volume is divided into a array of m equal subvolumes dV(rj ) (at same potential) at the space coordinate rj. At one point rref called reference point we assume to know the pressure P(rref) and the concentration xiref . Then, all other points in the volume are interconnected to their neighbor compartment by the equations for (hydrostatic) pressure using g(rj-1)=-dV(rj-1)/dr
For concrete programming the formula µi=µi0(p+,T)+
ln(fi /p+) + Mi .V(r) has to be applied.
The subroutine sums up as well the total number Ni of each particle in the total volume because it is needed in the next subroutine. Therefore, the algorithm proceeds as follows :
1) Define the number M of volume
array cell, i.e. the partition of the volume .
2) Give pressure P0, temperature T0 and concentrations xi at one point rref called the reference point
3) Using the subroutine V_PX invert numerically the equation of state at rref and calculate v
4) Initialize the particle numbers Ni=0
5) FROM volume array cell J=1 TO M
6) Determine all interesting thermodynamic data at rj using EQOFSTATE, especially µi and dni(rj),
i.e. the number of particles of each sort i in this subsection dV(rj) of V
7) Ni=Ni+dNi(rj ) Add up the particle number in the compartment to find the total number of the array
8) Determine P and µi in the next adjacent volume section dVj+1 acc. to equations (27) + (28)
9) Invert numerically the equation of state at rj+1 and calculate v,T,xi using the subroutine VTX_PMu
10) NEXT J
11) Give out all calculated values
3.3.7 Volume array calculation at conserved
particle numbers - the subroutine N_PX
Because the total particle number is conserved in the most problems, the volume array in the field has to determined under the constraint Ni0=consti. Acc. to the last section 3.3.6 the total particle numbers Ni in the volume is calculated in the subroutine VOLUME . This numbers can be regarded as numerical functions of the starting values Pref and xiref in VOLUME , i.e.
3.3.8 Fitting generalized Bender equation
to empirical mixture data -the subroutine LEASTSQUARES
Because only few published unreliable fit parameter exist for mixtures calculated using the Bender equation, the fit constants must be calculated from empirical data. Therefore, a known least square fit algorithm  and  was implemented in a subroutine LEASTSQUARES. This sub-routine fits the parameter kij, and to empirical values. The input of this routine contains the starting values of the fit constants, experimental values, fixed parameters like material constants, allowed errors and switches used for economical programming. The output of this routine consists of the results of the iteration, i.e. the fit constants and furthermore the least squares sum values and errors documenting the performance of the iteration. For a critical discussion of the fit method and its results click here .
We used Argon-Methan data  to test
our algorithm. The choice of this system was due to wrong expectation disproved
in the course of the investigations and was continued then as lab rat for
study. For practical purposes under environment conditions, mixtures of
Nitrogen, Argon, SF4, SF6 as superheated component
and Propane, Butane, CO, CO2 as condensable component are possible.
(We omit here the classical refrigerants.)
First, we calculated appropriate fit constants to the data using the least square routine. For results click here ! With these model data we calculated the rotator cycle. We choosed as field free initial state a mixture with a molar ratio of Argon of 56%, 55 bar and 170 K. This is very near in the neighborhood of the critical point in the gaseous ara of the phase diagram, comp. fig.8 . The rotator cycle is shown in fig.9. The closed volume of the mixture is set under rotation. At 4000 RPM the total volume is split unsymmetrically into two halves by a tap. The split point is chosen arbitrarily at the point where the specific volume in the field equals about the specific volume without field. Then, both volumes are decelerated again to the field free situation. Fig. 10a)- c) shows the pressure, concentration and specific molar volume profiles versus radius in the centrifugal field each during acceleration (upper diagrams) and deceleration (lower diagrams). Fig. 11 shows the work diagram of the cycle calculated from the diagrams of Fig.10a)-c) using a tabel calculation program.
The orientation of the isothermal cycle indicates a mechanical energy loss. This can be seen if we write down the definition of work.
Appendix 1: The Poynting energy conservation reduced to a energy conversion formula
The Poynting formula is written
Appendix 2: Momentum conservation of a many particle system with internal central forces
Inner forces do not contribute to the total momentum of a many particle system if they are central forces between the particles which can be derived from a potential.
We assume that the force between the molecules i and j is a potential function dependent only from distance, i.e. V(ri-rj). The Hamiltonian for the whole system including inner forces is
If we regard the total force on the whole system we get
Appendix 3: Second variation of gravitational and centrifugal fields
A) Centrifugal field
For thermodynamic potentials including centrifugal fields holds
B) Gravitational field
For thermodynamic potentials including force fields like gravitation holds
Appendix 4: The degrees of freedom of a multicomponent mixture in a potential field
In order to calculate a phase equilibrium we consider a tubular volume filled with fluid with n components in a field V(r) with the coordinate r. As an approximation the volume is divided into m compartments for calculation purposes. In each compartment the equation of state is fulfilled locally with defined values P(v(rj),xi(rj), T(rj)) and µi(v(rj),xi(rj), T(rj)), comp. fig.A4.1(last pic) .
Numbers of equations:
1) The next equation shows that the pressure equation is linearly dependent from the chemical potential equations.
Therefore, between two adjacent compartments
exist each n equations. It is recommended to take only the equations of
chemical potential for calculation purposes in order to avoid integrals
in the algorithm. Therefore, n.(m-1) equations (32) and (33) exist between
the m compartments, indicated by the double arrows in equ. 47 !
2) Additionally we have n equations of mass conservation (see equs. 35), i.e.
Number of variables:
Because temperature is constant over the volume we have n.m unknown variables v(m) and xi(m) over the volume.
Therefore, because the numbers of variables is equal to the numbers of equations, the system is determined completely.
5: Calculation of the thermodynamic state of a real fluid with a Bender
equation of state
 W.H. Müller, W. Muschik J.Non-Equilib.
Thermodyn. Vol.8 (1983) p. 29 - 46
Bilanzgleichungen offener mehrkomponentiger Systeme I. Massen und Entropiebilanzen (in German)
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Bilanzgleichungen offener mehrkomponentiger Systeme II. Energie und Entropiebilanz (in German)
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Table 1: different realizations of the
Table 2: different possible and postulated
realizations of non-conservative force fields
|windmill||angle||"wind" field energy||mechanical energy|
|"strange" iron ball
in magnetic field
|path x||magnetic field energy||mechanical energy|
|parametric changing inductivity||time t||magnetic field energy||electric current|
|time t||electric field energy||electric current|
Table 3: Overview of possible additional
thermodynamic variables with potential extremum behaviour
- H dM
Maximum (if µ>0)
- P dE
Maximum (if >0)
with ( F=m.g(r) )
- F dr