Integral sentences and numerical comparative calculations for the validity of the dispersion model for air pollutants AUSTAL2000

Schenk, Rainer

doi:10.1186/s40068-020-00181-6

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Published: 16 October 2020

Integral sentences and numerical comparative calculations for the validity of the dispersion model for air pollutants AUSTAL2000

Rainer Schenk ORCID: orcid.org/0000-0003-3759-5792²^nAff1

Environmental Systems Research volume 9, Article number: 28 (2020) Cite this article

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Abstract

Background

The authors (Janicke and Janicke (2002). Development of a model-based assessment system for machine-related immission control. IB Janicke Dunum) developed an expansion model under the name AUSTAL2000. This becomes effective in the Federal Republic of Germany with the entry into force of TA Luft (BMU (2002) First general administrative regulation for the Federal Immission Control Act (technical instructions for keeping air TA air clean) from July 24, 2002. GMBL issue 25–29 S: 511–605) declared binding in 2002. Immediately after publication, the first doubts about the validity of the reference solutions are raised in individual cases. The author of this article, for example, is asked by senior employees of the immission control to express their opinions. However, questions regarding clarification in the engineering office Janicke in Dunum remain unanswered. In 2014, the author of this article was again questioned by interested environmental engineers about the validity of the reference solutions of the AUSTAL dispersion model. In the course of a clarification, the company WESTKALK, United Warstein Limestone Industry, later placed an order to develop expertise on this model development, Schenk (2014) Expertise on Austal 2000. Report on behalf of the United Warstein Limestone Industry, Westkalk Archives and IBS). The results of this expertise form the background of all publications on the criticism of Schenk’s AUSTAL expansion model. It is found that all reference solutions violate all main and conservation laws. Peculiar terms used spread confusion rather than enlightenment. For example, one confuses process engineering homogenization with diffusion. When homogenizing, one notices strange vibrations at the range limits, which cannot be explained further. It remains uncertain whether this is due to numerical instabilities. However, it is itself stated that in some cases the solutions cannot converge. The simulations should then be repeated with different input parameters. Concentrations are calculated inside AUSTAL. In this context, it is noteworthy that no publication by the AUSTAL authors specifies functional analysis, e.g. for stability, convergence and consistency. Concentrations are calculated inside closed buildings. It is explained that dust particles cannot “see” vertical walls and therefore want to pass through them. One calculates with “volume sources over the entire computing area”. However, such sources are unknown in the theory of modeling the spread of air pollutants. Deposition speeds are defined at will. 3D wind fields should be used for validation. The rigid rotation of a solid in the plane is actually used. You not only deliver yourself, but also all co-authors and official technical supporters of the comedy. Diffusion tensors are formulated without demonstrating that their coordinates have to comply with the laws of transformation and cannot be chosen arbitrarily. Constant concentration distributions only occur when there are no “external forces”. It is obviously not known that the relevant model equations are mass balances and not force equations. AUSTAL also claims to be able to perform non-stationary simulations. One pretends to have calculated time series. However, it is not possible to find out in all reports which time-dependent analytical solution the algorithm could have been validated with. A three-dimensional control room is described, but only zero and one-dimensional solutions are given. All reference examples with “volume source distributed over the entire computing area” turn out to be useless trivial cases. The AUSTAL authors believe that “a linear combination of two wind fields results in a valid wind field”. Obviously, one does not know that wind fields are only described by second-degree momentum equations, which excludes any linear combinations. It is claimed that Berljand profiles have been recalculated. In fact, one doesn’t care about three-dimensional concentration distributions. On the one hand, non-stationary tasks are described, but only stationary solutions are discussed. In another reference, non-stationary solutions are explained in reverse, but only stationary model equations are considered. Further contradictions can be found in the original literature by the AUSTAL authors. The public is misled. The aim of the present work is to untangle the absent-mindedness of the AUSTAL authors by means of mathematics and mechanics, to collect, to order and to systematize the information. This specifies the relevant tasks for the derivation of stationary and non-stationary reference solutions. They can be compared to the solutions of the AUSTAL authors. These results should make it possible to make clear conclusions about the validity of the AUSTAL model.

Results

Using the example of deriving reference solutions for spreading, sedimentation and deposition, the author of this work describes the necessary mathematical and physical principles. This includes the differential equations for stationary and non-stationary tasks as well as the relevant initial and boundary conditions. The valid initial boundary value task is explained. The correct solutions are given and compared to the wrong algorithms of the AUSTAL authors. In order to check the validity of the main and conservation laws, integral equations are developed, which are subsequently applied to all solutions. Numerical comparative calculations are used to check non-stationary solutions, for which an algorithm is independently developed. The analogy to the impulse, heat and mass transport is also used to analyze the reference solutions of the AUSTAL authors. If one follows this analogy, all reference solutions by the AUSTAL authors comparatively violate Newton’s 3rd axiom. As a result, the author of this article comes to the conclusion that all reference solutions by the AUSTAL authors violate the mass conservation law. Earlier statements on this are confirmed and substantiated further. All applications with “volume source distributed over the entire computing area” turn out to be useless zero-dimensional trivial cases. The information provided by the AUSTAL authors on non-stationary solutions has not been documented throughout. The authors of AUSTAL have readers puzzled about why, for example, the stationary solution should have set in after 10 days for each reference case. It turns out that no non-stationary calculations could be carried out at all. In order to gain in-depth knowledge of the development of AUSTAL, the author of this article deals with his life story. It begins according to (Axenfeld et al. (1984) Development of a model for the calculation of dust precipitation. Environmental research plan of the Federal Minister of the Interior for Air Pollution Control, research report 104 02 562, Dornier System GmbH Friedrichshafen, on behalf of the Federal Environment Agency), according to which one is under deposition loss and not Storage understands. In the end, the AUSTAL authors take refuge in (Trukenmüller (2016) equivalence of the reference solutions from Schenk and Janicke. Treatise Umweltbundesamt Dessau-Rosslau S: 1–5) in incomprehensible evidence. How Trukenmüller gets more and more involved in contradictions can be found in (Trukenmüller (2017) Treatises of the Federal Environment Agency from February 10th, 2017 and March 23rd, 2017. Dessau-Rosslau S: 1–15).

Conclusion

The author of this article comes to the conclusion that the dispersion model for air pollutants AUSTAL is not validated. Dispersion calculations for sedimentation and depositions cannot be carried out with this model. The authors of AUSTAL have to demonstrate how one can recalculate nature experiments with a dispersion model that contradicts all valid principles. Applications important for health and safety, e.g. Security analyzes, hazard prevention plans and immission forecasts are to be checked with physically based model developments. Court decisions are also affected.

Background

By the authors Janicke et al. (2002) a dispersion model is developed under the name AUSTAL2000. In the Federal Republic of Germany, this became binding in 2002 when the Technical Instructions for Air Quality Control (TA Luft), BMU (2002), came into force. Other model developments have to prove their equivalence to the reference solutions of AUSTAL. Immediately after publication, individual employees of immission control and later also environmental engineers raise doubts about the validity of the reference solutions. For the purpose of clarification, the author of this article in 2014 was commissioned by the company WESTKALK, United Warstein Limestone Industry, to develop expertise on this expansion model according to Schenk (2014). The author of this article comes to the conclusion that all reference solutions from AUSTAL violate mass conservation and the second law of thermodynamics and are therefore not usable. The use of critical terms also leads to the conclusion that the AUSTAL authors are not very familiar with the theory of modeling the spread of air pollutants. The results of this expertise are published in Schenk (2015a). They form the background of all criticism. In Trukenmüller et al. (2015) is strongly contradicted. However, the authors of this publication are forced to publish the derivation of their reference solutions for the first time in 31 years. The development of the AUSTAL dispersion model is based on the work of Axenfeld et al. (1984). 31 years had passed until 2015. In the solution process of the reference solutions one refers to an alleged “usual Convention”, which could be found everywhere in “listed standard literature”. With this convention, which is later referred to as the Janicke Convention, the speed of deposition is mistakenly understood as a proportionality factor and not as a material constant. The following replica Schenk (2015b) demonstrates that the one described in Trukenmüller et al. (2015) specified algorithm is incorrect. The initial boundary value tasks responsible for spreading, sedimentation and deposition cannot be solved without contradiction. The authors resist again and claim in Trukenmüller (2016) that there is equivalence to the correct solutions described in Schenk (2015b). The author of this article is clearly against this claim. It is not credible that this claim can only be traced back to ignorance. It is more likely that one is pursuing an intention to deceive here, as will be understood later. For example, the claim that Venkatram et al. (1999) also proves to be devoid of purpose. The publication Schenk (2017) proves that it is solely an unfounded evidence. In Trukenmüller (2017) i.a. tried again to save Janicke’s Convention. One almost conjures up the author of this article that he should “… recognize the correct boundary condition, and this follows from the definition of the deposition speed”. It simply “… parameterizes the mass balance at the bottom of the model…”, which actually leads to a loss of mass, as was already the case in Axenfeld et al. (1984) must admit. “Worldwide, the dispersion models are based on the definition of the speed of deposition that is recognized in the literature”, you can read. However, studying literature has shown that the opposite is correct. You obviously only use the reputation of authorities to distract yourself from your ignorance. This allegation will also be justified later. Because of the demand for equivalence of other model developments to AUSTAL, non-university research is blocked rather than promoted. How should new model developments be able to demonstrate equivalence if the necessary reference solutions contradict all principles of mathematics and mechanics. The Schenk publication (2018a) shows which faults the demand for equivalence leads to. Not only is the AUSTAL dispersion model not validated. The authors of other model developments are forced to question their excellent algorithms, e.g. can be found in Schorling (2009). Finally, Schenk (2018b) proves, for example, that the authors of AUSTAL have compared the results of Venkatram et al. (1999) understand deposition as loss rather than storage. All incantations in Trukenmüller (2017) are questioned. At the request of authorities and other interested parties, the AUSTAL authors are currently spreading the Trukenmüller (2016) deception regarding the validity of the AUSTAL expansion model. They don’t care that this already contradicts Trukenmüller (2017). Because once Trukenmüller denies al al. (2015) the correctness of the solutions according to Schenk (2015a). And another time, Trukenmüller (2016) wants to demonstrate equivalence to it. The public is confused and misled. The aim of the present work is to untangle this embarrassment of the AUSTAL authors. For this purpose, all information provided by the AUSTAL authors in all available publications is collated, arranged and systematized. Optionally, stationary and non-stationary tasks are considered and the associated solutions are described. They can be compared to the solutions of the AUSTAL authors. Integral rates for mass balance and numerical comparative calculations are used for this. It turns out that all of Schenk’s criticism of the AUSTAL expansion model is justified and cannot be invalidated.

In Sect. “Methods and material” of this work an overview of the contents of the literature used is given. The author of this article studies past and current literature by the AUSTAL authors. The basic knowledge of mathematics and mechanics is described in textbooks and monographs. The fact that Trukenmüller (2016) is intended to deceive is deepened. The accusation that Trukenmüller (2017) tries to distract from one’s own ignorance and uses the reputation of other authors is justified. Section “Berljand’s boundary condition, initial boundary value problem and integral theorems” provides the mathematical and physical foundations for deriving, analyzing and evaluating AUSTAL’s reference solutions. This includes the derivation of the boundary conditions valid for spreading, sedimentation and deposition, the description of the relevant model equations as well as the development of integral sentences for the establishment of mass balances. A comparison of the contradictory solutions of the author of this article with the wrong algorithms is made in Sect. “Calculation of concentration, sedimentation and deposition for a one-dimensional spread of air pollutants”. This section also explains how the Janicke Convention was created and used. It is differentially emphasized that their use leads to a mass deficit. In Sect. “Reference solutions for dispersion, deposition, sedimentation and homogeneity” the contradictory and wrong solutions are optionally given for stationary and non-stationary considerations for all reference cases for dispersion, sedimentation, deposition and homogeneity. Their validity is checked using the developed integral rates. The reference solutions of the AUSTAL authors comparatively contradict Newton’s 3rd axiom. This statement is made in Sect. “The analogy to the impulse, heat and mass transfer”. How can it happen that the AUSTAL expansion model has been misleading the public from 1984 to the present? The author of this article deals with this question in Sect. “The life stories of the AUSTAL dispersion model”.

Methods and material

In the present case, it should be checked on the basis of generally valid integral sentences for each individual case of the reference solutions of the dispersion model AUSTAL whether the mass conservation law or the II. Law of thermodynamics are violated. It is also necessary to clarify how stationary and non-stationary calculations were carried out. For this purpose, numerical and analytical algorithms have to be developed and applied to the spread, sedimentation, deposition and homogeneity of the AUSTAL authors in each individual case. Mathematics and mechanics alone are the methods used for clarification.

Literature studies are required to get to know the mathematics and mechanics of the AUSTAL dispersion model.

The work of von Axenfeld et al. (1984) must be studied. In cooperation with the first author of AUSTAL, Janicke, a model for calculating the dust precipitation is developed. The so-called Janicke Convention, which can be explained later in Sect. “Contradictory solution using the Janicke Convention according to Janicke (2002) and the difference to Berljand’s boundary condition” is already there in the developed algorithms used. The thought model used describes deposition as loss and not as preservation.

With the scientific manual according to the VDI Commission for Air Pollution Control (1988) one wants to refer to the work Axenfeld et al. (1984) establish a new propagation theory.

The reference solutions and graphics belonging to the tasks for dispersion, sedimentation, deposition and homogeneity are explained in Janicke (2000).

With the intention of developing a national dispersion model, the model developed in 1984 for the calculation of dust precipitation in Janicke (2001) is further developed to the “mother model” LASAT.

The work Janicke (2002) describes tasks and tables for the calculation of dispersion, sedimentation, deposition and homogeneity.

The “Development of a model-based assessment system for immission control for companies” is described in Janicke et al. (2002) with the name AUSTAL2000 presented to the public. The BMU publication (2002) declares this model binding for all expert dispersion calculations. All other dispersion models have to prove their equivalence.

The work Trukenmüller et al. (2015) must be studied to get to know the derivation of the reference solutions declared binding for the first time. There the author of this article recognizes that all algorithms for this are wrong.

The reader has to laboriously collect the physical and mathematical foundations of model equations, tasks, solution algorithms, graphics and tables from seven publications individually. Other publications deal with applications and further developments at AUSTAL.

The publications Janicke (2009) and Janicke (2015) claim that the spread of radionuclides and aviation pollutants can be calculated. However, this would require non-stationary dispersion calculations, which AUSTAL is not able to do.

The author of this publication also studies Schorling (2009). With WinKFZ, the author develops an excellent model for calculating the spread of air pollutants, but it is discredited by court rulings because there is no equivalence to AUSTAL. The author subsequently wants to bring them about. However, it turns out that an approximate agreement can only be recognized visually. An actual equivalence cannot be inferred, since only unknown dimensionless pollutant concentrations are used. A clarification cannot be brought about. The author reckons with the superficiality of administrations rather than denying his excellent algorithms.

With the publications Trukenmüller (2016) one wants to achieve an equivalence to the correct reference solution according to Schenk (2018b). The author of this article looks at this publication and notes that it is simply a deception, as will be explained in more detail. The AUSTAL authors equate their wrong reference solution with the correct one. You get a simple algebraic equation and realize that there is no identity. You now rename variables and refer to the deposition rate $v_{d} \, \left[ {\text{m/s}} \right]$ of your wrong solution from now on $v_{d}^{Janicke} \, \left[ {\text{m/s}} \right]$. The algebraic equation is now solved after the second deposition rate $v_{d} \,$ of the correct solution. At the end of the invoice, it will be renamed $v_{d}^{Schenk} \, \left[ {\text{m/s}} \right]$. The accusation of an intention to deceive is well founded.

a.
According to Trukenmüller et al. (2015) is known that both solutions are different. With the intention of manipulation, they are still equated. Left and right of the algebraic equation are the deposition velocities twice $v_{d}$.
b.
The own deposition speeds $v_{d}$ are renamed with the intention to pretend equivalence in $v_{d}^{Janicke}$. After the second deposition rate $v_{d}$ the algebraic equation is solved.
c.
At the end the second deposition speed $v_{d}$ is cleverly renamed to $v_{d}^{Schenk}$.

The accusation of deception is well founded. This castling can be studied in detail in Schenk (2017).

The fact that the difference between a numerical and analytical solution was still not understood in 2017 can be seen in Janicke et al. (2017) read. The heading shows that analytical methods are used for approximate solutions and numerical algorithms for exact solutions. The opposite is true.

The publication Trukenmüller (2017) describes a summary of the exchange of views held with the UBA regarding the validity of all reference solutions. Because the AUSTAL dispersion model is used in all areas of the economy, such as city and community planning, traffic planning, landscape design and also to avert danger, there is a high level of public interest in correctly carried out immission forecasts. For this reason, the public also has a right to be involved in discussions about the validity of this model development. There are no objections to publications on this. In the publication mentioned, those responsible for dispersion calculations according to TA Luft develop their thoughts on how they are responsible for promoting and accompanying model developments. In scientific discussions, however, they obviously rely more on the reputation of other well-known and valued authors than on their own competence. So you want to distract from your own ignorance. This wording is not very friendly. However, it is correct in every respect and affects not only the content but also the form of this publication. As far as the content is concerned, in connection with the definition of the deposition speed, one refers sequentially to authors such as Pasquill, Chamberlein, Berljand, Wiedensohler, Zhang, Slinn, Kumar, Cunningham, Monin, Kasanski, Bonka, Sehmen, Hodgson, Seinfeld, Pandis, Nicholson, Simpson and Travnikov. If one adds the work Trukenmüller (2016), the list is to be completed by the authors Venkatram and Pleim. Without a doubt, these authors have earned varying degrees of merit in the modeling of spreading, deposition and sedimentation and can point to an excellent reputation. However, they would definitely object if their research results on Trukenmüller (2017) were assumed to be equivalent. In the case of the first and the last of the authors cited, the ignorance of the AUSTAL authors can easily be demonstrated. Pasquill (1962), for example, is an excellent description of atmospheric diffusion, but in the relevant section “6.2 Deposition of airborne material” on 14 pages and 19 formulas, not a single statement can be found which indicates the violation of the Mass conservation and the Janicke’s Convention could justify. The ignorance of the AUSTAL authors is that they are unable to use the excellent physics described there to develop a suitable thought model that would be accessible to a contradictory mathematical description. In the last of the cases cited, the author of this work deals intensively with the publication Venkatram et al. (1999) in Schenk R (2018b). The ignorance of the AUSTAL authors is that they did not understand that the one in Venkatram et al. (1999) found connection between sedimentation and deposition is only applicable for the special case of a disappearing soil concentration, $c_{0} \left[ {\mu {\text{g/m}}^{3} } \right] = 0$, which consequently with $F_{c} = v_{d} \cdot c_{0} \equiv 0$ not only questions all dispersion calculations, but also all other explanations by the authors of the AUSTAL on the validity of the Janicke Convention. According to the authors of AUSTAL, $F_{c} \left[ {\mu g/(m^{2} \cdot s)} \right]$ means the total emission in the study area. It is also unlikely that the authors cited in the list believe that “… a column standing on the surface of the earth, which contains the material capable of deposition, runs empty through deposition”, as in Axenfeld et al. (1984) is claimed. Also in the work Simpson et al. (2012) and Travnikov et al. (2005) there is no indication with which one could conclude that the Janicke Convention is valid. The accusation of ignorance is well founded. With regard to form, the style and expression of Trukenmüller (2017) snub every German authority.

In UBA (2018) the authors of AUSTAL complacently describe their history of the AUSTAL expansion model.

The publications, research projects, papers and studies mentioned here form the material that was to be analyzed using the methods described.

Basic knowledge can be found in the literature references Albring (1961), Бepлянд (1975), Boŝnjakoviĉ (1971), Graedel et al. (1994), Gröber et al. (1955), Janenko (1968), Kneschke (1968), Naue (1967), Stephan et al. (1992), Schlichting (1964) and for, Schüle (1930), Truckenbrodt (1983) example also in Westphal (1959). These references are given to show that traditional mathematics and mechanics can be used as much as possible. Important physical basics and mathematical algorithms from AUSTAL are part of school knowledge.

External literature was also studied. For example, in Abas et al. (2019) brilliantly described that environmental protection is an international task. The calculation of cross-border pollutant flows allows a scientifically based cause analysis and promotes international cooperation. Cross-border pollutant flows can only be calculated using high-quality, scientifically based and validated dispersion models. The work by Schenk et al. (1979) and Schenk (1989) are of interest.

In the work Rafique et al. (2019) shows convincingly that population growth, energy policy and environmental protection are to be seen in a close connection. Political decisions cannot ignore this link. The development of the AUSTAL expansion model was also accompanied by political decisions.

If air quality monitoring is required, active measurement methods are often used. Using a pump, ambient air is drawn into the mini-volume collector (Mini-VS) and the dust contained in it is separated.

Results

Berljand’s boundary condition, initial boundary value problem and integral theorems

Boundary condition

The spread of air pollutants is described by the initial boundary value task of the impulse, heat and mass transport. This includes the differential equation of mass transport (1).

$$\frac{{\partial {\text{c}}}}{{\partial {\text{t}}}} + {\text{v}}_{{\text{i}}} \cdot \frac{{\partial {\text{c}}}}{{\partial {\text{x}}_{{\text{i}}} }} = \frac{\partial }{{\partial {\text{x}}_{{\text{i}}} }}\left( {K \cdot \frac{{\partial {\text{c}}}}{{\partial {\text{x}}_{{\text{i}}} }}} \right) + \dot{q}({\text{t}}),$$

(1)

which can be solved with suitable starting and boundary conditions. In this equation, $c\left[ {\mu g/m^{3} } \right]$ explain the concentration, $x_{i} \left[ m \right]$ the coordinates in the different spatial directions, $K\left[ {m^{2} /s} \right]$ the diffusion coefficient in the free atmosphere, $\dot{q}(t) \, \left[ {\mu g/(m^{3} \cdot s)} \right]$ the source term, $v_{i} \left[ {m/s} \right]$ the flow velocity and $t\left[ s \right]$ the time coordinate.

In the case of a one-dimensional and non-stationary propagation, the differential Eq. (2).

$$\frac{\partial c}{\partial t} - v_{s} \cdot \frac{\partial c}{\partial z} = K \cdot \frac{{\partial^{2} c}}{{\partial z^{2} }} + \dot{q}(t)$$

(2)

is obtained according to Eq. (1), and in the stationary case if the source term $\dot{q}(t) = 0$ is missing, the relationship (3).

$$- v_{s} \cdot \frac{\partial c}{\partial z} = K \cdot \frac{{\partial^{2} c}}{{\partial z^{2} }}.$$

(3)

In these equations, besides the already known quantities $v_{s} \left[ {m/s} \right]$ means the sedimentation speed and $z\left[ m \right]$ the vertical position coordinate. With a view to later applications, it is negative. For further considerations, various simplifications are of interest for Eq. (1).

$$\frac{dc}{dt} = \dot{q}(t) \,$$

(4)

Equation 4 describes the simple further development of an equally distributed initial concentration $c_{A} \, \left[ {\mu {\text{g/m}}^{3} } \right]$, neglecting all convective and conductive material flows. This equation can be obtained, for example, if spatial concentration changes are not observed, $\partial /\partial x_{i} = 0$.

In the case of a time-independent source term $\dot{q}(t) = \dot{q} = const.$, the relationships of (5).

$${\text{c(t)}} = {\text{c}}_{\text{A}} + \int\limits_{0}^{{T}} {\dot{q}} \cdot dt\quad{\text{ c(t)}} = {\text{c}}_{\text{A}} + \dot{q} \cdot t$$

(5)

explain a linear increase in concentration as a solution of (4), where $T_{E} \left[ s \right]$ denotes the end of emission.

The boundary condition belonging to Eq. (1) is derived from the mass constancy at the control limits between atmosphere and soil. It is known as the Berljand boundary condition. The relationships to this are described in Fig. 1. All representations have been selected so that they can be applied directly to the study areas of the AUSTAL authors to derive the reference solutions. The ordinate $x_{i}$ is directed into the free atmosphere and the coordinate $x_{i}^{*} \left[ m \right]$ points from the depth of the earth towards the boundary. With $x_{i} (0)$ and $x_{i}^{*} \left[ T \right]$ the soil and atmosphere touch. In order to establish a relationship with the reference solutions of the AUSTAL authors, the coordinate notation $x_{3} = z$ and $x_{3}^{*} = z_{{}}^{*}$ is used for $i = 3$ below. This is how $\dot{m}^{A} = \dot{m}_{z}^{A} = \int {d\dot{m}_{z}^{A} = \dot{m}^{A} [\mu g/(m^{2} \cdot s)]}$ designates the conductive material flow in the free atmosphere and $\dot{m}_{{}}^{B} = \dot{m}_{z}^{B} = \int {d\dot{m}_{z}^{B} = \dot{m}^{B} [\mu g/(m^{2} \cdot s)]}$ in the depth of the earth. There is a surface source in the atmosphere. The pollutants emitted there move convectively and conductively towards the ground. The sedimentation flow $\dot{m}_{{}}^{S} [\mu g/(m^{2} \cdot s)]$ is calculated as the product of concentration and sedimentation rate, $\dot{m}_{{}}^{S} (z) = - c \cdot v_{s}$. The conductive material flows are represented as products between the diffusion coefficients and the concentration gradients, $\dot{m}_{{}}^{A} (z) = - K \cdot \partial c/\partial z$ and $\dot{m}_{{}}^{B} (z^{*} ) = - K_{B} \cdot \partial c/\partial z^{*}$. The $K_{B} \left[ {m^{2} /s} \right]$ is the diffusion coefficient in the soil. At the lower boundary of the study area, the identical conductive material flows are obtained for $z_{{}}^{*} = T$ and $z = 0$.

$$\dot{m}_{{}}^{A} (z = 0) = \dot{m}_{{}}^{B} (z^{*} = T)$$

(6)

$T\left[ m \right]$ means the depth in the ground.

The sedimentation rate in the soil itself is identical to zero. With $v_{s} = 0$ according to Eq. (3) one obtains the simple relationship $\partial^{2} c/\partial z^{*2} = 0$. The boundary conditions $c(z^{*} = T) = c_{0}$ and $c(z^{*} = 0) = c_{T}$ result in a linear concentration distribution in the soil, which is described by Eq. (7).

$$c = \frac{{c_{0} - c_{T} }}{T} \cdot z^{*} + c_{T} .$$

(7)

Under $c_{T} [\mu g/m^{3} ]$ is to be understood the concentration in great depth of the soil and under $c_{0} [\mu g/m^{3} ]$ the soil concentration. Because of the constant mass, the conductive material flows on the floor must be identical. This gives

$$\begin{aligned} K \cdot \frac{\partial c}{\partial z}(0) &= K_{B} \cdot \frac{\partial c}{{\partial z^{*} }}(T) = \frac{{K_{B} }}{T} \cdot (c_{0} - c_{T} ) \\ &\approx \frac{{K_{B} }}{T} \cdot c_{0} = v_{d} \cdot c_{0} \end{aligned}$$

(8)

and

$$v_{d} = \frac{{K_{B} }}{T}.$$

(9)

Equation (8) also gives the definition of the deposition rate, as can be seen from Eq. (9). Equation (8) assumes that the soil can absorb material capable of deposition without restriction, which is why one can set $c_{T} \approx 0$. Equation (8) finally gives Berljand’s boundary conditions (10).

$$K \cdot \frac{\partial c}{\partial z}\left( 0 \right) - v_{d} \cdot c_{0} = 0 .$$

(10)

It is identical to Eq. (11)

$$K \cdot \frac{\partial c}{{\partial x_{i} }}\left( 0 \right) - \beta_{i} \cdot c_{0} = 0,$$

(11)

as can be found in Бepлянд (1975). The mass transfer rate is to be understood under $\beta_{i} \left[ {m/s} \right]$. In the case of deposition, the deposition speed means $v_{d} = \beta_{3}$, which means $i = 3$ is the direction in which the pollutants are deposited.

The Berljand boundary condition is well known and is used in particular to describe the spread, deposition and sedimentation. This preferably affects the research area of the spread of air pollutants, but it is also not unknown in other disciplines, such as fluid mechanics, thermodynamics and process engineering, for the calculation of convective and conductive material flows.

Initial boundary value task

The boundary value task for one-dimensional spreading, sedimentation and deposition is described by the balance Eq. (2) and by the boundary condition (10). In the case of an initial boundary value task, the initial condition (12)

$$c\left( {x_{1} ,x_{2} ,x_{3} ,t = 0} \right) = c_{A} \left( {x_{1} ,x_{2} ,x_{3} ,} \right)$$

(12)

to be added. $x_{1,2,3} \left[ m \right]$ mean the three-dimensional spatial coordinates.

The closed initial boundary value task is therefore explained by the formula (13)

$$\begin{aligned} & \frac{\partial c}{\partial t} - v_{s} \cdot \frac{\partial c}{\partial z} = K \cdot \frac{{\partial^{2} c}}{{\partial z^{2} }} + \dot{q}(t) \\ & {\text{Model equation}} \\ & K \cdot \frac{\partial c}{\partial z}\left( 0 \right) - v_{d} \cdot c_{0} = 0 \\ & {\text{Boundary condition}} \\ & c\left( {x_{1} ,x_{2} ,x_{3} ,t = 0} \right) = c_{A} \left( {x_{1} ,x_{2} ,x_{3} ,} \right) \\ & {\text{Initial condition}} \\ \end{aligned}$$

(13)

Volume and area integrals

To prove mass conservation, one starts from the differential Eq. (1) and forms volume integrals according to Eq. (14)

$$\begin{aligned} & \int\limits_{V} {\frac{\partial c}{\partial t} \cdot dV} + \int\limits_{V} {v_{i} \cdot \frac{\partial c}{{\partial x_{i} }} \cdot dV} \\ &\quad= K \cdot \int\limits_{V} {\frac{{\partial^{2} c}}{{\partial x_{i} \partial x_{i} }}} \cdot dV + \int\limits_{V} {\dot{q}(t) \cdot dV} . \end{aligned}$$

(14)

Using the Gaussian theorem, volume integrals can be converted into surface integrals. This leads to the relationship (15) with two orbital integrals.

$$\int\limits_{V} {\frac{\partial c}{\partial t} \cdot dV} + \oint_{A} {v_{i} \cdot c} \cdot dA_{i} = K \cdot \oint_{A} {\frac{\partial c}{{\partial x_{i} }}} \cdot dA_{i} + \int\limits_{V} {\dot{q}(t) \cdot dV} .$$

(15)

In this equation, $A\left[ {m^{2} } \right]$ means the surface and $V\left[ {m^{3} } \right]$ the volume of the control area. Integration over the surface of the study area leads to Eq. (16).

$$\begin{aligned} \int\limits_{V} {\frac{\partial c}{\partial t} \cdot dV} + v_{i} (0) \cdot c(0) \cdot \int\limits_{U} {dA_{i} } + v_{i} (h) \cdot c(h) \cdot \int\limits_{O} {dA_{i} } . \hfill \quad \\ = K \cdot \frac{\partial c}{{\partial x_{i} }} ( 0 )\cdot \int\limits_{U} {dA_{i} } + K \cdot \frac{\partial c}{{\partial x_{i} }} ( {\text{h)}}\int\limits_{O} {dA_{i} } + \int\limits_{V} {\dot{q}(t) \cdot dV} \hfill \\ \end{aligned}$$

(16)

The integration can be carried out because the integrants are constant over the respective boundary surfaces below (U) and above (O). An integration over side surfaces can be dispensed with, since due to the lack of flow velocities and concentration levels, no mass transfer can take place. If you consider that all surface vectors are directed positively outwards, the scalar products can also be formed. This results in Eqs. (17).

$$\begin{aligned} &\int\limits_{V} {\frac{\partial c}{\partial t} \cdot dV} + v_{s} \cdot c_{0} \cdot A_{U} - v_{s} \cdot c_{h} \cdot A_{O} \\ &\quad= - K \cdot \frac{\partial c}{\partial z} ( 0 )\cdot A_{U} + K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} \cdot A_{O} + \int\limits_{V} {\dot{q}(t) \cdot dV} . \end{aligned}$$

(17)

Here, $A = A_{O} = A_{U} \left[ m^2 \right]$ mean the control areas at the top and bottom edges and $c_{h} \left[ {\mu g/m^{3} } \right] = c(z = h)$ the concentration at the top. $h\left[ m \right]$ is to be understood as the vertical extent of the study area. Also note that Eq. (18) is obtained.

$$\begin{aligned} & \int\limits_{0}^{h} {\frac{\partial c}{\partial t} \cdot dz} + v_{s} \cdot c_{0} - v_{s} \cdot c_{h} + v_{d} \cdot c_{0} \\ &\quad- K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} - \int\limits_{0}^{h} {\dot{q} \cdot dz = 0} . \end{aligned}$$

(18)

This equation can be used to check the validity of all reference solutions with regard to mass conservation.

For later considerations, Eq. (19)

$$Q = 1/A \cdot \int\limits_{V}^{{}} {\dot{q} \cdot dV} \cdot = \int\limits_{0}^{h} {\dot{q} \cdot dz}$$

(19)

of interest. The source term is to be understood as $Q\left[ {\mu g/(m^{2} \cdot s)} \right]$.

In the case of steady-state expansion, the mass balance (20)

$$v_{s} \cdot c_{0} - v_{s} \cdot c_{h} + v_{d} \cdot c_{0} - K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} = 0.$$

(20)

is obtained from a comparison between Eqs. (2) and (3) because of $\partial {\text{c/}}\partial {\text{t = 0}}$.

Calculation of concentration, sedimentation and deposition for a one- dimensional spread of air pollutants

Conflict-free solution using the Berljand boundary condition according to Schenk (2018b)

The correct solution of the differential Eq. (3) can be found in Schenk (2018b). It is described by Eqs. (21)

$$c\left( z \right) = c_{0} \cdot \frac{{v_{s} + v_{d} }}{{v_{s} }} \cdot \left[ {1 - \frac{{v_{d} }}{{v_{s} + v_{d} }} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot z} \right)} \right] \,$$

(21)

and (22). Equation (21) explains the course of the solution as a function of the deposition and sedimentation velocities $v_{d}$ and $v_{s}$, the height coordinate $z$, the diffusion coefficient $K$ and the soil concentration $c_{0}$, which can be determined using Eq. (22).

$$c_{0} = \frac{Q}{{(v_{s} + v_{d} )}} \, .$$

(22)

With known model parameters, concentration distributions, deposition and sedimentation flows as well as soil concentrations can be calculated.

For later use, Eq. (21) also gives the first derivative.

$$\frac{\partial c}{\partial z} = c_{0} \cdot \frac{{v_{d} }}{K} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot z} \right) \, .$$

(23)

and for $z = 0$.

$$\frac{\partial c}{\partial z}(0) = c_{0} \cdot \frac{{v_{d} }}{K} \, .$$

(24)

Equation (24) proves that solution (21) fulfills Berljand’s boundary condition (10). With this boundary condition one understands deposition storage and not loss.

Contradictory solution using Janicke’s Convention according to Janicke (2002) and the difference to the Berljand boundary condition

The incorrect solution is described in Trukenmüller et al. (2015) given by the relationships (25) and (26).

$$c(z) = c_{0} \cdot \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right) + \frac{{F_{c} }}{{v_{s} }} \cdot \left[ {1 - \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right)} \right],$$

(25)

and

$$F_{c} = c_{0} \cdot v_{d} .$$

(26)

The authors of AUSTAL use Eq. (25) to calculate the wrong concentration distribution, and Eq. (26) begins the confusion. First, $F_{c} \left[ {\mu g/(m^{2} \cdot s} \right]$ according to Eq. (26) has the meaning of a deposition and later again according to Eq. (30) that of a sedimentation stream. The authors of AUSTAL do not realize that both interpretations are wrong. In the end, you make a decision and mean according to VDI 3945 Sheet 3 (2000) and Janicke (2002) “the mass flow density deposited on the ground” according to Eq. (6) and Eq. (27).

$$\dot{m}_{{}}^{B} = F_{c} .$$

(27)

Equation (26) is used to calculate the soil concentration.

$$c_{0} = \frac{{F_{c} }}{{v_{d} }},$$

(28)

and does not care what happens if there is no deposition stream with $v_{d} \equiv 0$.

It is of interest to learn how to understand the Janicke Convention. In the course of the derivation of Eq. (25), the authors of AUSTAL receive the relationship (29).

$$F_{c} = K \cdot \frac{\partial c}{\partial z} + v_{s} \cdot c.$$

(29)

It results from the one-time integration of the differential Eq. (3), where $F_{c}$ has the meaning of an integration constant, which would have been determined using Berljand’s boundary condition. Instead, the authors of AUSTAL use a constant concentration distribution as a special solution for $F_{c}$ according to Eq. (30).

$$F_{c} = v_{s} \cdot c_{i} = const.$$

(30)

With the specification of this special solution $c_{i} \left[ {\mu g/m^{3} } \right]$ it can subsequently be seen from Eq. (31).

$$c_{i} = c(z) = c(0) = c_{0} = const.$$

(31)

that the concentration value of $c_{i}$ also means the soil concentration $c_{0}$. This gives the relationship (32).

$$F_{c} = v_{s} \cdot c_{0} .$$

(32)

Equations (25) and (30) can be used to prove the worthlessness of the solution function (25) according to Eq. (33).

$$\begin{aligned} c(z) = c_{0} \cdot \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right) + \frac{{c_{i} \cdot v_{s} }}{{v_{s} }} \cdot \left[ {1 - \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right)} \right] = \hfill \\ c_{0} \cdot \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right) + \frac{{c_{0} \cdot v_{s} }}{{v_{s} }} \cdot \left[ {1 - \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right)} \right] = c_{0} \hfill \\ \end{aligned}$$

(33)

already mentioned. This integral of the differential Eq. (3) cannot be used to perform simulations for determining concentration distributions. The AUSTAL authors recognize the uselessness of the special solution used (31). Instead of changing the solution method, for example, according to Kneschke (1968), they swap the sedimentation stream $v_{s} \cdot c_{0}$ with the deposition stream $v_{d} \cdot c_{0}$ without reason and refer to their self-written convention in VDI 3945 Part 3 (2000) and claim that it would be universal. Instead of Eq. (30), Eq. (26) $F_{c} = v_{d} \cdot c_{0}$ is used for no reason. After criticism, Trukenmüller (2017) assures that this castling would also be used by the authors Simpson et al. (2012) and Venkatram et al. (1999) used. “Worldwide, the dispersion models are based on the definition of the deposition speed that is recognized in the literature”, the AUSTAL authors in Trukenmüller (2017) affirm, but this is not confirmed.

One should know that sedimentation and deposition flows, $v_{s} \cdot c_{0}$ and $v_{d} \cdot c_{0}$, can be explained physically differently. They cannot be exchanged at will. Incidentally, the castling of $v_{s} \cdot c_{0}$ by $v_{d} \cdot c_{0}$ differentially violates the mass conservation rate, as was demonstrated in Schenk (2018b). With this poorly thought-out knowledge, the authors of AUSTAL finally obtained the wrong Janicke Convention (34) from Eqs. (23) and (29) for $z = 0$, which is used as a boundary condition.

$$F_{c} = K \cdot \frac{\partial c}{\partial z} + v_{s} \cdot c = v_{d} \cdot c_{0} .$$

(34)

Trukenmüller (2017) later asserts that Eq. (34) is the “true” definition of the deposition rate. It would represent deposition flows parameterized. If you add the two other definitions given initially in Trukenmüller (2016), it is now the third definition. One does not want to learn that the deposition speed $v_{d} = K_{B} /T$ according to Eq. (9) can be regarded as a material constant.

Here, too, the first derivatives of Eq. (25) are of interest,

$$\frac{\partial c}{\partial z} = \frac{1}{K} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot z} \right) \cdot (F_{c} - c_{0} \cdot v_{s} )$$

(35)

and for $z = 0$.

$$\frac{\partial c}{\partial z}(0) = \frac{1}{K} \cdot (F_{c} - c_{0} \cdot v_{s} )$$

(36)

for subsequent use.

Equation (36) proves that the solution (25) by the AUSTAL authors does not meet Berljand’s boundary condition (10). Taking Eq. (26) $F_{c} = c_{0} \cdot v_{d}$ into account, Eq. (36) is identical to the Janicke Convention.

The difference between Janicke’s Convention on Berljand’s boundary condition can be seen in the comparison of Eqs. (10) and (34). It is described with the formula (37). It can be seen that this convention results in a mass deficit of $- c_{0} \cdot v_{s}$ at the area boundary from atmosphere to ground.

$$\begin{aligned} & K \cdot \frac{\partial c}{\partial z}(0) - v_{d} \cdot c_{0} = 0 \\ & {\text{Berljand's boundary condition}} \\ & K \cdot \frac{\partial c}{\partial z}(0) - v_{d} \cdot c_{0} = - c_{0} \cdot v_{s} \\ & {\text{Janicke Convention}} \\ \end{aligned}$$

(37)

Reference solutions for dispersion, deposition, sedimentation and homogeneity

Assessment of the tasks

The authors of AUSTAL have failed to explain their tasks, model parameters and algorithms for deriving the reference solutions uniformly in a publication. The reader must collect all information about the task, the solution algorithms as well as numerical and graphical evaluation from various publications. With this confusion and trust in the authority of the administration, one can justify why in the past only a few critics have found themselves concerned with the theoretical foundations of AUSTAL. It is only after 31 years that Trukenmüller et al. (2015) read about the derivation of the reference solutions for the first time.

In this section, examples of “sedimentation without deposition”, ”Deposition with sedimentation”, and “homogeneity” are used to check all information provided by the AUSTAL authors for credibility and to show contradictions. The tasks explained by the authors of AUSTAL are only slightly different. A uniform three-dimensional control volume is considered, although it is only a matter of zero-dimensional and one-dimensional propagation processes. Time-dependent simulation results are given uniformly. In all cases, it is said that time series over 10 days were expected. The emission occurs only in the first hour of the first day, and the stationary solutions would have appeared after 10 days in all cases. Algorithms and graphics for non-stationary calculations are not described. The AUSTAL authors provide incorrect stationary solutions for all reference cases. Non-stationary calculations are not carried out at all, although simulation results are also given for this. In order to be able to provide credible evidence for this, stationary and non-stationary calculations are carried out for all case studies.

The first and second options distinguish between non-stationary and stationary bills. The correct solutions are compared to the wrong ones.

Sedimentation without deposition

Task

The task for the “sedimentation without deposition” propagation process according to Fig. 2 is taken from the literature reference Janicke (2002).

The model parameters and simulation results can be summarized.

a)
“The emission occurs only in the first hour of the first day”, which means $T_{E} = 3600$.
b)
The simulation is completed on the “10th day” with $t = 240{\text{ h}}$.
c)
One calculates a “time series over 10 days”.
d)
The size of the control volume is specified with the geometric lengths $L_{x} \left[ m \right] = 1000$, $L_{y} \left[ m \right] = 1000$ and $L_{z} \left[ m \right] = 200$.
e)
There is no “mass flow density forced by the source”, $F_{c} = 0$,
f)
“Volume source distributed over the entire computing area”.
g)
In the literature reference Janicke (2000) one learns for this case of spread that the mean concentration is $\bar{c}\left[ {\mu g/m^{3} } \right] = 500$.
h)
The sedimentation rate and the diffusion coefficient are $v_{s} = 0,01$ and $K = 1$.
i)
Because “A mass flow density enforced by the source” does not exist, $F_{c} = 0$, the deposition velocity $v_{d} = 0$ disappears, since only $c_{0} \ne 0$ can be valid for the soil concentration.

From the task described it appears authentically that one means a non-stationary propagation process, for which only the differential Eq. (2) is responsible. Regardless of this, the authors of AUSTAL assume in their solution process according to Eq. (3) a stationary propagation process. A solution (25) is also given for this. It is not known who should understand this.

Correct non-stationary and stationary solution taking into account the Berljand boundary according to Schenk (2018b)

First option, non-stationary consideration

In the case of a non-stationary consideration, the differential Eq. (2) with the initial condition (12) with $c_{A} = 0$ applies. The total emission $m_{E} \left[ {kg} \right]$ can be determined from the specified mean concentration $\overline{c}$. The geometric information d) for the size of the study area then gives the numerical expression (38).

$$\begin{aligned} m_{E} &= \overline{c} \cdot V = \overline{c} \cdot L_{x} \cdot L_{y} \cdot L_{z} \\ &= 500 \cdot 1000 \cdot 1000 \cdot 200 \cdot \frac{1}{{10^{9} }} = 100 \end{aligned}$$

(38)

With the specification a) the emission is ended according to $1{\text{ h}}$. This enables the source term $\dot{q} = const.\,{ 0} < {\text{t}} \le {\text{T}}_{\text{E}}$ of the differential Eq. (2) to be determined. Together with $V = L_{x} \cdot L_{y} \cdot L_{z} = 2 \cdot 10^{8}$, the numerical value can be given using Eq. (39).

$$\begin{aligned}& \dot{q} = \frac{{m_{E} }}{{V \cdot T_{E} }} = \frac{100}{{2 \cdot 10^{8} \cdot 3600}} \cdot 10^{9} = 0,139 \quad 0 \le t \le T_{E} \,\\ & \dot{q} = 0\quad{\text{ t}} > {\text{T}}_{\text{E}} . \end{aligned}$$

(39)

In addition, the specification f) must be taken into account that the “volume source is distributed over the entire computing area”, which means that there are no spatial concentration gradients, $\partial c/x_{i} = \partial c/z = 0$. This simplifies Eq. (2) to Eq. (4). A simple integration with the initial condition $c_{A} = 0$ gives the calculated value and Eq. (40).

$$\begin{aligned}& {\text{c(t)}} = {\text{c}}_{\text{A}} + \dot{q} \cdot t\\ &{\text{ c(T}}_{\text{E}} )= {\text{c}}_{\text{A}} + \dot{q} \cdot T_{E} = 0 + 0,139 \cdot 3600 \approx 500 . \end{aligned}$$

(40)

For $t > T_{E}$ and because of Eq. (39) as well as Eq. (4), $\dot{q} = 0$ and $dc/dt = 0$, this solution cannot be developed further. The concentration of $c = 500$ reached remains constant over time. Equation (40) describes zero-dimensional propagation with the time coordinate as the only independent variable.

The results are shown in graphs A and B in Fig. 3. The graphic A describes the time-dependent course of the filling in the interval $0 \le t \le T_{E}$ and for $t > T_{E}$. Graph B further explains that there is no vertical concentration gradient, $\partial c/z = 0$. The concentrations remain spatially and temporally unchangeable for all simulation times. The results prove that the statement b) by the AUSTAL authors that the steady state would only be reached after 10 days is not correct. The concentration value of $c = 500$ has already set after 1 h, $T_{E} = 3600$. According to c) it is said that a time series of 10 days was expected, which cannot be confirmed either. The trivial solution (40) is comparable to filling different containers with different media.

It would have to be proven that the solution fulfills the mass conservation law after the first option. The integral Eq. (18) is used for this. According to h), the sedimentation rate in the entire control room is $v_{s} = 0,01$. The concentrations are spatially constant at all simulation times, so that the identity $c_{0} (t) \equiv c_{h} (t)$ can be assumed. In addition, the integrals of Eq. (18) can be calculated with $\partial c/ \partial t = \dot{q}$.

According to task i) no deposition should take place, $v_{d} = 0$. Because of the spatially constant concentration, $\partial c/dz(h) = 0$ is also valid. The integral Eq. (41).

$$\begin{aligned} & \int\limits_{0}^{h} {\frac{\partial c}{\partial t} \cdot dz} &+ &\,\,v_{s} \cdot c_{0} \,& - &\,\,v_{s} \cdot c_{h} \, & + &\,\,v_{d} \cdot c_{0} \, &-&\,\, K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} &- &\,\,\int\limits_{0}^{h} {\dot{q} \cdot dz = 0}\\ & \dot{q} \cdot h & + &\,\,v_{s} \cdot c_{0} (t) &-&\,\, v_{s} \cdot c_{h} (t) \, &+ &\,\,0 \cdot c_{0} (t) \, &- &\,\,K \cdot 0 \,& - &\,\,\dot{q} \cdot h\quad \quad\equiv 0 \\ \end{aligned} ,$$

(41)

gives that the mass conservation law is fulfilled for all simulation times if the solutions are correct. Because of $\partial c/\partial z(z,t) \equiv 0$ according to Eq. (40) and Eq. (6), $\dot{m}_{{}}^{A} = \dot{m}_{{}}^{B}$, no deposition takes place according to Eq. (42)

$$\dot{m}_{{}}^{B} (t) = - K \cdot \frac{\partial c}{\partial z}(0,t) \equiv 0 .$$

(42)

The deposition stream $\dot{m}^{B}$ and the conductive material stream $- K \cdot \partial c/\partial z(0)$ are identical zero.They coincide according to amount and direction. The second law of thermodynamics is fulfilled.

Second option, stationary viewing

The second option alternatively considers a stationary propagation process. The corresponding correct stationary solution is described by Eqs. (21) and (22). After e) the task, there are no mass flow densities, which means $F_{c} = Q = 0$. According to this, no pollutant can be found in the study area, which is also confirmed by the trivial solutions (43).

$$c_{0} = \frac{Q}{{(v_{s} + v_{d} )}} \, = \frac{ 0}{(0,01 + 0)} = 0$$

(43)

and (44).

$$\begin{aligned} c\left( z \right) = c_{0} \cdot \frac{{v_{s} + v_{d} }}{{v_{s} }} \cdot \left[ {1 - \frac{{v_{d} }}{{v_{s} + v_{d} }} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot z} \right)} \right] \, = \hfill \\ 0\cdot \frac{0,01 + 0}{0,01} \cdot \left[ {1 - \frac{0}{(0,01 + 0)}\left( {\exp \left( { - \frac{0,01}{1} \cdot z} \right)} \right)} \right] = 0 \hfill \\ \end{aligned}$$

(44)

The result is shown in Fig. 3, graphic C.

For the sake of completeness alone, it should be proven that the mass conservation law is fulfilled. Equation (20) can be assumed.

$$\begin{aligned}& v_{s} \cdot c_{0} \, &-&\,\, v_{s} \cdot c_{h} \,& +&\,\, v_{d} \cdot c_{0} \,& -&\,\, K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} = 0\hfill \\ &0,01 \cdot 500& - &\,\,0,01 \cdot 500& +&\,\, 0 \cdot 500 \,& -&\,\, 1 \cdot 0 \quad \quad\equiv 0 \hfill \\ \end{aligned}$$

(45)

With the appropriate calculation parameters, mass conservation is guaranteed.

Because of $\partial c/\partial z(0) = 0$ and Eq. (44) and considering Eq. (6) $\dot{m}_{{}}^{A} = \dot{m}_{{}}^{B}$, Eq. (46).

$$\dot{m}^{B} = - K \cdot \frac{\partial c}{\partial z}(0) = - 1 \cdot 0 = 0$$

(46)

results. After that, there is no conductive mass transfer, $\dot{m}^{B} = 0$. After i) the task, no deposition should take place $F_{c} = 0$. Because of a missing potential gradient $\partial c/\partial z(0) = 0$, this does not take place either, $\dot{m}^{B} = - K \cdot \partial c/\partial z(0)$. The second law of thermodynamics is fulfilled.

Faulty non-stationary and stationary solution taking into account the Janicke Convention according to Trukenmüller et al. (2015)

First option, non-stationary consideration

First, a non-stationary view is assumed. After a) “The emission only occurs in the first h of the first day”, b) the stationary solution is reached on the “10th day” and c) a “time series over 10 days” is calculated, it is a non-stationary task. However, no solution algorithms and concentration profiles are described for this. For this reason, the integral Eqs. (18) and (20).

$$\begin{aligned} & \int\limits_{0}^{h} {\frac{\partial c}{\partial t} \cdot dz} + v_{s} \cdot c_{0} - v_{s} \cdot c_{h} + v_{d} \cdot c_{0} \\ &\quad- K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} - \int\limits_{0}^{h} {\dot{q} \cdot dz = 0} \end{aligned}$$

(47)

cannot be used. The authors of AUSTAL remain guilty of the answer, which is why a stationary concentration distribution should have set in after 10 days.

Second option, stationary viewing

The second option describes a stationary view. The authors of AUSTAL assume the stationary differential Eq. (3) and state the solution functions (25) and (26). First, the soil concentration would have to be calculated again according to Eq. (26). However, due to i), $F_{c} = 0$ and without deposition, $v_{d} = 0$, an indefinite expression is obtained for calculating the soil concentration $c_{0}$, $c_{0} = 0/0$. Equation (25) is simplified because of e) to the exponential function (48).

$$c(z) = c_{0} \cdot \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right).$$

(48)

Because the soil concentration $c_{0}$ cannot be calculated according to Eq. (26), a volume source is introduced without further ado after f). According tog), the pollutant particles with a concentration of $\overline{c} = 500$ are in a thermodynamic equilibrium. Speculatively, these are now redistributed so that they follow the exponential function (48). The second law of thermodynamics is already violated, because mass transport only takes place against the concentration gradient and not vice versa. You don’t necessarily have to have doctrine, such as according to Westphal (1959), page 265, cite that mass transport “… never by itself in the reverse sense” can be observed. The authors of AUSTAL reverse all basic knowledge to the contrary and calculate speculatively with Eq. (49) a soil concentration of $c_{0} = 1100,6$

$$\begin{aligned} c_{0} &= c(z = 5) = \bar{c} \cdot \frac{{v_{s} \cdot L_{z} }}{K} \\ &\quad\cdot \frac{1}{{\left[ {1 - \exp \left( { - \frac{{v_{s} \cdot L_{z} }}{K}} \right)} \right]}} \cdot \exp \left( { - \, \frac{{v_{s} }}{K} \cdot 5} \right) \\ &= 500 \cdot \frac{0,01 \cdot 200}{1} \cdot \frac{1}{{\left[ {1 - \exp \left( { - \frac{0,01 \cdot 200}{1}} \right)} \right]}} \\ &\quad\cdot \exp \left( { - \, \frac{0,01 \cdot 5}{1}} \right) = 1156,52 \\ &\quad\cdot \exp \left( { - 0,05} \right) = 1 1 0 0 , 6\\ \end{aligned} .$$

(49)

This concentration value can also be found in Fig. 2, column V.

The calculation Eq. (49) has been hidden for 31 years and is left to the public to solve this puzzling algorithm. Its development is described in Schenk (2018b).

The course of the solution to this is shown in Fig. 4. According to e) and i), no deposition should take place, but this contradicts the course of the solution function. Because of the negative concentration gradient, there is a conductive mass transfer on the ground towards the free atmosphere.

Equation (20) can be used to show that the physics of the authors of AUSTAL prevent mass conservation. The soil concentration is $c_{0} = 1100,6$, and the concentration at the upper boundary of the study area is calculated according to Eqs. (25) and (50), $c_{h} = 148,95$.

$$\begin{aligned} c_{h} & = c_{0} \cdot \exp \left( { - h \cdot \frac{{v_{s} }}{K}} \right) + \frac{{F_{c} }}{{v_{s} }} \\ &\quad\cdot \left[ {1 - \exp \left( { - h \cdot \frac{{v_{s} }}{K}} \right)} \right] \\ &= 1100,6 \cdot \exp \left( { - 200 \cdot \frac{0,01}{1}} \right) + \frac{0}{0,01} \\ &\quad\cdot \left[ {1 - \exp \left( { - 200 \cdot \frac{0,01}{1}} \right)} \right] = 148,95 \\ \end{aligned}$$

(50)

The concentration gradient at the upper limit is.

$$\begin{aligned} \frac{\partial c}{\partial z}(h) &= \frac{1}{K} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot h} \right) \cdot (F_{c} - c_{0} \cdot v_{s} ) \\ &= \frac{1}{1} \cdot \exp \left( { - \frac{0,01}{1} \cdot 200} \right) \\ &\quad\cdot (0 - 1100,6 \cdot 0,01) = - 1,48 \end{aligned}$$

(51)

to $\partial c/\partial z(h) = - 1,48$ according to Eqs. (35) and (51). With further parameters according to h) and i) of the task, the mass balance (52).

$$\begin{aligned}& v_{s} \cdot c_{0} \,& -&\,\, v_{s} \cdot c_{h} \,& +&\,\, v_{d} \cdot c_{0} \,& - &\,\, K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} = 0\hfill \\ &0,01 \cdot 1100,6 &- &\,\,0,01 \cdot 148,9 &+ &\,\,0 \cdot 1100,6 \, &+&\,\, 1 \cdot 1,489 \, \ne 0 \hfill \\ \end{aligned} .$$

(52)

is obtained with Eq. (20). The mass conservation law is violated.

Taking into account Eqs. (36) and (6), $\dot{m}_{{}}^{A} = \dot{m}_{{}}^{B}$, Eq. (53).

$$\begin{aligned} \dot{m}^{B} &= - K \cdot \frac{\partial c}{\partial z}(0) = - (F_{c} - c_{0} \cdot v_{s} )\\ & = - \left( {0 - 1100,6 \cdot 0,01} \right) = 11,006 , \end{aligned}$$

(53)

results for the calculation of the deposition current. Then there is a conductive mass transfer, $\dot{m}^{B} = 11,006$. However, the AUSTAL authors stipulate that no deposition should take place after e) and i), $F_{c} = 0$. This contradiction can only be clarified in such a way that one would have to assume that the diffusion coefficient would be identical to zero, $K = 0$ or, on the other hand, that despite an existing potential gradient, $\partial c/\partial z(0) \ne 0$, it would be contrary to $\dot{m}^{B} = - K \cdot \partial c/\partial z(0)$ no material flow take place. The first case is excluded because the diffusion coefficient is a substance parameter. The second case is applicable and justified, why the second law of thermodynamics is violated. If contradicted, then the pollutant particles would have to be contrary to the one in Häfner et al. (1992) described Fick’s law can be rearranged so that there would be $\partial c/\partial z(0) = 0$ on the ground.

Deposition with sedimentation

Task

Figure 5 describes the task for the spreading case “Deposition with sedimentation”. The input parameters are described by the following information a) to g). The task and parameters have been taken from the literature reference Janicke (Janicke (2002).

a)
“The emission is continuous at 1 g/s”.
b)
The simulation is completed on the “10th day” with $t = 240{\text{ h}}$.
c)
The size of the control volume is specified with the geometric lengths $L_{x} = 1000$, $L_{y} = 1000$ and $L_{z} = 200$.
d)
The sedimentation and deposition rate are $v_{d} = 0,05$ and $v_{s} = 0,05$. The diffusion coefficient is $K = 1$.
e)
There is a “mass flow density forced by the source”, $F_{c} = Q = 1$.
f)
The source is at an altitude of $h\left[ m \right] = 200$.
g)
One calculates a “time series over 10 days”.

According to f) the area source is at a height of $h = 200$. As described under e), the emission takes place through a “mass flow density forced by the source” with $F_{c} = 1$. According to b) and g) the task is again based on a non-stationary approach. In contrast, the AUSTAL authors only carry out stationary examinations. Algorithms and solution functions for non-stationary examinations are also unknown here. The AUSTAL authors relate their calculations to the validity of the differential Eq. (3) and use the wrong solution functions (25) and (26). In order to gain certainty about the validity of all approaches, non-stationary and stationary simulations are also carried out here and the results compared with Janicke’s solutions.

Correct non-stationary and stationary solution taking into account the Berljand boundary condition according to Schenk (2018b)

First option, non-stationary consideration

According to the first option, it is a non-stationary task, which is described by the differential Eqs. (2) with the initial condition (12) $c_{A} = 0$. Analytical solutions are not available for this, which is why the solution method according to Schenk (1980) was used here.

The results for this are shown in Fig. 6 with the graphs A and B. The deposition and sedimentation speed as well as the diffusion coefficient are specified according to d) the task. The source height is taken into account according to f) and is at a height of 200 m. At the lower boundary, the Berljand boundary condition according to Eq. (10) was fulfilled. At the upper limit, it was assumed that pollutant concentrations can no longer be measured at a sufficiently high level, $c_{H} = c(H) = 0$. So that the height of the source can be included with sufficient accuracy, the height of the study area was increased from to $H\left[ m \right] = 400$. In graph A, the temporal development of the concentration distribution in the interval $0 \le t \le T_{E}$ was evaluated. After a simulation time of $T_{E} = 2,6h$ the stationary solution is reached. The maximum concentration at source height is $c_{h} = c(200) \approx 20$ and the soil concentration is $c_{0} \approx 10$. The error deviation $\varepsilon = \left| {c_{An} - c(2,6h)} \right|/c_{An} \cdot 100\left[ \% \right]$ compared to analytical solutions is below $\varepsilon < 0,1$. The analytical solution is to be understood under $c_{An} \left[ {\mu g/m^{3} } \right]$.

High demands are placed on reference solutions. The mass consistency must be demonstrated for all simulation times. For this purpose, all required balance sheet quantities according to Eq. (18) must be carried along during the calculation. These include the production term $\int {\partial c/\partial t} \cdot dz$, the convective and conductive material flows at the boundary surfaces $v_{s} \cdot c_{0} , \, v_{s} \cdot c_{H} , \, v_{d} \cdot c_{0} {\text{ und K}} \cdot \partial {\text{c/}}\partial {\text{z(H)}}$ as well as the source term $Q = F_{c} = \int {\dot{q}} \cdot dz = 1$ according to Eqs. (19) and e). These terms were determined numerically as a function of time and their course is shown graphically in graphic B in Fig. 6.

The integral Eq. (54)

$$\begin{array} {llll} X) \, \int\limits_{0}^{H} {\frac{\partial c}{\partial t} \cdot dz} \, &+ \,\,\, v_{s} \cdot c_{0} \,\,\quad\quad - \, v_{s} \cdot c_{H} \, &+\,\, v_{d} \cdot c_{0} \, \quad\quad - \,\,K \cdot \frac{\partial c}{\partial z} ( {\text{H) }} &- \,\int\limits_{0}^{H} {\dot{q} \cdot dz = 0} \\ Y){\text{ 5,32E - 01 }} \,&+ \,\,{\text{ 2,31E - 01 }} - \, 0,05 \cdot 0 \,&+ {\text{ 2,31E - 01 }} + {\text{ 2,58E - 05 }} &- \,\, 1 \,\quad\quad \approx 0 \\ Z){\text{ 1,75E - 02 }} \,&+ \,\,{\text{ 4,85E - 01 }} - \, 0,05 \cdot 0 \, &+ {\text{ 4,85E - 01 }} + {\text{ 4,94E - 05 }} &- \,\, 1 \,\quad\quad \approx 0 \\ \end{array}$$

(54)

explains an exemplary numerical evaluation of the mass balance (18) for two different simulation times. In it, X) describe the integral mass balance (18) and Y) and Z) the evaluation at real time $t = 1,16h$ and $t = 2,60h$. The steady state is reached approximately after $t = 2,60h$ and not only after 10 days, as the AUSTAL authors claim.

The mass conservation law is fulfilled.

Because of $\partial c/\partial z(0,t) \ne 0$ according to Fig. 6, graphic A, the deposition current is non-zero at all simulation times, $\dot{m}_{{}}^{B} (t) = - K \cdot \partial c/\partial z(0,t) \ne 0$. For $t \to \infty$ one obtains a concentration distribution that does not change over time. It is approximately identical to the later stationary solution according to Eq. (24) if it is transformed according to $- K \cdot \partial c/\partial z(0)$. This gives Eq. (55).

$$\begin{aligned} & \dot{m}^{B} (z = 0,t \to \infty ) = - K \cdot \frac{\partial c}{\partial z}(0) \\ &\quad\cong - c_{0} \cdot v_{d} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot z} \right) = - 1 0\cdot 0,05 \\ &\quad\quad\cdot \exp \left( { - \frac{0,01}{1} \cdot 0} \right) = - 0,5 . \end{aligned}$$

(55)

In the stationary case, $t \to \infty$, Eqs. (55) and (60) are identical. The deposition current $\dot{m}_{{}}^{B} < 0$ is directed against the positive potential gradient $\partial c/\partial z(0) > 0$ at all simulation times. The second law of thermodynamics is fulfilled.

Second option, stationary viewing

In the stationary case, Eqs. (21) and (22) must be assumed. First, the soil concentration is calculated according to Eq. (56).

$$c_{0} = \frac{Q}{{(v_{s} + v_{d} )}} = \frac{ 1}{ ( 0 , 0 5+ 0 , 0 5 )} = 1 0 { }$$

(56)

with the information on d) and e). Equation (57) gives the maximum concentration $c_{h} = 2 0$ at source height $h = 200$.

$$\begin{aligned} c_{h} &= c\left( {200} \right) = c_{0} \cdot \frac{{v_{s} + v_{d} }}{{v_{s} }} \\ &\quad\cdot \left[ {1 - \frac{{v_{d} }}{{v_{s} + v_{d} }} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot 200} \right)} \right] \, \hfill \\&= 1 0\cdot \frac{ 0 , 0 5+ 0 , 0 5}{ 0 , 0 5} \\ &\quad\cdot \left[ { 1 { - }\frac{ 0 , 0 5}{ 0 , 0 5} \cdot \exp \left( { - \frac{0,05}{1} \cdot 200} \right)} \right] \, = 2 0\hfill \\ \end{aligned} .$$

(57)

The concentration curve calculated with this equation can be seen in the graph C of Fig. 6. The excellent agreement between the analytical and numerical solution in the scope $z \le 200$, which was achieved with the Schenk (1980) method, should be emphasized.

Here too it must be demonstrated that the mass conservation law is fulfilled. The integral Eq. (20) is again responsible. Approximately, no convective material flow is observed at the upper limit of the investigation area for $h = 200$ according to Eq. (23), which is proven with Eq. (58).

$$\begin{aligned} K \cdot \frac{\partial c}{\partial z}(h) &= v_{d} \cdot c_{0} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot h} \right) \,\\ & = 0,05 \cdot 1 0\cdot { \exp }\left( { - \frac{0,05}{1} \cdot 200} \right) \\ &= 2 , 2 7 {\text{E - 05}} \approx 0 \end{aligned}$$

(58)

In addition, the specification $v_{s} = 0,05$ according to d) must be observed

$$\begin{aligned} & v_{s} \cdot c_{0} \, &- &\,\, v_{s} \cdot c_{h} \, &+&\, \, v_{d} \cdot c_{0} \, &-&\, \, K \cdot \frac{\partial c}{\partial z} ( {\text{h) }}\quad\quad = 0 { } \hfill \\& 0,05 \cdot 10 \, &-&\, \, 0,05 \cdot 20 \, &+&\, \, 0,05 \cdot 10 \, &-&\, \, 1 \cdot 2,27 \cdot 10^{ - 5} \, \approx 0 \hfill \\ \end{aligned}$$

(59)

The balance Eq. (59) proves that constant mass is guaranteed.

Equation (23) gives the relationship (60)

$$\begin{aligned} \dot{m}_{{}}^{B} &= - K \cdot \frac{\partial c}{\partial z}(0) = - c_{0} \cdot v_{d} \cdot \exp \left( { - \frac{{v_{s} }}{K} \cdot z} \right) \\ &= - 1 0\cdot 0,05 \cdot \exp \left( { - \frac{0,01}{1} \cdot 0} \right) = - 0,5 . \end{aligned}$$

(60)

Then there is a conductive mass transfer, $\dot{m}^{B} = - 0,5$. Deposition should take place after (e), $F_{c} \ne 0$. Due to an existing potential gradient $\partial c/\partial z(0) \ne 0$ there is also a deposition, $\dot{m}^{B} = - K \cdot \partial c/\partial z(0) \ne 0$. The second law is the thermodynamics is fulfilled.

Faulty non-stationary and stationary solution taking into account the Janick’s Convention according to Trukenmüller et al. (2015)

First option, non-stationary consideration

Again, according to b) and g) of the task, it must be assumed that the AUSTAL authors considered non-stationary conditions. However, no solution algorithms and concentration profiles are given for this. The AUSTAL authors did not perform non-stationary calculations.

$$\begin{aligned} & \int\limits_{0}^{h} {\frac{\partial c}{\partial t} \cdot dz} + v_{s} \cdot c_{0} - v_{s} \cdot c_{h} + v_{d} \cdot c_{0} \\ &\quad- K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} - \int\limits_{0}^{h} {\dot{q} \cdot dz = 0} \end{aligned}$$

(61)

Due to the lack of non-stationary solution courses, the integral Eq. (18) cannot be used to control mass conservation. The AUSTAL authors do not provide any simulation results. If the authors of AUSTAL state that a stationary solution would have appeared after 10 days, the public will be deceived as well.

Second option, stationary viewing

The AUSTAL authors only provide stationary solutions for this task. To do this, they use their incorrect solutions (25) and (26). Using Eq. (26), $F_{c} = v_{d} \cdot c_{0}$, one calculates the soil concentration according to Eq. (62).

$$c_{0} = \frac{{F_{c} }}{{v_{d} }} = \frac{1}{0,05} = 20$$

(62)

and with Eq. (25) a constant concentration distribution in the entire study area from $c(z) = 20 = const.$ according to Eq. (63).

$$\begin{aligned} c(z) &= c_{0} \cdot \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right) + \frac{{F_{c} }}{{v_{s} }} \cdot \left[ {1 - \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right)} \right] \\ &= c_{0} \cdot \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right) + \frac{{0,05 \cdot c_{0} }}{0,05} \\ &\quad\cdot \left[ {1 - \exp \left( { - z \cdot \frac{{v_{s} }}{K}} \right)} \right] = c_{0} = 20 \hfill \\ \end{aligned} .$$

(63)

The result of this calculation is shown in Fig. 7. In contrast to the correct solution with $c_{0} = 10$, the authors of AUSTAL calculate the wrong concentration distribution in the amount of $c(z) = 20 = const.$. This untrue result is also highlighted in column V of Fig. 5. As can be seen with the specification f), a source should have been in 200 m, which, however, contrary to Fig. 6, Graph C, cannot be seen in Fig. 7 of the AUSTAL authors.

It can easily be demonstrated that Eq. (25) is an incorrect solution of differential Eq. (3). For this purpose, Eq. (20) is used again. With the simulation results already described and taking into account Eq. (36), $\partial c/\partial z(h) \sim (F_{c} - c_{0} \cdot v_{s} ) = 1 - 20 \cdot 0,05 = 0$, the expression (64).

$$\begin{aligned} &v_{s} \cdot c_{0} \,& - &\,\, v_{s} \cdot c_{h} \, &+&\, \, v_{d} \cdot c_{0} \,& - &\,\, K \cdot \frac{\partial c}{\partial z} ( {\text{h)}} = 0\hfill \\& 0,05 \cdot 20 \,& - &\,\, 0,05 \cdot 20 \,& + &\,\, 0,05 \cdot 20 \, &- &\,\, 1 \cdot \, 0 \quad\quad \ne 0 \hfill \\ \end{aligned}$$

(64)

results. The mass conservation law is therefore also violated for the “sedimentation with deposition” propagation case.

Equation (6), $\dot{m}^{A} = \dot{m}^{B}$, can be used to prove that the second law of thermodynamics is also violated. Equation (36).

$$K \cdot \frac{\partial c}{\partial z}(0) = (F_{c} - c_{0} \cdot v_{s} ) = (1 - 20 \cdot 0,05) \equiv 0$$

(65)

is required to calculate the conductive current. The deposition current

$$\dot{m}_{{}}^{B} = - K \cdot \frac{\partial c}{\partial z}(0) = - (F_{c} - c_{0} \cdot v_{s} ) = - \left( {1 - 20 \cdot 0,05} \right) = 0 .$$

(66)

is calculated using Eq. (65).

After that, there is no conductive mass transfer. However, the authors of AUSTAL state that deposition should take place after e), $F_{c} \ne 0$. This contradiction can only be clarified in such a way that one would have to assume that the diffusion coefficient would strive towards infinity, $K \to \infty$. On the other hand, contrary to $\dot{m}^{B} = - K \cdot \partial c/\partial z(0)$, the material flow would follow a non-existent potential gradient $\partial c/\partial z(0) = 0$. The first case is excluded because the diffusion coefficient is a finite material parameter. The second case is correct and justified. Therefore the second law of thermodynamics is violated. If contradicted, the pollutant particles would have to be contrary to Fick’s law according to Häfner et al. (1992) that the concentration gradient at the bottom is not equal to zero, $\partial c/\partial z(0) \ne 0$.

Homogeneity tests