Open access peer-reviewed chapter

The Theory of Random Transformation of Dispersed Matter

By Marek Solecki

Submitted: February 8th 2012Reviewed: August 14th 2012Published: July 24th 2013

DOI: 10.5772/52369

Downloaded: 1147

1. Introduction

The development of civilization, with regard to its consequences, requires appropriate research tools. This is the reason of significant progress in different fields in mathematical modelling of processes involving mass transfer. For their description there are often used widely known models which can be written in a general form of nonlinear differential equations. Woltera [1], who conducted the research of oscillation level of selected fish species in the Adriatic, described the interaction system in the population of predator-victim type. It is compatible with the obtained by Lotka [2] description of the reaction with the expected oscillations in the concentrations of chemical compounds. The process of destruction of organisms during disinfection is included in the model of Chick [3] developed by Watson [4]. The kinetics of the course of simple enzymatic reactions was described by Michaelis and Menten [5]. Mathematical description of the process of microorganisms disintegration in high-pressure homogenizer was developed on the basis of experimental data by Hetherington et al. [6]. After over twenty years since revealing the effect of the position of enzymes in a cell on their release rate [7], Melendres et al. [8] proposed a nonlinear description of the process as a consequence of the following events: cell disruption and release of intracellular compounds. The results of pioneering protease inhibitor therapy in HIV infection were the basis for the development by Ho et al. [9] of a simple model of treatment and solving the mystery associated with the quasi-stationary phase of infection. These models are still used to describe the processes or are often the basis for future studies of more complex mathematical problems.

The essence of all the above processes is random transformation of material objects dispersed in a limited space. Their relationships may help to increase the pace of knowledge development. A uniform theory of the presented issues was developed by Solecki [10]. The common general concept based on the knowledge and understanding of important factors shaping the specified sphere of events will facilitate particularistic analysis as well as transfer of knowledge and experience. The aim of the presented study was to develop a unified theory of the presented issues, allowing for phenomenal and mathematical modeling of various processes based on mass transfer.

2. Paradigm

This section presents a coherent conceptual system of uniform theory which includes various processes of random transformation of dispersed matter in a limited space.

2.1. Space and material objects

Let there be a set N consisting of n elements that are material objects. We assume that n is a natural number.

Figure 1.

The set of identified properties of a material object belonging to set N.

Identified properties of material objects belonging to set N are the elements of set Nt (notum) (Fig. 1). The property of an object is the feature which characterizes it. Objects can have χ same properties (ct1, …, ctχ) belonging to set Ct (constans) which is a subset of set Nt. Objects from set N can differ with ξ properties (vr1, …, vrξ). They belong to set Vr (variabilis) which is a subset of identified features of object Nt. We assume that the ψ- th property vrψi of any i-th object belonging to set N is included within the range described by the relationship

vrψminvrψivrψmax.E1

Changing the feature of an object belonging to set Vr involves a change of the parameter vrψi described by the relationship (1). An object from set N may lose some features from set Vr or Ct. Any element belonging to set N must have all the φ features belonging to set Im (inamissibilem). They are the basic features of any object from set N and satisfy the relationship

0<φχ+ξ.E2
.

Set Im is a subset of set Nt. We assume that after the loss of at least one basic feature of set Im the object is no longer what it was, that is it no longer belongs to set N.

Objects from set N have limited duration after which they lose at least one basic feature from set Im and they do not belong to set N anymore.

It is assumed that the fact of existing of objects in set N results in a possibility of generating new elements which are objects of set N.

Let there be a limited medium of volume V described by the relationship according to the equation

V=f1t.E3

In his medium there is set N consisting of n dispersed material objects described by the relationship

n=f2t.E4

We assume that the volume of material object V is incomparably greater than volume Vni of any i-th object of set N, according to the formula

Vni<<V.E5

At the initial moment t0 = 0 the number of objects from set N dispersed in space V amounts n0, where n0 is large natural number. We assume that after time Δt the number of objects belonging to set N may be changed. Changes may be caused by:

  • addition of new objects from the outside to volume V – introduction,

  • exclusion of some objects from volume V to the outside – removal,

  • generating new objects resulting from the fact of existing of material objects from set N in volume V – multiplication,

  • natural exclusion of objects from set N, resulting from its duration appropriate for the elements of this set, caused by the loss of at least one feature from set Im – exclusion.

Change in number of objects belonging to set N, resulting from introduction, removal, multiplication and exclusion, described by functions respectively f31 (t), f32 (n, t), f33 (n, t) and f34 (n, t), at the moment t is described by the relationship consistent with the formula

ΔnA=f31(t)f32(n,t)+f33(n,t)f34(n,t).E6

2.2. Converting of objects

The conditions of the existence of objects in the environment are determined by the interaction of physical, chemical, biological and psychical factors. We assume that for each material object from set N there exists such set of environmental conditions in which there is a loss of ς object properties (ς>0) belonging to set Nt. If at least one of ς lost features belongs to set Im we are dealing with the transformation of an element from set N. In this case the object from set N after the loss of at least one feature from set Im does not belong to set N. A set of environment conditions which may influence the transformation of an object from set N is called transformation conditions. Their intensity, at which the object is transformed, is called transformation intensity and is denoted as Γt. Locally occurring environmental conditions of transformation intensity were designated γt. The features of an object from subset Ct have no effect on the variation of transformation intensity of material objects. We assume, however, that the combination of ξ features of set Vr affects the value of the smallest transformation intensity γti of the i-th object from set N, according to the formula

γti=f4(vr1,,vrξ).E7

If none of ς lost properties by the i-th element belongs to set Im we are dealing with the formation of object from set N. After the loss of features not belonging to set Im an object from set N still belongs to set N. A set of environment conditions which may influence the formation of an object from set N is called formation conditions. Their intensity, at which the object is formed, is called formation intensity and is denoted as Tt. Locally occurring environmental conditions of formation intensity were denoted by τt. The features of an object from subset Ct have no effect on the variation of formation intensity of material objects. However, the combination of ξ features of set Vr affects the value of the smallest transformation intensity τti of the i-th object from set N, according to the formula

τti=f5(vr1,,vrξ).E8

Later in this paper the theory of transforming objects is described. The formation of material objects is an important issue because of the possibility of changing their susceptibility to transformation. The theory of random formation of matter is analogous to the described theory.

2.3. Types of volume

The main parts of space V are volumes Vαi. Environmental conditions occurring in them are safe for the individual objects from set N. Thus volume Vαi is safe for the i-th object from the set N. We assume that this volume there is intensive mixing. Its purpose is to homogenise the concentration of objects present in the volume Vαi.

Volume Vαi consists of two parts:

  1. Vαci is a part of volume Vαi, whose subsets are never transformed to other types of volumes,

  2. Vαti is a part of volume Vαi, in which subsets can be transformed to other types of volumes (Vγji and Vβji).

Between the components of volume Vαi for the i-th object there are relationships which are described by the following formula

Vαi=VαtiVαciE9

and

VαtiVαci=.E10

In the space Vαti there are generated transformation volumes Vγji. Possible cases of generating transformation volumes are shown in Figure 2. Volume Vγji is the j-th transformation volume of the i-th material object from set N. We assume that transformation volumes generated for the i-th element are uniformly dispersed in space Vαti. Space Vαti is incomparably greater than any j-th volume Vγji, which is described by the formula

VγjiVαti.E11
.

Volume Vγji is limited from the outside with surface Fγαji and from the inside with surface Fγβji (Fig. 2). Both surfaces belong to volume Vγji according to the formula

FγαjiVγji,E12

and

FγβjiVγji.E13

Figure 2.

Schematic of formed transformation volume Vγji mach larger then (Vγji)min: general case.

Component factors of transformation conditions of the i-th material object can affect it locally or even pointwise. Component factors of transformation conditions of the i-th material object can affect it locally or even pointwise. However, the effects of their action affect the whole object. We assume that the smallest transformation volume of the i-th object is equal to its volume at the moment of transformation and amounts (Vγji)min (Fig. 3). Occurring in his volume set of transformation conditions of intensity at least γti ensures transformation of the i-th object from set N. Volume (Vγji)min is a value characterizing the i-th material object from set N.

Figure 3.

Schematic of formed transformation volume Vγji= (Vγji)min: special case.

The range of variation of generated in space V transformation volume Vγji by the size of the volume is described by the following formula

(Vγji)minVγji(Vγji)max,E14

and by the intensity of transformation conditions by a formula

Vγji(γt=γtmin)Vγji(γt)Vγji(γt=γtmax).E15

By γtmin and γtmax denoted accordingly minimum and maximum intensity of transformation conditions which can be generated in space Vγji.

Transformation volume Vγji is generated at time t (t >t0) of process transformation duration. It can exist in any time interval Δt > 0. In this time interval Vγji can increase; if it is greater than (Vγji)min it can decrease or stay unchanged. It can also be displaced randomly to any available place of volume Vαti. In general form Vγji is described by the relationship in the following formula

Vγji=D1[f61ji(x,y,γ,t)f62ji(x,y,γ,t)]dxdyD2[f71ji(x,y,γ,t)f72ji(x,y,γ,t)]dxdyE16

of course at the assumptions described by the relationships

f61ji(x,y,γ,t)0E17

and

f71ji(x,y,γ,t)0.E18

Flat areas D1 and D2 describe relationships defined by formulas respectively

y=f63ji(x,γ,t)E19

and

y=f73ji(x,γ,t),E20

the range of changes in x value is described by the relationships defined in the formulas respectively

x6jiminxx6jimaxE21

and

x7jiminxx7jimax.E22

Inside volume Vγji volume Vβji is generated. It is not available for non-transformed i-th object from set N.

Axiom 1.

If linear dimensions of volume Vγji are smaller than doubled linear dimensions of the i-th object belonging to set N, then the area of Fγβji reached by the i-th object from the inside does not belong to volume Vαi according to

FγβjiVαi.E23

Since for limit case (Vγji)min there is

Vβji(Vγji)min,E24

that is

FγβjiFγαjiE25

then

FγαjiVαi.E26

and

FγαjiVβji.E27

Axiom 2.

If the linear dimensions of the volume Vγji are greater than doubled linear dimensions of the i-th object belonging to N then area Fγβji does not belong to Vαi volume according to the formula

FγβjiVαi.E28

For the case for which occurs a relationship described by a formula (28), volumes closed by area Fγβji do not belong to space Vαi.

Volume Vβji is closed by external surface Fγαji.

Axiom 3.

If the linear dimensions of the volume Vγji are smaller than doubled linear dimensions of the i-th object belonging to N then area Fγβji reached by the i-th object from the inside belongs to Vβji volume according to the formula

FγβjiVβji.E29

Axiom 4.

If the linear dimensions of the volume Vγji are greater than doubled linear dimensions of the i-th object belonging to N then area Fγβji does not belong to Vβji according to the formula

FγβjiVβji.E30

For the case for which occurs relationship described by the formula

Vγji>(Vγji)min,E31

volumes closed by area Fγαji do not belong to Vβji volume according to formula

FγαjiVβji.E32

If the linear dimensions of the volume Vγji are equal to the doubled linear dimensions of the i-th object belonging to N then area Fγβji may be replaced by a segment (for example for cylindrical surfaces) or even a point (for example for spherical surfaces). They will belong to volume Vβji and they will not belong to volume Vαi.

Volume component of the space V, apart from the above mentioned kinds of volumes, is Vδi. This volume meets the following conditions:

  1. is safe for the i-th object from set N,

  2. there is not mixing in it,

  3. there is a possibility of object migration from volume Vαi to Vδi and vice versa,

  4. subsets of this volume are not transformed to other volumes.

We assume that there is a relationship

Vδi<Vαi.E33

Volumes Vδi may be dispersed in space V as volumes Vγji.

2.4. Transformation process

We assume that the above described events: introduction, removal, multiplication, and exclusion are not the events of the investigated process of random transformation of dispersed material objects.

In a given material medium V occurs process of random transformation of objects belonging to set N. It runs as follows:

At any moment t (t > 0) for the i-th object in space Vαti there are generated randomly p transformation volumes Vγji, where p is a natural number. The i-th object belonging to set N is in the space safe volume Vαi described by a relationship

Vαi=j=1pVαji.E34

We assume that random transformation of objects is independent events.

It is assumed that during transformation process the number of produced volumes Vγji is large and may change over time, according to the formula

p=f8(t).E35

Due to relative displacement the i-th material object is introduced for time tti to appropriate volume for its transformation Vγji. This is done by the surface Fγαji which limits volume Vγji (Fig. 2 and 3). The considered element of set N remains unconverted if at least its one point is beyond volume Vγji. Transformation of the i-th object occurs simultaneously with its complete introduction to transformation volume Vγji. According to the conditions given in Section 2.2 transformation occurs when at least one of the ς features belonging to set Ba is lost. Only transformed in volume Vγji i-th object may be dislocated to volume Vβji.

Theorem about transformation of dispersed matter:

For each material object ni from set N there exist such local conditions of transformation γti belonging to set Γt, that if there is time tti, in which object ni is included in volume Vγji of transformation properties γti then starting from time tti object ni does not belong to set N.

This theorem recorded by means of quantifiers has the following form:

(niN)(γtiΓt)((tt)niVγji(ttt)niN),E36

and its falsification is included in the record

(niN)(γtiΓt)((tt)niVγji(ttt)niN).E37

Proof of the theorem

The properties of an object from set N belonging to set Ba (Fig.1) are marked by ba. The relationship is introduced

Z1n;baE38

which expresses the statement that object n has the property ba. We can record the following observations:

object ni has all the properties ba from set Ba

(baBa)(Z1(ni;ba)),E39

and there is a property ba, which object ni does not have

(baBa)(¬Z1(ba;ni)).E40

Let N be the set of all elements fulfilling condition W(n) described by a logical statement (39)

niN:baBaZ1ni;ba.E41

The relationship is introduced

Z2Vγ;γtE42

expressing the statement that volume Vγ has the property γt of object transformation n. We can write the following observation:

object Vγji has got a property γti of object transformation ni

(γtiΓt)(Z2(Vγji;γti)).E43

Let P be the set of all elements fulfilling the condition W2(Vγji) described by a logical statement (39)

VγjiP:baBaZ1ni;ba.E44

Let tti denotes time in which object ni was introduced to volume Vγji. From the moment tti is true the statement about object ni

(ttti)(baBa)(¬Z1(ba;ni)).E45

It is contrary to the assumption (31) so that from the moment tti the object ni cannot be the element of set N according to

niN.E46

Theorem about creating families of transformation volumes

If at any point 2 belonging to space Vαti environmental conditions of the transformation would function γt2 and in set N there would be r objects of the transformation conditions γti fulfilling the relations

γt2γtiE47

Then in point 2 there will be generated family consisting of r transformation volumes for r objects from set N.

Conclusions:

  1. If set N consists of n elements and conditions of transformation γti of any element from set N satisfy the relation

γtminγtiγtmaxE48

then the family can include from 1 to n transformation volumes.

  1. If in any point 2 belonging to space Vαti would function conditions of transformation γt2 fulfilling the relationship

γt2<γtmaxE49

and in set N there would exist s objects of transformation conditions greater then γt2 then in space V there are s objects from set N which are not subjected to transformation in point 2.

Volume Vγji is a transformation volume of the i-th material object which belongs to j-th family.

3. General phenomenological model

Figure 4 shows the general set of possibilities for generating volumes associated with the transformation of objects from set N distributed in space V (t). Each vertical segment with the opposite ends located on segments AB and CD denotes space V (t). On each subsequent vertical section there are marked up divisions of volume V in result of generated one transformation volume. The whole rectangle ABCD includes a set of all possible divisions of space V according to one defined system.

For any i-th object from set N rectangle ABCD is divided into five parts with four segments. The first part, contained between segments LM (marked with blue dashed line) and IJ (marked with red dashed line) is a set of possible volumes Vαji. From the bottom a single volume Vαi is limited by surface Fγβji. Depending on the size of generated volume Vγji this surface according to formula (23) conditionally does not belong or according to formula (28) at all does not belong to volume Vαi. Space Vαi is safe for both the i-th object and other objects from set N of the smallest transformation volume not greater than that which is appropriate for the i-th object. In volume Vαji takes place intensive mixing. Its purpose is to homogenize the dispersion of objects present in it.

Figure 4.

General set of space divisions V(t) resulting from generating volume Vγji for one hypothetical generating system.

In Figure 4 the area included between segments GH (marked with orange dashed line) and IJ (marked with red dashed line) is a set transformation volumes that can be generated. Single volume Vγji is limited by surfaces: Fγαji from the top and Fγβji from the bottom. Both these surfaces belong to volume Vγji, which is the result of assumptions accepted in formulas (12) and (13). In volume Vγji occurs a transformation if the i-th object and other objects from set N of minimal transformation volume in accordance with that which is appropriate for the i-th object i.e. (Vγi)min. During the process at time t there are p transformations volumes of the i-th object according to relationship (34) are they are uniformly dispersed in space Vαti. The size of the volume Vγji in Figure 4 is defined by two parameters: (Vγji)min and red colour intensity. The parameter – red colour intensity is a visualization of functional dependence.

Another area in Figure 4, contained between segments GH (marked with orange dashed line) and CD (marked with brown solid line) is a set of volumes Vβji. Such single volume is limited by surface Fγαji. Depending on the size of generated volume Vγji this surface according to (27) conditionally belongs to Vβji or according to formula (32) does not belong to Vβji. During the process at time t volume Vβji is no more than p. They are linked to appropriate volumes Vγji and dispersed in space Vαti. In volume Vβji the existence of unconverted objects from set N is impossible, for which minimal transformation volume is not smaller than minimal transformation volume appropriate for the i-th object. The size of the volume Vβji in Figure 4 is defined by two parameters: vertical segment with ends located on segments GH and CD and brown colour intensity. The consequence of increasing volume Vγji is increased volume Vβji and decreased volume Vαti. The size of volume Vαti in Figure 4 is defined by two parameters: vertical segment with ends located on segments RS and IJ and blue colour intensity.

Between segment AB and segment LM there is volume Vδi.

Division of vertical segment 16 reflects the division of space V which appears after generation of the j-th volume for the i-th object ni:

  • segment 1,3)corresponds to volume Vβji,

  • segment 2,3corresponds to volume Vγji,

  • segment (2,4corresponds to volume Vαtji,

  • segment (4,5corresponds to volume Vαcji,

  • segment (5,6corresponds to volume Vδi.

In Figure 4, as for the i-th object, it is possible to determine components of space volume V for the objects of the highest and smallest transformation volume marked (Vγb)min and (Vγs)min respectively.

Between segments AC and BD it is possible to lead infinite number of vertical lines. Their division into sections, presented above, describes the field of possibilities of generation, in space V, the transformation volume Vγji and connected with them volumes Vβji. For segment AC transformation volumes are generated according to Figure 3. The divisions are on the right side of Figure 4 together with the segment BD is the general case shown in Figure 2.

At any moment of the transformation process in volume V a finite number transformation volume is generated.

Phenomenological model of random transformation of dispersed matter is constructed as follows:

  1. The possibilities of generating transformation volumes in space V are shown on the system map appropriate for the given transformation process which is analogous to that presented in Figure 4. In space V transformation volumes according to many system maps can be generated simultaneously.

  2. Phenomenological model consists of p layers formed by p transformation volumes generated at a given moment in volume V according to system maps.

  3. The model is constructed in such a way that the sum of sets of all volumes is equal to V. The sum of all volumes safe for the i-th objects located above the generated in a given moment its transformation volume is equal to Vαi, according to the equation (33).

  4. Material objects belonging to set N, in space V(t), were reduced to a point. One of their features is volume Vni. On system maps (Fig. 4) and in the phenomenological model the size of volume is defined by vertical segments, so the i-th object will be a vertical segment of the length corresponding to volume (Vγji)min. This object can be displaced in space Vαi and Vδi till the transformation moment in volume Vγji.

4. General mathematical model

In general, the concentration of elements of set N in volume V(t) is determined by the number of not transformed objects n per volume V according to the formula

S(t)=n(t)V(t).E50

At the initial moment t0 = 0 it will amount

S0=n0V0.E51

We assume that for the process duration moment t0 = 0 in space V volumes Vγji, Vβji, Vαi and Vδi are generated. Not transformed i-th object can occur only in the appropriate volumes Vαi and Vδi. From formulas (1) and (7) results the relationship between the volumes for the individual objects from set N (Fig. 4) given in the formula

VαbVαiVαs.E52

For relationship (52) equivalence

(Vγs)min=(Vγb)minVαs=VαbE53

or

(Vγs)min(Vγb)minVαbVαsE54

can be true. In the case included in formula (53) in the whole volume Vα defined by formula

Vα=i=1nj=1pVαjiE55

occurs uniform dispersion of the elements of homogeneous set N. However, from the given in formula (54) alternative equivalence result the uneven dispersion of inhomogeneous objects from set N in volume Vα. Each object ni characterized by feature vri which distinguishes it from other objects from set N results in introduction of the additional area Fγβi (Fig. 2 and 3). Each additional area Fγβi introduced in space V divides volume set Vα (Fig. 4). The resulting parts can differ with the concentration of the objects contained in them. The process of ideal mixing ensures homogeneity of the dispersion only within a volume limited by neighboring areas e.g. Fγβ(i-1) and Fγβi, Fγβi and Fγβ(i+1). In the case covered by formula (54) due to the relationships given in formulas (5) and (11) as well as uniform dispersion of volume Vγ in space Vαt we can use average concentration of dispersed material objects. Thus, after starting the process, to the generated volumes Vγji there will be introduced randomly appropriate objects ni of set N dispersed in volume Vα. Elements from set N being in volumes Vδi do not participate directly in the transformation process. In moment t0 = 0 of process duration the number of unconverted objects n(t) in volume Vα is defined by the equation

nα0=n0nδ0E56

Average concentration of untransformed objects in volumes occupied by them volume Vα is described by formula

(Sα¯)t0=0=n0αVαE57

where nδ0 denotes initial number of elements from set N in volumes Vδi. Number of objects which were transformed nd0 for time t0 is defined by the formula

nd0=0E58

The degree of objects transformation X(t) determined by the quotient of the number of transformed objects nd to the initial number of unconverted objects n0 for t = 0 equals 0, according to the formula

X=nd0n0=0.E59

At any moment of process duration t the number of transformed objects amounts nd(t). Converted objects may be located in any place of volume V. The number of unconverted objects, being exclusively in volume Vα, is defined by the difference of the initial number of objects nα0 and converted objects nd after time t of process duration corrected by a number of objects transferred between volumes Vαi and Vδi and the number of objects described by the formula (6), which is described by equality

nα=n0αnd+nδαnαδ+ΔnA,E60

where:nδα – means number elements of set N transferred from volume Vδi to Vαi,

nαδ – means number of elements of set N transferred from volume Vαi do Vδi.

After time t of process duration, to transformation volumes Vγji there are introduced untransformed objects of average concentration (Sα¯)tdefined by the number of unconverted nα, per volume Vα. This is shown in formula

(Sα¯)t=nαVα.E61

Of course general concentration of unconverted objects in volume V will be defined by the relationship

S=nV,E62

where

n=nα+nδ.E63

It will be shown, for example, after stopping the process. The degree of transformation of objects after time t of the transformation process is described by the formula

X=ndn0+ΔnA.E64

The increase of transformed objects dnd in all transformation volumes Vγji after the lapse of any small time interval dt is defined by the formula

dnd=(Sα¯)tdV.E65

The external area limiting the generated for the i-th object the j-th volume Vγji, consists of active and inactive part. Through the active part there can be introduced or removed the i-th element of set N while the inactive part is not available for such transfer. The active part of the surface is divided into surface Fγα→, through which the i-th object can be introduced to Vγji and surface Fγα←, through which the transformed or not transformed i-th object can be introduced from Vγji according to the formula

Fγαji=Fγαji+Fγαji.E66

The sum of surfaces limiting the transformation volumes Vγji generated in space V for the i-th object is described by the formula

Fγαi=j=1pFγαji.E67

The sum of surfaces limiting the volumes Vγji generated in space V for all objects form set N is described by the formula

Fγα=i=1nj=1pFγαji.E68

Volume dV displaced from space Vαji to volume Vγji in time increase dt depends on the size of limit area Fγα→, through which dV is displaced from volume Vαji to volume Vγji and on the average speed of its displacement u, according to the formula

dV=uFγαdt.E69

After substituting equations (61) and (69) to equation (65) we get the equation

dnd=k(nα)dtE70

describing the increase of objects transformed in generated volumes Vγji. Process rate constant k is described by the relationship

k=FγαVαu.E71

According to formulas (16), (19) and (20) parameters Fγα→ and Vα depend on time t. Dislocation rate u can also be a function of process duration

u=f9(t).E72

On the basis of formula (70) the loss of unconverted objects in set N can be expressed by the equation

dn=k(nα)dt.E73

After substituting (60) to the relationship (73) we get

dn=k(n0αnd+nδαnαδ+ΔnA)dt.E74

The total balance of the loss of objects dnd in time interval dt will be equal on the right side of the formula (74) with the opposite sign

dnd=k(n0nd+nδα+nαδ+ΔnA)dt.E75

In case

ΔnA=0E76

the relationship (76) will be simplified to

dnd=k(n0nd)dt.E77

After separation of variables in equation (77) and integration of both sides we get the relationship

ln(n0n0nd)=kt,E78

often used to describe the kinetics of the course of transformation of various material objects. In case of homogeneity of set N (Vr is an empty subset) relationship (78) is a final description of the transformation process. Let’s consider the case when the object of set N differ with only one feature vr1, which has a considerable effect on the course of the process. In order to analyse it, the whole range of changes should be divided into m such intervals in which changes in features have no significant effect on the course of the process. Then the course of transforming objects from any interval ζ - th such that

ζ1,m,E79

describes the relationship

n0ζn0dndζ=n0ζn0kζ(n0ζndζ)dtE80

taking into account the existence of a whole set N.

Description of the transformation of the entire set of objects including the division of set N into intervals is described by the relationship

ζ=1mn0ζn0dndζ=ζ=1mn0ζn0kζ(n0ζndζ)dt.E81

After the separation of variables (80) and integration of both sides we get

ζ=1mn0ζn0lnn0ζn0ζndζ=ζ=1mn0ζn0kζt.E82

Relationship between speed constants in equations (77) and (80) is defined by a formula

Φζ=1mn0ζn0kζ=k.E83

Coefficient Φ describes the relationship

Φ=lnn0n0nd(ζ=1mn0ζn0lnn0ζn0ζndζ)1.E84

5. Examples of application

The technological process based on the theory of random transformation of dispersed matter is the disintegration of microbial cells. At present many compounds coming from the inside of the microorganism cells have commercial application. They are used among others in the food, pharmaceutical, cosmetic, chemical industry as well as medicine and agriculture. In order to isolate the desired compounds it is usually necessary to destroy the cell walls and cytoplasmic membranes. The process of disintegration of microorganisms is carried out by different methods: physical, chemical and biological.

The technical means used to implement the process on an industrial scale are bead mills and high pressure homogenizers. On a laboratory scale there are often used vibrating mixers, ultrasonic homogenizers and enzymatic methods. All the listed methods of the process involve the random effects of the factor which destroys the cell walls of microorganisms dispersed in the liquid. The difference between them depends mainly on the method of generating the transformation volume of dispersed matter.

During the disintegration of microorganisms the suspension occupies volume V. It is constant in time. The process is carried out usually for the optimal initial concentration of biomass. The initial number of microorganisms is determined n0. We consider the case of batch operation (constant charge). During the process microorganism cells are not added from the outside (we assume the sterility of conditions of disintegration) nor are they removed outside. The process duration compared to the lifetime of microorganisms and the time needed to form new cells is very short. So it can be assumed that the change in the number of objects described by the formula (6) satisfies the relationship (76).

In the case of disintegration of microorganisms in of bead mills the transformation consists in the disruption of microbial cell walls. Destruction volumes Vγji are generated by circulating filling beads. To carry out the process in the bead mill, high degree of filling the working chamber with beads and high rotational speed of the agitator are applied. It can be assumed that at any time during the process for fixed working conditions number p (Eq. (35)) is constant in time and its value is high. Thus the relations given in the formulas

j=1pVγji=const,E85
j=1pVβji=constE86

are fulfilled.

General diagram of cell disruption between the spherical surfaces is shown in Figure 5. It concerns the range of the distribution of suspension volume V to Vαji, Vγji and Vβji shown on the right in Figure 4 in the area adjacent to the segment BD. Volumes Vαc occur close to the inside surface of the mill chamber in the case when filler elements have a diameter substantially greater than the dimensions of the cells of microorganisms. In a bead mill volumes Vαc occur at all surfaces of the working chamber and agitator. They are distant by a distance similar to the size of the largest cells and the thickness of their layer is slightly smaller than the radius of the smallest filling beads. In these volumes cells are never disrupted.

For a properly constructed mill chamber volumes Vδ (may be slots at the interface between two structural elements) are negligibly small and insignificant in terms of technology, especially when conducting sterilization of equipment between the processes. In the analyzed case it was assumed that disintegrated microorganisms have an ellipsoidal shape. After the limit deformation of the i-th cell, its walls are disrupted. (Fig. 5). The generated transformation volume Vγji is limited by surfaces: active Fγαji (orange dashed line), Fγβji (red dashed line) and two inactive spherical of filling elements. Limiting surfaces Fγαji and Fγβji belong to volume Vγji. Its axis of symmetry is axis OO. To the presented in Figure 5 volume Vαi, in which i-th living cell can be present, does not belong the volume limited by two spherical surfaces and surface Fγβji. Straight line OO is a symmetry axis of this volume. Surfaces limiting volume Vαji do not belong to it. The volume unavailable for the living i-th cell Vβji is limited by two spherical surfaces and surface Fγαji. Limiting surfaces do not belong to Vβji. The axis of symmetry of volume Vβji is also straight line OO. In the special case the line dividing volume V in an AC position (Fig. 4), there is generated volume (Vγji)min shown in Figure 6. It is limited by two spherical surfaces and active surface Fγαji (orange dashed line). Volume symmetry axis (Vγji)min passes through points O and O. Surface Fγαji does not belong to generated volume Vβji. If all points of the i-th object are not introduced to (Vγji)min it will not be transformed.

Figure 5.

Model of cell disruption during non-axial hitting with spherical elements – general case [10].

Figure 6.

Model of cell disruption during axial hitting with spherical element and plane – special case [14].

For monogenic set N (Vr is an empty subset) cell disintegration process, taking into account the above assumptions, is described by differential equation (77). The case when the objects of set N differ with only one feature vr1 that has a significant impact on the course of cell disruption is included in equation (80) and (81). The process of release of intracellular compounds, using the theory of transformation of dispersed matter, has been widely described by Heim at all [12], Heim and Solecki [13] as well as Solecki [10 and 14]. These works included more complex cases of the course of the process, causes of nonlinearity of kinetics were given, and the effect of concentration of microorganism suspension was explained as well as disappearance of the largest size fraction during the process for very low concentrations of the suspension.

There are many devices implementing the process of disintegration of microorganisms due to critical stresses in the cell walls caused by stress in the liquid. Best known are: high pressure homogenizers [6, 15-18], French press [19], Ribie press [20], Chaikoff press [21]. The technical method of process realization in the above mentioned equipment is similar and consists in disruption of microorganisms during pumping suspension through a valve under high pressure. High performance in continuous operation and fairly wide range of diversity of disintegrated microorganisms made that high pressure homogenizers are widely used on a technical scale. The general model of disintegration of mechanisms, developed by Hetherington et al. [6], and then modified by Sauer et al. [15] is included in a formula

ln(RmRmR)=kNb2Pb1,E87

where: R concentration of released proteins [mg/g],

Rm - maximum concentration of released proteins; [mg/g],

k - process rate constant; [1/s],

N - number of suspension passage cycles through homogenizer; [-],

P - suspension pumping pressure, [MPa].

b1 - exponent depending on the type of microorganisms and their growth conditions; [-],

b2 - exponent including the effect of suspension concentration on the course of the process; [-].

Pumping pressure of the suspension of microorganisms is within the range from 50 to 120 MPa. Transformation volumes Vγji are generated, according to the results of research conducted by Keshavarz Moore at al. [16] and Engler [17] on the cell disruption mechanism, in homogenization zone within the valve unit and with the impingement in the exit zone. The results of research conducted by Lander at al. [18] showed that disruption of cells is mainly due to shearing of the liquid in the valve unit and as a result of cavitation occurring in the impingement section, where the stream of suspension hits the impact ring and follows an implosion of bubbles caused by the increased pressure.

High-frequency ultrasounds (in the supersonic wavelength range 15 - 25 kHz) are used for disruption of dispersed in a liquid microbial cells and releasing contained in them intracellular compounds [22-25]. The mechanism of microorganisms disintegration is associated with the occurrence of cavitation induced by ultrasound and hypothetically runs as follows:

  1. Passing sound wave causes the thickening and thinning of the liquid.

  2. During the thinning of the liquid occurs nucleation and growth of gas and vapor bubbles.

  3. During the thickening of the liquid bubble implosion occurs at a rate not less than the speed of sound (hence the loud roar accompanying cavitation).

  1. Bubbles which are not adhering to the cells of microorganisms are sinking evenly in all directions. Bubbles adhering to cells are sinking from the free side so that the surface of the liquid with a powerful force strikes the cell wall breaking it and releasing intracellular compounds. The striking force may be so powerful that the released compounds are often destroyed and free radicals are formed.

Daulah [22], using the theory of local isotropic turbulence Kolmogorov [26 and 27], presented a description of the process of ultrasonic disintegration of cells of baker's yeast as a model

1Sp=exp(kt),E88

where: Sp – concentration of released proteins, [g/kg]

t – duration of the process; [min],

k – process rate constant; [1/min].

Dependence of constant k on energy dissipation Pd is described by the relationship

k=ζ(PdPc)0,9E89

where: ζ - constant; [kg/J],

Pc - threshold energy dissipation ensuring occurrence of cavitation; [J/kgs].

Confirming the above results experimentally for brewing yeast showed no effect of suspension concentration on constant k and its proportionality to the energy dissipation level [25].

The process of disinfection consists in an impact of physical or chemical agents in a limited gas, liquid or solid medium on biological contaminants. They may be it viruses, bacteria and their spores, protozoa and their cysts and eggs of parasites. The purpose of disinfection is to destroy the above mentioned objects and to prevent in a required, limited period of time their re-growth. The disinfectants selected according to pathogens cause: an irreversible destruction of cells, disruption of metabolic processes, disruption of biosynthesis and growth. An example of such process is chemical disinfection of water [28]. Its kinetics is often described by Chick’s model [3] in form of equation (77). The dependence of the rate constant

k=ACstE90

taking into account the disinfecting power coefficient A and the concentration of disinfectant C was given by Watson [4]. Exponent s is dependent on the type of disinfectant and medium’s pH.

The developed theory can be used to model the impact of population of predator-prey or competition type. If we assume that:

  • change objects from the set of N (f31 from Eq. (6)) in equation (75) is described under the law of Malthus by the formula

f31=a1NE91
  • rate constant from equation (75) will be marked as kγ and is written in the form

kγ=a2P,E92

directly dependent on the generated families of transformation volume (see formula (71)),

  • unconverted objects N cause an increase in the families of transformation volume of families in the form of a3NP,

in the absence of objects N (N is an empty set), there is an exponential decrease in the number of families of transformation volume – a4P,

the received equations in the form

dN=N(a1a2P)dtE93

and

dP=P(a3Na4)dtE94

are a model for predator-prey for the Lotka-Volterra system [1 and 2]. The coefficients a1, a2, a3 and a4 are constants with positive values.

On the basis of the presented theory it is possible to build other more realistic models of predator-prey system taking into account e.g. natural selection, nutrient profile, the effect of age on the activity of predators and many other factors. It is also possible to model chemical reactions, and assuming the generation in space V of different types of transformation volume, including reversible transformation, enzymatic reactions can be modeled [5].

The theory can be used to study and model the action of immune system [10]. It can also be employed in constructing artificial immunology and controlling the support of various therapies. Simple applications cover responses of the immune system to viral, bacterial, fungal and parasitic infections. The immune response depends on the type, properties, portal of entry and severity of infection and the state of organism. Application of the theory is illustrated by viral infection. Virus replication in the host cells in space V is represented by the function in Eq. (6). The aim of an immune response is to inhibit virus replication in the cell and its spread to other cells. Next goals include the elimination of transformed objects from the organism and development of long-lasting immunity. Initially, transformation volumes Vγji are formed as a result of action of the complement system, interferon (IFN) and natural killer cells (NK) within non-specific anti-viral response. At the next stage, neutralizing antibodies, mainly of class IgG (immunoglobulin G), prevent infection of other cells. Their fragment Fab (fragment antygen binding) binds to antigens of the virus, while fragment Fc (fragment crystallizable) to relevant receptors on NK cells, macrophages and others. This enables phagocytosis and cellular cytotoxicity dependent on antibodies, but probably also immediate destruction with the use of the complement system. In the case of infections through mucous membranes of the intestinal tract and respiratory system the same role is played by IgA (immunoglobulin A) antibodies. In development of humoral immune response the active role is played by CD4+ lymphocytes. In this case antigen proteolysis occurs in endosome. Cytotoxic CD8+ lymphocytes play the crucial role in the response to viral infections. They function inside the cells. They recognize a viral antigen on cells transformed by the infection in association with MHC (major histocompatibility complex) class I molecules. In the infected cells, cytotoxic lymphocytes can induce a synthesis of nucleases destroying genetic material of the virus and enhance IFN synthesis. Responses of the immune system occurring at subsequent stages of antigen destruction according to different mechanisms can be described by the relation analogous to Eq. (81).

Many pathogens developed such properties which allow them to avoid non-specific and specific immune response. High variability of a pathogen caused by differences in the gene sequence region leads to a delay of specific immune response. This is so in the case of influenza, hepatitis C and HIV (human immunodeficiency virus) viruses. Ho et al. [9] studied the effects of HIV treatment with inhibitors. A linear model based on experimental data in the form concordant with Eq. (77) well described the healing process at the first stage of the therapy. At the final stage, however, control over the virus population was lost due to multiplication of drug-resistant strains. A result of genetic modifications is the formation of virus mutations in transformation conditions γlb resulting from their properties which belong to set Pr. In volumes Vγji(γ<γlb)formed by the drug, the concentration of transformation conditions γ is smaller than that required for the transformation of so mutated viruses. Similar effects are observed in presently used, much more efficient highly active antiretroviral therapy (HAART). It includes the interactions of protease inhibitors combined with reverse transcriptase inhibitors. A mathematical model of the combined therapy was developed by Perelson et al. [29]. The same relations can be generated from Eq. (81). Objects from set N according to their properties may be vulnerable to any of the means used in combination therapy (usually five) or to none. In the case of AIDS (acquired immunodeficiency syndrome), HAART does not result in patient’s recovery despite a decrease of virus concentration below the detectability threshold. One of the reasons can be hiding of the virus in the milieu of limited immune response, such as brain and testes. They are unreachable for therapy just like memory lymphocytes and dendritic cells which are also a target of HIV attack. In the phenomenological model (Fig. 3), regions inaccessible for transformations are volumes (brain, testes) and Vαc (dendritic cells, memory lymphocytes). Objects can remain in them until producing mutation which is resistant to the applied drug combination, capable of regaining the whole space V.

Now, to model an artificial immune system, the shape space concept proposed by Perelson and Ostera [30] is often used. Antigen and antibody were determined as a point in the L-dimensional space of complementary traits. The notion of threshold ε determines the level of imperfectness of fitting of the antigen-antibody activation.

The idea of the transformation process preceded by a process of forming objects from a set N (Section 2.2) can be used to model the epidemic taking into account such factors as vaccinations or immunization. The solution achieve for the simple SIR model (division of the individuals: susceptible, infecting and convalescents with acquired resistance) is consistent with a simple epidemic model developed by Kermack and McKendrick [31].

6. Summary

The presented theory relates to the physical, chemical and biological processes of random transformation of dispersed matter. It has the interdisciplinary significance allowing the phenomenological and mathematical modeling of mass transfer processes in many areas. These include among others: industrial technology, ecology and environment protection, medicine, veterinary medicine, immunology, oncology, epidemiology, hygiene and agriculture. Specific descriptions of the processes create the possibility of linking phenomena, mechanisms and factors determining the process. Obtaining a correct modeling effect must be preceded by knowledge and deep understanding of the nature of the problem. However, full success depends on the proper, conducted on the basis of areas considered process, final interpretation of the results. The presented general concept of transformation of matter can be used both to study and describe the processes as well as for their management and control. It systematizes a range of knowledge concerning the transformation of dispersed matter whose nature has not previously been combined into unity. Together with the given methodology of building phenomenological and mathematical models it makes a platform on which it seems possible to achieve significant scientific development among others through analogies, critical comparisons and transfer of knowledge.

Acknowledgments

The study was carried out within the frames of the grant W-10/1/2012/Dz. St.

© 2013 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution 3.0 License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Marek Solecki (July 24th 2013). The Theory of Random Transformation of Dispersed Matter, Mass Transfer - Advances in Sustainable Energy and Environment Oriented Numerical Modeling, Hironori Nakajima, IntechOpen, DOI: 10.5772/52369. Available from:

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