Arch. Min. Sci., Vol. 53 (2008), No 2, p. 271–292

Arch. Min. Sci., Vol. 53 (2008), No 2, p. 271–292
271
MARIAN KOLARCZYK*
THE ANALOGY BETWEEN THE CONDITIONS CONCERNING AIRFLOW OUTPUTS SENSITIVITY
SIGNS TO CHANGES IN THE RESISTANCE OF SIDE BRANCHES AND H. CZECZOTT’S CONDITIONS
CONCERNING THE FLOW DIRECTION IN A SIMPLE DIAGONAL NETWORK
ANALOGIA W ZAKRESIE POSTACI WARUNKÓW DOTYCZĄCYCH ZNAKÓW WRAŻLIWOŚCI
WYDATKÓW POWIETRZA NA ZMIANY OPORÓW BOCZNIC DO WARUNKÓW H. CZECZOTTA NA
KIERUNEK PRZEPŁYWU W SIECI PRZEKĄTNEJ PROSTEJ
The scope of the paper is the problem of the sensitivity of airflows to changes in the resistance of
side branches in simple diagonal networks. Sensitivity εir of air output Vi to an increase of side branch
resistance Rr is the limit of the differential quotient ΔVi /ΔRr, where ΔRr is approaching zero. Reference
is made to the methods of designating sensitivity illustrated by examples of an ancillary network. The
ancillary network describes the propagation of a disturbance that appears in a basic network, evoked by
changes in the resistance of a selected side branch. In diagonal networks sensitivity signs are not always
explicit. This feature distinguishes diagonal networks from normal networks, due to the so called: “relative
diagonality” of side branches. In an ancillary network other side branches are diagonal in relation to the
source branch that enforces flows. By employing the ancillary network the conditions concerning the
sensitivity signs in a simple diagonal network were derived for the couples of side branches belonging to
the same cross-section and for the couples belonging to the same route. For some couples in the crosssection the signs were determined as positive or negative. Likewise, for some couples belonging to the
same route the sings can also be positive or negative. Accordingly, some dependencies between airflow
outputs and resistance may be increasing, very small (insignificant) or decreasing, which may be resolved
on the grounds of sensitivity. A regularity distinguishing simple diagonal networks from normal networks
was indicated. Hence, the problem of side branches diagonality concerns not only airflow directions
but also sensitivity signs. The form of the conditions determining the sensitivity signs is very similar to
H. Czeczott’s conditions describing the direction of airflows in a diagonal network.
Keywords: colliery ventilation network, changes in side branches resistance, diagonal side branches,
disturbance in airflow, airflow outputs sensitivity to changes in the resistance of side branches
*
SILESIAN TECHNICAL UNIVERSITY, UL. AKADEMICKA 2, 44101 GLIWICE, POLAND
272
W artykule przeanalizowano zagadnienie wrażliwości prądów powietrza na zmiany oporów bocznic
w sieci przekątnej prostej. Przez wrażliwość εir wydatku powietrza Vi na przyrost oporu bocznicy Rr rozumie się granicę ilorazu różnicowego ΔVi /ΔRr przy ΔRr dążącym do zera. Przypomniano znane metody
wyznaczania wrażliwości oraz ilustrację tych metod z wykorzystaniem sieci dołączonej. Sieć dołączona
obrazuje rozchodzenie się zaburzenia, które powstaje w sieci podstawowej przy zmianie oporu w wybranej
bocznicy. W sieciach przekątnych znaki wrażliwości nie zawsze są jednoznaczne. Odróżnia to sieci przekątne od normalnych. Istotna w tym zakresie jest tzw. względna przekątność bocznic. W sieci dołączonej
inne bocznice są przekątne względem bocznicy źródłowej wymuszającej przepływy. Korzystając z sieci
dołączonej wyprowadzono warunki dotyczące znaków wrażliwości w sieci przekątnej prostej dla par
bocznic należących do tego samego przekroju przez sieć oraz dla par bocznic należących do tej samej
drogi. Stwierdzono, że znaki te dla niektórych par w przekroju mogą być dodatnie lub ujemne. Podobnie
dla niektórych par bocznic należących do tej samej drobi znaki te mogą być dodatnie lub ujemne. Tym
samym niektóre zależności wydatków powietrza od oporów mogą być rosnące, bardzo słabe lub malejące. Można to rozstrzygnąć znając wrażliwości. Stwierdzone prawidłowości odróżniają sieć przekątną
prostą od sieci normalnych. Zagadnienie przekątności bocznic w sieci dotyczy więc nie tylko kierunków
przepływu powietrza lecz także znaków wrażliwości. Postać podanych warunków rozstrzygających
o tych znakach jest bardzo podobna do warunków H. Czeczotta, które rozstrzygają o kierunku przepływu
powietrza w bocznicy przekątnej.
Słowa kluczowe: kopalniana sieć wentylacyjna, zmiany oporów bocznic, bocznice przekątne, zaburzenia
w przepływie, wrażliwości wydatków powietrza na zmiany oporów bocznic
1. Czeczott’s conditions describing the direction of airflows
in a diagonal network
To provide the background for the issues discussed in this paper, reference should
be made to H. Czeczott’s conditions describing the direction of airflows in diagonal
side branches of simple diagonal networks (Czeczott, 1957; Budryk 1961). After some
modifications, these conditions may be expressed as (Fig.1):
if
R3 R4
>
R1 R2
then
V6 > 0
(1a)
if
R3 R4
=
R1 R2
then
V6 = 0
(1b)
if
R3 R4
<
R1 R2
then
V6 < 0
(1c)
where:
Ri, for i = 1,2,3,4 — aerodynamic resistances of side branches,
V6 — airflow volume output in diagonal side branch 6.
In the case of the orientation of a diagonal side branch (Fig. 1) from node 3 to node 2,
V6 > 0 signifies the airflow in the side branch in accordance with the assumed orientation,
273
H5(V5)
_
4
4
_
3
2
V6
_
6
_
2
3
_
1
1
Fig. 1. Canonical diagram of a simple diagonal network
Rys. 1. Schemat kanoniczny sieci przekątnej prostej
V6 = 0 signifies the stop of the airflow, whereas V6 < 0 demarks the airflow opposite to
the assumed orientation (reverse airflow).
As Czeczott’s conditions do not specify resistance R6 of a diagonal side branch and
main fan dam H5, the airflow direction is derived solely on the grounds of the relations
between the neighbouring side branches.
The graphic illustration of Czeczott’s conditions was discussed in (Kolarczyk, 1980;
Frycz & Kolarczyk, 1985), where the authors indicated that the direction of airflows
is determined by equivalent characteristic Hz(Vz) of the neighbourhood of the diagonal
of side branch 6. The function of this characteristic is influenced by the side branches
resistances Ri = 1,2,3,4. The changes in the resistance lead to the displacement of the
characteristic to the beginning of the system (Vz, Hz) and to the 3rd quadrant. The intersection of this characteristic with characteristic W6 = R6 · W62 of side branch 6 in
the 1st quadrant of the system (Vz, Hz) signifies a straight airflow direction (V6 > 0), the
intersection at the beginning of the system signifies the stop of the airflow, whereas
the intersection in the 3rd quadrant signified the reverse airflow (V6 < 0), where: R6, W6
denote resistance and decrease in pressure in side branch 6. As mentioned above, the
illustration of Czeczott’s conditions and the conclusions drawn from their analysis were
discussed in papers (Kolarczyk, 1980; Frycz & Kolarczyk 1985).
The main difference between the networks containing diagonal side branches and the
networks containing the standard ones is not only the possibility of the reversal of the
diagonals of airflows, but also other distinguishing regularities in terms of the sensitivity
of airflow outputs to the changes in the resistance of side branches (Kolarczyk, 1980,
274
1993; Frycz & Kolarczyk 1988). This problem will be analysed below for a simple
diagonal network. The conditions concerning sensitivity shall be presented, the form of
which is very similar to Czeczott’s conditions. The analysis is pertinent for the assumptions concerning the so called passive networks.
2. Sensitivity of airflow outputs to changes in the resistance of side branches
The process of underground coal mine airing under ordinary and emergency circumstances takes advantage of many coefficients concerning the stability of airflows
directions and volumes, as well as of the sensitivity of airflow outputs to changes in the
resistance of side branches. Sensitivity εir of air output Vi (response) to an increase of side
branch resistance Rr is understood as the limit of the following differential quotient:
e ij = lim
R j ® R ja
Vi - Via
D Vi
¶ Vi
( R ja ) = tg a
= lim
=
R j - R ja DR j ® 0 D Rj
¶R j
(2)
where:
Vi, Via — airflow output in i-th side branch
ΔVi = Vi – Via — change in airflow output in i-th side branch evoked by j-th change
in resistance Rj
Rj, Rja — resistance of j-th side branch,
ΔRj = Rj – Rja — change in resistance Rj in j-th side branch,
i, j = 1, 2,3…, m — number of side branches in the network,
tgα — direction coefficient of the tangent towards the graph of the relation: Vi (Rj)
Index α refers to the current condition of the airflow in a ventilation network, which
is also shown in Fig. 2. The direction coefficient tgα of the tangent towards the Vi (Rj)
graph is the searched sensitivity and illustrates increasing, insignificant or decreasing
nature of relation Vi (Rj) in the neighbourhood of the current operation point Rja. To
simplify the notation, sensitivity shall be demarked as εij.
In general, matrix E of the airflow outputs sensitivity to changes in the resistance of
side branches may be considered, which is expressed in the following way:
E = [εij]
i, j = 1,2, …, m
(3)
The above matrix is a square one, with the dimensions: m × m. The values in the
particular rows of the matrix contain the information on the manner in which i-th airflow
output Vi responds to changes of all resistances Rj (j = 1,2,…, m), the values in the columns describe the manner in which selected resistance Rj influences all airflow outputs
Vi (i = 1,2,…, m). The values of sensitivity εij and, especially, their signs, are not always
explicit in networks of various structures.
275
Vi
eij @0
Vi (Rj)
Vi (Rj)
Via
Vi (Rj)
tga =e ij >0
tga = e ij <0
a
a
Rja
Rja
Rja
Rj
Fig. 2. Sensitivity εij as a directional coefficient of tangential tgα to the graph of Vi (Rj)
Rys. 2. Wrażliwość εij jako współczynnik kierunkowy stycznej tgα do wykresu zależności Vi (Rj)
3. Outline of the methods of designating the sensitivity of airflow outputs
to changes in the resistance of side branches
Authors of professional publications devoted to airflows through coal mine ventilation networks noticed the relevance of using the concept of the sensitivity of airflow
outputs to changes in the resistance of side branches. This variable, together with other
associated quantities, were employed by E. Simode (1976) to illustrate and analyse
changes in airflow outputs evoked by changes in the resistance of side branches and fan
dams. However, Simode did not identify effective methods of designating sensitivity
coefficients for the side branches in the entire network. In other publications (Bojko
et al., 1975; Roszczynialski, 1988) proposals were made to designate sensitivity coefficients by means of the incremental method from two network solutions, or by means
of approximating relation Vi (Rj) derived from many network solutions and followed by
the calculation of the value of differential coefficient dVi /dRj.
An interesting method of designating an arbitrary row of sensitivity matrix E was
proposed by J. Chojcan (1975). The designation of an arbitrary column of this matrix, on
the grounds of some modifications of Chojcan’s method, was carried out at the Institute
of Mining Aerology, Silesian University of Technology (Kolarczyk, 1980, 1993).
Chojcan’s method of designating an arbitrary i-th row of matrix E is based on
Tellegen’s theorem well recognized in electro-technology. Ancillary flow network S^
(Fig. 3a, b) is constructed in a special way, its structure identical with basic network S.
The elements (side branches) of network S^ have resistances Ri^ equal to:
276
RiÙ =
dWi
(Via )
dVi
(4)
where the value of the differential coefficient is derived for the current operation point
of i-th side branch of basic network S. Relations Wi (Vi) are the characteristics of the side
branches in basic network S and assume the following form:
– for side branches with resistance
Wi = Ri × Vi × Vi
(5)
– for side branches containing the fan with characteristic Hi (Vi)
Wi = - Hi × ( Vi )
(6)
Under the operation corresponding to the declining graph of the fan characteristic
Hi (Vi), resistance Ri^ of this side branch in ancillary network S^ assumes a positive
value. For the side branches, which in the basic network have the assumed airflow output Vi = constant, in ancillary network Chojcan allocated flow Vi^ equal to zero (which
is equivalent to the existence of infinitely big resistance Ri^ in this side branch). The
constructed ancillary network S^ is a linear one. The characteristics of the passive elements of this network are the following:
Wi Ù = RiÙ × Vi Ù
(7)
where:
Wi^ is the equivalent of voltage decrease in the side branch,
Ri^ is the equivalent of resistance,
Vi^ is the equivalent of the current in a linear power network.
To designate i-th row of matrix E, i.e. the sensitivity of selected airflow output Vi to
changes in the resistances of all side branches i, i.e. to designate the values of differential
coefficients ∂Vi /∂Rj, i = 1, …, m, Chojcan installed the source; Hi^ = 1 in i-th side branch
of ancillary network S^ (Fig. 3b), enforcing the flow in this network. By solving network
S^, flows (currents) Vi^ are obtained, which, in the next step, are used in the calculations
of sensitivity ∂Vi /∂Rj. After that, the sensitivity of i-th airflow output to the change in
i-th resistance is derived from the following equation (Chojcan, 1975):
¶Vi
= -Vi Ù × Vi × Vi
¶ Rj
j = 1, 2, ... m
(8)
Detailed substantiation of the method of designating the sensitivity of airflow outputs
to changes in the resistance of side branches and to changes of other parameters (fan
dams, given air outputs) was provided by Chojcan. The description of the method was
also discussed in (Kolarczyk, 2004).
277
a)
b)
S
H
_ 6
4 5
3
S^
_
2
_
3
_
6
_
4
_
1
dR1
4
S*
2
4 _
5
3
_
2
_
3
H 1^ =1
_
1
1
_
5
3
_
2
2
c)
_
3
_
6
_
4
2
H1* = -V1
2
_
1
1
_
6
_
4
1
Fig. 3. Basic normal network S (a), ancillary network S^ for designating the row of the sensitivity
matrix in accordance with Chojcan’s proposal (b), ancillary network S* for designating
the column of the sensitivity matrix (c)
Rys. 3. Sieć podstawowa S normalna (a), sieć dołączona S^ do wyznaczania wiersza macierzy
wrażliwości wg Chojcana (b), sieć dołączona S* do wyznaczania kolumny wrażliwości (c)
To designate another row of sensitivity matrix E the calculations must be reiterated.
In such case, ancillary network S^ is almost identical, the only difference being that
source Hi^ enforcing airflows Vi^ in network S^ (Fig. 3b) is installed in the successive
side branch.
To derive the values of the terms in any r-th column of sensitivity matrix E describing the airflow outputs to the change in the resistance of r-th side branch, the system
of equations of the network equilibrium (Kolarczyk, 1980, 1993) may be transformed
by differentiating them in relation to resistance Rr, by constructing ancillary network
S*, which is identical to the one proposed by Chojcan, and, consequently, by providing
the solution of such network. Detailed substantiation of this method was discussed by
Kolarczyk in (Kolarczyk, 1993, 2002, 2003; Frycz & Kolarczyk, 1988).
The constructed ancillary network S* used for the determination of r-th column of
matrix E may be described by the following system of equations (9):
ìm
j = 1, 2, ..., n –1
ïå sji × eir = 0
ï i =1
ím
when k-th cycle does not con tain r -th side branch
ï c × R * × e = ì0
í
å
ki
i
ir
2
ï i =1
î- cki × Vr when k -th cycle contains r -th side branch
î
(9a)
(9b)
(9c)
278
where:
– sji, cki are the elements of the nodular-side branch incidence matrix S and cyclicalside branch matrix C in basic network S and ancillary network S*. These elements
assume the values of –1, 1, 0, following the manner in which i-th side branch
belongs to j-th node and to k-th cycle,
– Ri* are resistances of the side branches in ancillary network S* designated by
means of the method identical to the one proposed by Chojcan (1975), i.e. Ri* = Ri^
(eq. 4),
dVi
( Rra ) is airflow Vi* (current) in the ancillary network equal to the sensi– e ir =
dR r
tivity of i-th airflow output to the change of r-th resistance,
– n is number of nodes in the network,
– k = 1,2,…, v, is number of cycles in the network,
– v is cyclomatic number of the network,
– Vr is airflow output in r-th side branch, where an elementary change of resistance
dRr occurs.
The emerged system of equations (9), in view of the unknown values of sensitivity
εij, is a system of linear equations with m unknowns. The number of the equations is:
n–1+v=n–1+m–n+1=m
(10)
and is equal to the number of the unknowns εir = Vi*. The matrix of the equation system
coefficients consists of numbers sji, cki, Ri*. The three terms column contains zeros for the
equations derived from the node equations and cycle equations that do not contain r-th
side branch; whereas numbers –ckr Vr2 for the equations formed from the cycle equations
containing r-th side branch. Such system may easily be solved by commonly recognized
methods. However, huge networks require computer-aided calculations.
The system of equations (9) derived to designate sensitivity εij to the change in r-th
resistance (r-th column of matrix E) may be given a network interpretation (Fig. 3),
describing the airflow in a network which shall be referred to as ancillary network S*.
The structure of this network is the same as that of basic network S, and as Chojcan’s
ancillary network S^, because structural matrices S and C of the two networks are identical. Specific groups of the system of equations (9a, 9b, 9c) should be interpreted as
the equations for the nodes and cycles in ancillary network S*. The values of sensitivity
Vi* = εij are the searched airflows in this particular network. Due to the characteristic
of side branches: Wi* Ri* = εij for constant coefficients Ri* (resistances in network S*)
this network is linear. The free terms: ckr·Vr2 in some of equations (9c) for the cycles
containing r-th side branch should be treated as source Hr* = –Vr2 located in r-th side
branch, enforcing airflows Vi* in ancillary network S*. The analysis of the sign, under
the condition: Hr* = –Vr2 indicates that the source in r-th side branch enforces flow εrr in
279
ancillary network S* (Fig. 3c) the direction of which is reverse to the airflow in r-th side
branch of basic network S (Fig. 3a). Sensitivity εrr of airflow output Vr to an increase in
resistance Rr is negative, which is obvious, since any increase of Rr evokes a decrease
of airflow output Vr in basic network S.
It is interesting to compare the directions of airflow εir in ancillary network S* enforced
by source: Hr* = –Vr2 installed in r-th side branch to the directions of the airflows in basic
network S. This is illustrated in Fig. 3c, where the dotted arrows denote the direction of
airflow Vi in basic network S, whereas the broken arrows denote the direction of airflow
Vi* in ancillary network S*. The example shown in Fig. 3 represents a normal network,
which contains only parallel and series connections of the side branches or network
fragments. For only one airflow source: Hr* = –Vr2 in ancillary network S* (Fig. 3c)
the directions of airflows εir are explicit. The consistence of the broken arrow (ancillary
network S*) with the one marked on the side branch (basic network S) means a positive
value of εir. An increase in resistance Rr in the basic network leads to an increase in
airflow output Vi. The opposite directions of the arrows indicate a negative value of εir;
accordingly, an increase in Rr in basic network S leads to a decrease in airflow output
Vi. Ancillary network S* and the values of εir illustrate the propagation of a disturbance
in network S caused by an elementary increase in resistance dRr.
The example of basic network S and ancillary network S* shown in Fig. 3 was
intentionally simplified, as the mutual relations are clear. In more complex networks
with diagonal side branches, the signs of sensitivity εir are difficult to estimate without
making relevant calculations.
4. Sensitivity of airflow outputs to changes in the resistance of side branches
in a simple diagonal network
In (Kolarczyk 1993, 2005) the regularities concerning the signs of sensitivity εir of
airflow outputs Vi to changes in r-th resistance in a side branch in normal and certain
diagonal networks were discussed. As mentioned above, in normal networks sensitivities are always negative for the side branches belonging to the route that contains r-th
side branch (Fig. 3) and these sensitivities are always positive for the other airflows
referred to as: “split currents”. The regularities in normal networks do not always occur
in diagonal ones. The analysis of this issue shall be carried out for the simplest example
of a diagonal network with one diagonal side branch (Fig. 4a and 7a). In the course of
the analysis, ancillary network S* will be useful (Fig. 4b, c and 7a, b), which, as discussed in (Kolarczyk, 1993, 2003), illustrates the propagation of elementary changes in
airflow outputs dVi in the network evoked by a change in dRr of the resistance of r-th
side branch.
280
a)
b)
H5(V5)
S
3
3
5
e 51
S*
4
e 31
3
5
1
e 61
1
1
2
2
e 61
2
? e 41
2
e 11
1
e 21
H1* = V1
o
e 11
1
2
e 31
H1* = V1
?
6
2
R1®R1 + dR1
4
R3*
R6*
4
e 41
6
4
S*
3
4
c)
e 21
R1*
4
R4*
e 51
R2*
R5*
2
1
Fig. 4. Basic network S with one diagonal branch (a), ancillary network S*
used for determining sensitivity εi1 (b, c)
Rys. 4. Sieć podstawowa S z jedną bocznicą przekątną (a), sieć dołączona S*
przy wyznaczaniu wrażliwości εi1 (b, c)
4.1. Analysis of the sensitivity for the side branches belonging
to the same network cross-sections
In this part of the paper, the author shall prove that in a simple diagonal network the
sensitivity signs are not always explicit for the side branches belonging to the same crosssection, or, in other words, are not always unequivocal for the so called split airflows.
It is well known that in diagonal networks the directions of airflows in certain side
branches, even if only one operation source is working, depend on the pressure-resistance conditions. In a generalized case, for many side branches it is difficult to determine, without calculations, the direction of the airflow. Such is a specific property of
the network due to its structure. This property is transferred to ancillary network S*,
which, as proved before, has an identical structure as basic network S. In ancillary network S*, there will be diagonal side branches, for which the direction of airflow V*, i.e.
the sign of sensitivity εir, shall be dependent on resistances Ri* of the side branches of
the network. The diagonal nature of the side branches is inferred from the position of
the source that enforces the flow in a given network (Budryk, 1961; Kolarczyk, 1993,
2002). In the ancillary network, source: Hr* = –V 2 enforcing airflows εir is in r-th side
branch. In a generalized case, this side branch is different from the side branch with
fans in basic networks. In ancillary network S*, other side branches may have diagonal
properties towards source Hr. The use of intuition not substantiated by calculations in
the course of detecting the side branches, and, consequently, in determining the signs
of sensibility εir may be unreliable.
281
Let us assume that in basic network S of a simple diagonal in side branch 1 (Fig. 4a)
there occurs an increase in the resistance of the side branch with the value of dR1. Ancillary network S* used for designating sensitivity εir of airflows Vi, i = 1, …, 6, derived
from the occurred change in resistance dR1 is shown in Fig. 4b. In this network, the side
branches have resistance Ri* designated in accordance with equations (4, 4a); whereas
in side branch 1, the airflow source: H1* = –V12 was installed. In Fig. 4c, network S*
was represented in a manner that highlights the importance of source side branch 1. The
analysis of the nature of the side branches in reference to source branch 1 (Fig. 4c) indicates that in ancillary network S* side branches 2, 3, 5 and 6 are normal. The directions
of the airflows in these side branches in network S*, i.e. the signs of the corresponding
sensitivities are explicit. The positive values of ε21, ε61 and the negative values of ε31,
ε51 (Fig. 4c) comply with the expectations. The airflow outputs in these side branches
in the basic network (Fig. 4a) increase; correspondingly, and decrease when resistance
R1 increases. Side branch 4 in network S* is diagonal to source H1* installed in side
branch 1 (Fig. 4c). This means that the sign of sensitivity ε41 depends on resistances Ri*
of the remaining side branches in network S*. For a simple case with one diagonal side
branch 4 in ancillary network S* it is easy to specify the relevant conditions of the sign
of sensitivity ε41, i.e. the sign of differential coefficient dV4 /dR1 (Fig. 4c):
if
R6* R3*
>
R2* R5*
then
e 41 =
dV4
>0
dR1
(11a)
if
R6* R3*
=
R2* R5*
then
e 41 =
dV4
=0
dR1
(11b)
if
R6* R3*
<
R2* R5*
then
e 41 =
dV4
<0
dR1
(11c)
where:
Ri* — resistances of the side branches in ancillary network S*
R i* = 2 × Ri × Via
R5* = -
for i = 2, 3, 6
dH 5
( V5a )
dV5
In the case of the operation corresponding to the descending graph of the fan characteristic: H5(V5) the value of its differential coefficient is negative; accordingly, resistance
R5* is positive.
In such conditions the values of resistances Ri and airflow outputs Via are essential,
as well as the slope of the curve of the fan: (dH5 /dV5) in side branch 5 in current operation state A of network S. The relations between the corresponding resistances Ri* in
282
ancillary network S* determine the sign of sensitivity ε41, and, accordingly, the impact
of the increase in resistance R1 on the increase or decrease in airflow output V4. Normal
networks do not have such property of the ambiguity of the sign of sensitivity. In diagonal
networks, however, apart from the changes in airflow directions, there occur unclear
impacts of changes in the resistance of side branches on changes in airflow outputs.
The form of equations (11a, b, c) is similar to conditions (1a, b, c) proposed by Czeczott, but it does not refer to the sign of airflow outputs in a diagonal side branch, but to
the sign of sensitivity for the side branch diagonal to the branch in which a disturbance
arose. Czeczott’s conditions (1a, b, c) and conditions (11 a, b, c) result from the properties of a diagonal network. At the same time, it should be noticed that the issue of the
diagonality of side branches in the network is important in view of the sensitivity of
airflow outputs to changes in the resistance of side branches. In such case, the signs of
sensitivity are not always explicit.
If the selected part of the network is a sub-network, in view of conditions (11a, b, c)
the slope of the characteristic of the neighbourhood of Hz(Vz) is important (Kolarczyk,
1980, 1993), as it functions as an abstract fan, enforcing the flow in this network. It was
indicated in the above mentioned publications that the characteristic of the neighbourhoods of sub-network Hz(Vz) may have diverse angular coefficients tgα, which confirms
that the slope of this characteristic towards the axis of airflow output V is not uniform.
Moreover, the signs of sensitivity ε41 in the network shown in Fig. 4a should be considered in the case when the fan or the characteristic of the neighbourhood represented by
source side branch 5 is treated as an ideal source of pressure rise H5 = const. irrespective
of the air output, or as an ideal source of airflow output V5 = const. irrespective of the
pressure. As discussed in (Kolarczyk 1980, 1993), in such cases, resistances R5* of the
corresponding side branches in ancillary network S* assume specific values. If in basic
network S the source has the characteristic H5 = const. (Fig. 4a), in ancillary network S*
resistance R5* = 0 (Fig. 5b). R3*/R5* quotient in equation (11 c) is approaching infinity,
which results in meeting the condition of ε41 < 0. This is also clear in ancillary network
S*, where, in view of R5* = 0, extreme nodes 1 4 of the side branch may be joined – see
Fig. 5b. Accordingly, ancillary network S* becomes normal, whereas airflow ε41, i.e. the
corresponding sensitivity, is always negative.
The above conclusions may also be drawn in another way. Considering the constant
difference of aerodynamic potentials Φ1 – Φ4 , an increase in resistance R1 leads to an
increase in airflow output V2 (Fig. 4a). Accordingly, a decrease in thrust W2 rises, and
the difference of potentials Φ2 – Φ4 falls. Thus, airflow output V4 decreases.
If the source enforcing the airflow in basic network S has the characteristic V5 = const.
(Fig. 4a), then resistance R5* = ∞ in ancillary network S* (Fig 5c). R3*/R5* quotient in
condition (11a) is approaching zero, which results in meeting the condition of ε41 > 0.
When R5* = ∞ in ancillary network S*, side branch 5 can be eliminated. Accordingly,
ancillary network S* becomes normal, whereas flow ε41, i.e. the corresponding sensitivity, is always positive.
283
a)
b)
4
e 61
3
3
2
5
H1*
4
6
2
2
1
e 41
e 31
2
H1*
4
4
1
1
e 41
1
e 31
6
e 61
4
3
3
S*
S*
3
c)
3
2
2
e 21
e 21
1 4
e 11
1
e 11
Fig. 5. Example of a diagonal network with two main fans (a), reduction of ancillary
network S* from fig. 4c, with H5 = constant (b), reduction of ancillary network S*
from fig. 4c, with V5 = constant (c)
Rys. 5. Przykład sieci przekątnej z dwoma wentylatorami głównymi (a),
redukcja sieci dołączonej S* z rys. 4c przy H5 = constant (b), redukcja
sieci dołączonej S* z rys. 4c przy V5 = constant (c)
The above conclusions are consistent with the following interpretation of the relations:
an increase in resistance R1 (Fig. 4a) leads to a decrease in airflow output V3 which, in
consideration of constant airflow output V5, results in an increase of airflow output V4.
Examples of relation V4(R1) obtained from computer-aided simulation calculations of
airflow propagation in the analysed network were shown in Fig. 6. In the course of the
calculations R1 varied from zero to infinity. The resistances of the remaining side branches
were constant: R2a = 0,25 kg/m7, R3a = 0,16 kg/m7, R4a = 1,0 kg/m7, R6a = 0,3 kg/m7. The
values of other variables concerning the side branches in basic network S and ancillary
network S* are compiled in Table 1.
For the fan H5 = 486,4 Pa = const relation V4(R1) is diminishing (Fig. 6). If R1 is approaching infinity, airflow output R1 diminishes asymptotically to the value of V4(R1 = :)
= 13,8 m3/s. Conversely, for the fan V5 = 51,36 m3/s = const relation V4(R1) is increasing
(Fig. 6). If R1 is approaching infinity, airflow output R1 dincreases asymptotically to the
value of V4(R1 = :) = 20,7 m3/s. The third relation, V4(R1), derived from the calculations,
is also shown in Fig. 6., where the characteristic of the fan (or of the neighbourhood
of sub-network Hz(Vz) can be approximated by means of equation H5 = 1000 – 10 ·V5.
In such case, V4(R1) is also diminishing and assumes an intermediate position (Fig. 6).
If R1 is approaching infinity, airflow output R1 decreases asymptotically to the value of
V4(R1 = :) = 15,6 m3/s.
wp wk
3
2
4
4
1
3
—
—
—
—
—
—
—
—
—
1
1
3
2
4
2
bi
Ri
H
wp
Vi
Ri*
wk
Wi
εi1
bi
1
2
3
4
5
6
Vi
m3/s
21.18
30.17
35.28
16.08
51.36
14.09
Wi
Pa
287.2
227.6
199.1
258.7
0.0
59.5
Hi
Pa
0.0
0.0
0.0
0.0
486.4
0.0
27.117
15.088
11.289
32.169
0.0
8.456
Ri*
–1.313E+01
3.363E+00
–8.112E+00
–1.577E+00
–9.774E+00
4.941E+00
εi3
For
H5 = const. = 486.4 Pa
Resistance
Sensitivity
in S*
number of branch
side branch resistance, g/m7
fan pressure, Pa
initial node
airflow output, m3/s
side branch resistance in ancillary network S*
final node
decrease in pressure, Pa
sentivity of i-th airflow output to an increase of R1 in branch 1
Ri
g/m7
640.0
250.0
160.0
1000.0
0.0
300.0
Parameters of side branches under
current operation of basic network S
27.117
15.088
11.289
32.169
10.0
8.456
Ri*
–1.174E+01
5.348E+00
–5.877E+00
–5.199E-01
–6.397E+00
5.898E+00
εi3
For
H5 = 1000 – 10 ·V
Resistance
Sensitivity
in S*
TABLICA 1
TABLE 1
27.117
15.088
11.289
32.169
:
8.456
Ri*
–9.106E+00
9.106E+00
–1.483E+00
1.486E+00
0.0
7.623E+00
εi3
For
V5 = const. = 51.36 m3/s
Resistance
Sensitivity
in S*
Wyniki obliczeń rozpływu i wrażliwości εi1 wydatków powietrza na przyrost oporu w bocznicy 1 w sieci z Rys. 4
Results of the calculations of air distributions and sensitivity εi1 of airflow outputs to an increase in the resistance
in side branch 1 in the network from Fig. 4
284
285
V4 [m3/s]
20.7
21
asymptote
20
for H5 = const
19
for V5 = const
18
17
A
for H5 = 1000 – V5
16
15.6
15
13.8
14
asymptote
13
R1a
12
0
0.5
R1 [kg/m7]
1.0
1.5
2.0
2.5
Fig. 6. Example of the ambiguity of relation V4(Ri) in the diagonal network presented in Fig. 4
Rys. 6. Przykład niejednoznacznej zależności V4(Ri) w sieci przekątnej z Rys. 4
Surely, any change in resistance R1 may evoke changes in the direction of airflow V6.
In the analysed example, this occurs for R1 = 0,04 kg/m7. Accordingly, relations V4(R1)
shown in Fig. 6 were derived when V6 < 0 for R1 <0,04 kg/m7 and when V6 > 0 for
R1 > 0,04 kg/m7. The conclusion concerning the nature of V4(R1) in the neighbourhood
of a current operation point of the network can be drawn if the value of sensitivity ε41
is known. For the above mentioned input data and initial (current) resistance R1a = 0,64
kg/m7 the following values of ε41 were derived from the calculations of airflow propagations and sensitivities (see Table 1 – compilation of the results):
– if the fan operating in the network has the characteristic: H5 = 486.4 Pa = const
(Fig. 6, Table 1), ε41 = –1.58 m10/kg . s,
– if the fan operating in the network has the characteristic: H5 = 1000 – 10 . V (Fig. 6,
Table 1), ε41 = –0.52 m10/kg . s,
– if the fan operating in the network has the characteristic: V5 = 51.36 m3/s = const
(Fig. 6, Table 1), ε41 = 1.48 m10/kg . s.
In previously mentioned publications (Kolarczyk 1993, 2002, 2004) the following
conclusions were substantiated: relativity of the nature of some side branches for the
bottom position of the source side branch, symmetry of the matrices of the mutual relations of side branches, symmetry of sensitivity matrix E. In the example analysed above
(Fig. 4c) side branch 4 in ancillary network S* was diagonal when the source disturbing
the flow was located in side branch 1. This means that when the change in the resistance
occurs in side branch 4, and not in side branch 1, side branch 1 in the equivalent ancillary network is diagonal in relation to source side branch 4. Sensitivity ε14 of airflow
output V1 to change in resistance R4 is also ambiguous. The derivation of the equivalent
286
conditions concerning the sign of ε14 is possible following the same procedure as the one
used for designating conditions (11a, b, c) concerning the sign of sensitivity ε41.
Consequently, as indicated in the above analysis, in a simple diagonal network the
impact of increased resistance on the airflow output in the split that does not belong to
the same cycle – cell is not explicit. Such feature distinguishes diagonal networks from
the normal ones, where the sensitivity signs in the splits (or in the splits that belong to
the same cross-section of the network) are always positive. In a diagonal network, the
sign of sensitivity depends on the variables characterising the current state of airflow in
the network (conditions 11a, b, c).
Ancillary network S* shown in Fig. 5b is constructed from basic network S shown
in Fig. 5a. In this case, side branch 5 (closing branch) in basic network S has resistance
R5 = 0. Accordingly, resistance R5* in ancillary network S* also assumes the value of
zero. On these grounds, nodes 1 4 may be joined (Fig. 5b). Thus, ancillary network S*
becomes normal, and sensitivity ε41 is always negative. Relation V4(R1) in the network
shown in Fig. 5a is always decreasing. The signs of sensitivity εir may also depend on
the organization of the process or airing a given coal mine (several downcast and upcast
shafts).
4.2. Analysis of sensitivity for the side branches belonging
to the same route
In this part of the paper it will be proved that in a simple diagonal network the signs
of sensitivity are not always explicit for all couples of side branches belonging to the
same route. This feature distinguishes diagonal networks from the normal ones.
On the grounds of the regularities in the fist and fourth side branch couple, and in
consideration of the symmetry of straight diagonal (Fig. 4a), it may be inferred that the
identical ambiguousness of the signs of sensitivity also concerns the second and third
side branch couple (Fig. 7a). Basic network S shown in Fig. 7a, in which an increase of
resistance by the value of dR3 occurs in side branch 3, is compiled with ancillary network
S*, where in side branch 3, in accordance with the construction of this network, source
H3* = –V32 is installed, enforcing airflows εi3. Ancillary network S* was transformed to
highlight the importance of source side branch 3 (Fig. 7b). The directions of airflows
εi3 in side branches 1, 4, 5, and 6 in ancillary network S* are explicit, which confirms
the explicit nature of the signs of sensitivity. Sensitivity ε43 is positive; the other ones
are negative, which results from the consistency or inconsistency of the airflow directions in basic network S (arrows on the side branches) and ancillary network S* (broken
arrows). In particular, sensitivity ε63 is negative. An increase in resistance R3 in view
of a diagonal nature of side branch 6 in basic network S may cause, as already proved,
a decrease in airflow output V6 to negative values. The diagonal nature of side branch 2
in ancillary network S* should also be pointed out (Fig. 7b). Thus, the sign of sensitivity
ε23 is not explicit. Likewise before, on the grounds of the properties of this network the
conditions concerning the sign of sensitivity ε23 (Fig. 7b) may be designated:
287
a)
R5* R4*
>
R1* R6*
then e 23 =
dV2
>0
dR3
(12a)
if
R5* R4*
=
R1* R6*
then e 23 =
dV2
=0
dR3
(12b)
if
R5* R4*
<
R1* R6*
then e 23 =
dV2
<0
dR 3
(12c)
c)
b)
4
H5
4
S
3
2
H3* = V3
5
if
2
3
H3* = V
4
d)
S*
2
3
H3* = V
4
5
3
1
H3* = V
4
6
1
3
S*
4
2
2
3
1
2
2
2
2
2
3
3
?
1
1
4
S*
2
3
6
4
1
6
1
2
6
1
3
3
Fig. 7. Basic network S with one diagonal branch and ancillary network S*
used for determining sensitivity εi3 (a, b), reduction of the ancillary network presented
in Fig. 7b with H5 = constant (c) and V5 = constant (d)
Rys. 7. Sieć podstawowa S z jedną bocznicą przekątną i dołączona S* przy wyznaczaniu
wrażliwości εi3 (a, b), redukcja sieci dołączonej S* z Rys. 7b przy H5 = constant (c),
redukcja sieci dołączonej S* z Rys. 7b przy V5 = constant (d)
Quantities Ri*, likewise above (equations 4, 4a) are the resistances of the side
branches in ancillary network S*. Thus, it is the current state of airflow in the network,
i.e. resistances Ri of the side branches, airflow outputs Via and the slope of the characteristic of fan H5(V5) or the characteristic of the neighbourhood of sub-network Hz(Vz)
that determine the sign and value of sensitivity ε23. The relation between the quotients
of equivalent resistances Ri* in ancillary network S* is therefore essential and decisive
as far as the sign of sensitivity ε23 is concerned, and, accordingly, as far as the impact of
an increase in resistance R3 on an increase or decrease in airflow output V2. In addition,
the property of the ambiguity of the sensitivity sign does not occur in normal networks.
In diagonal networks, however, apart from changes in the airflow directions, ambigu-
288
ous impacts of changes in the resistance of side branches to changes in air distribution
may be expected.
Likewise, the form of equations (12a, b, c) is similar to conditions (1a, b, c) proposed
by Czeczott, although it does not concern the sign of airflow output in a diagonal side
branch but the sign of sensitivity for the side branch diagonal to the side branch in which
a disturbance occurred. Czeczott’s conditions (1a, b, c) and the discussed conditions
(12a, b, c) are implied from the properties of diagonal networks. So, it should be stated
that the issue of the diagonality of side branches in the network is important also in view
of the sensitivity of airflow outputs to changes in the resistance of side branches. In the
analysed case, the sensitivity signs are not always explicit.
In the next step, conditions (12a, b, c) can be analysed for the cases in which the fan
characteristic: H5(V5) in network S (Fig. 7a) is specific. If the fan has horizontal characteristic H5 = const, or vertical V5 = const, resistance R5* in ancillary network assume
extreme values: R5* = 0, R5* = ∞, respectively. Ancillary network S* is also subject of
reduction (Fig. 7c, d) and becomes a normal network. Side branch 2 loses its diagonal
nature in relation to airflow source H3*. In such specific cases the sign of sensitivity ε23
becomes explicit. If H5 = constant, sensitivity ε23 is always negative (Fig. 7c). This means
that airflow output V2 decreases with an increase in resistance R3. With V5 = constant,
sensitivity ε23 is always positive (Fig. 7d). This means that airflow output V2 increases
with an increase in R3. These conclusions may also be inferred from conditions (12a,
b, c). Examples of relations V2(R3) were shown in Fig. 8.
Considering the fan: H5 = 486,4 Pa = const, relation V2(R3) is diminishing (Fig. 8).
If R3 is approaching infinity, airflow output V2 diminishes asymptotically to the value:
V2 [m3/s]
for H5 = const
36
34
33.9
for V5 = const
asymptote
A
32
30
for H5 = 1000 – V5
28
24
22
20
18
16
14
12
17.1
asymptote
13.8
asymptote
R3a
10
0.1
R3 [kg/m7]
0.2
0.3
0.4
0.5
0.6
0.7
Fig. 8. Example of the ambiguity of relation V2(R3) in the diagonal network presented in Fig. 7
Rys. 8. Przykład niejednoznacznej zależności V2(R3) w sieci przekątnej z Rys. 7
289
V2(R3 = :) = 13.8 m3/s. Conversely, for the fan: V5 = 51.36 m3/s = const. relation V2(R3)
is increasing (Fig. 8). If R1 is approaching infinity, airflow output V2 increases asymptotically to the value: V2(R3 = :) = 33.9 m3/s. The third relation: V2(R3) derived from
the calculations, is also shown in Fig. 8, in which the characteristic of the fan (or of the
neighbourhood of sub-network Hz(Vz) can be approximated by means of equation of
straight line: H5 = 1000 – 10 .V5. In such case V2(R3) is also diminishing and assumes an
intermediate position (Fig. 8). If R3 is approaching infinity, airflow output V2 decreases
asymptotically to the value: V2(R3 = :) = 17.1 m3/s.
Under the current operation conditions of the network, when R3a = 0.16 kg/m7 and
with the resistances of the remaining side branches: R1a = 0.64 kg/m7, R2a = 0.25 kg/m7,
R4a = 1.0 kg/m7, R6a = 0.3 kg/m7 the corresponding sensitivities ε23 derived from the calculations on the grounds of the proposed method are the following (also see Table 2):
– if H5 = constant
ε23 = –22.4 m10/kg .s,
– if H5 = 1000 – 10 .V5 ε23 = –13.2 m10/kg .s,
– if H5 = constant
ε23 = 4.11 m10/kg .s.
As explicated above, the sensitivities are the direction coefficients of the tangential
lines in relation to the corresponding relations V2(R3) under current operation state A of
the network (Fig. 8). The information on sensitivities make it possible to forecast the
nature of relations V2(R3). In the course of calculations, these quantities may be analysed
by means of a software program and used in further research.
Identical remarks concerning the ambiguity of the impact of resistance changes can
also be applied for the case in which a change in the resistance occurs in side branch 2
(Fig. 7). In equivalent ancillary network S*, the third side branch is diagonal. Thus, the
sign of sensitivity ε32 depends on side branches resistance R* in ancillary network S*. In
this case, the derivation of relevant conditions is very easy. The ambiguity results from
the mutual diagonal nature of the second and third side braches and the symmetry of
the signs in sensitivity matrix E (Kolarczyk, 1993, 2002, 2004).
The second and third side branches at the initial state of the network operation belong
to the route from the initial node to the final node (wps → wks), (in accordance with the
nodes numbers: 1 2 3 4 ) – see Fig. 7a. The ambiguity of the sensitivity signs in diagonal
networks may also concern the side branches belonging to the same route. Such property
does not occur in normal networks, where sensitivities to an increase in resistance for
all side branches belonging to the same route are negative.
5. Conclusions
1. In a simple diagonal network the sign of airflow outputs sensitivity to changes in
the resistance of side branches, for certain couples of side branches that belong
to the same cross-section, may be positive, zero or negative. Accordingly, the
1
1
3
2
4
2
1
2
3
4
5
6
Ri
g/m7
640.0
250.0
160.0
1000.0
0.0
300.0
Vi
m3/s
21.18
30.17
35.28
16.08
51.36
14.09
Wi
Pa
287.2
227.6
199.1
258.7
0.0
59.5
Hi
Pa
0.0
0.0
0.0
0.0
486.4
0.0
27.117
15.088
11.289
32.169
0.0
8.456
Ri*
–2.272E+01
–2.240E+01
–5.564E+01
1.050E+01
4.513E+01
–3.291E+01
εi3
For
H5 = const, = 486.4 Pa
Resistance
Sensitivity
in S*
εi3 — sentivity of i-th airflow output to an increase of R3 in branch 3
3
2
4
4
1
3
wp wk
bi
Parameters of side branches under
current operation of basic network S
27.117
15.088
11.289
32.169
10.0
8.456
Ri*
–1.629E+01
–1.324E+01
–4.493E+01
1.539E+01
–2.953E+01
–2.863E+01
εi3
For
H5 = 1000 – 10·V
Resistance
Sensitivity
in S*
TABLICA 2
TABLE 2
27.117
15.088
11.289
32.169
:
8.456
Ri*
–4.113E+01
4.113E+00
–2.464E+01
2.464E+01
0.0
–2.053E+01
εi3
For
V5 = const, = 51.36 m3/s
Resistance
Sensitivity
in S*
Wyniki obliczeń rozpływu i wrażliwości εi3 wydatków powietrza na przyrost oporu w bocznicy 3 w sieci z Rys. 7
Results of the calculations of air distributions and sensitivity εi1 of airflow outputs to an increase in the resistance in side branch 3
in the network from Fig. 7
290
291
relations between the airflow outputs and the resistance of side branches for these
couples may be increasing, insignificant or decreasing. This feature distinguishes
a simple diagonal network from normal networks, where the relations between
the side branch couples are always increasing.
2. In a simple diagonal network the sign of airflow outputs sensitivity to changes in
the resistance of side branches, for certain couples of side branches that belong to
the same route through the network, may be positive, zero or negative. Accordingly,
the relations between the airflow outputs and the resistance of side branches for
these couples may be increasing, insignificant or decreasing. As already mentioned
above, this feature distinguishes a simple diagonal network from normal networks,
where the relations between the side branch couples are always decreasing.
3. The form of the proposed conditions concerning the signs of airflow outputs
sensitivity to changes in the resistance of side branches is similar to Czeczott’s
conditions, which determine the direction of airflows in a diagonal side branch. The
diagonality of side branches in diagonal networks is important not only in view of
the direction of airflows, but also in view of the signs and values of sensitivity.
4. The above analysis of the signs of sensitivity in a simple diagonal network does
not offer a complete in-depth description of the issue of ambiguity of sensitivity
in other diagonal networks, which should be subject of further research.
5. The knowledge of the signs and values of the sensitivity of airflow outputs to
changes in the resistance of side branches may be useful in the process of airing
underground coal mines, as well as in the calculations of air distribution and control
of ventilation networks.
REFERENCES
B o j k o B.A., P i e t r e c z u k A.A., C h a m a s u r i d z e W.D., 1975. Wzaimoswiazannost wentilacjonnych sietej szacht
pri otricatielnom regułirowanii wozduchorazpriedielenijem w odnoj iz wietwiej sieti. IWUZ Gornyj Zurnal, 9.
B u d r y k W., 1961. Wentylacja kopalń, cz. I, Przewietrzanie wyrobisk, WGH, Katowice. Czeczott H., 1957. Wybór
pism, PWN Warszawa.
C h o j c a n J., 1975. Sposób obliczania współczynników wpływu zmian wartości elementów sieci na natężenie przepływu
powietrza w bocznicach z wykorzystaniem tzw. „sieci dołączonej“. Przegląd Górniczy, 1975/1.
F r y c z A., K o l a r c z y k M., 1985. Classical criteria of direction in simple mine ventilacion network – An environment interpretation, Proceedings of the „21st International Conference of Safety in Mines Research Institutes“.
Sydney. 339-345.
F r y c z A., K o l a r c z y k M., 1988. Sensitivity of air volume outputs to changes in resistance of side branches in
a mine ventilation network. Proceedings of the „Fourt International Mine Ventilation Congres“. Brisbane, Oueensland. 125-131.
F r y c z A., K o l a r c z y k M., 1989. The suitability of methods used to detetmine the sensitivity of air the volume output
to changes in the resistance of side – branches in the ventilation control deep collieries. Proceedings of the „23rd
International Conference of the Safety in Mines Research Institutes“. Washington, DC, 337-349.
G e h e r K., 1976. Teoria tolerancji i wrażliwości układów elektronicznych. WNT, Warszawa.
K o l a r c z y k M., 1980. Własności i zastosowania charakterystyki podsieci i otoczenia w kopalnianej sieci wentylacyjnej.
Politechnika Śląska. ITEZ. Gliwice (niepublikowane).
292
K o l a r c z y k M., 1993. Wpływ struktury kopalnianej sieci wentylacyjnej na wrażliwości prądów powietrza przy zmianach oporów bocznic. Zeszyty Naukowe Politechniki Śląskiej, nr 1206, s. Górnictwo, z. 214. Gliwice, s. 202.
K o l a r c z y k M., 2002. Diagonality and relative diagonality of the all side branches in the ventilation network of mines.
Proceedings of the North American/Ninth US Mine Ventilation Symposium. Kingston. Ontario. Canada. 259-265.
K o l a r c z y k M., 2002. Метод спределения чувствительности расходов воздуха на изменение сопротивлений
ветвей в шахтной вентиляционной сети. Горный информационно-аналитический бюллетень. Аєрология,
Wyd. Moskiewski Państwowy Uniwersytet Górniczy. Moskwa, 54-59.
K o l a r c z y k M., 2003. Wrażliwościowa ilustracja rozchodzenia się zaburzenia w rozpływie powietrza spowodowanego
zmianą oporu bocznicy w kopalnianej sieci wentylacyjnej. W materiałach X Międzynarodowej Konferencji Naukowo-Technicznej na temat „Problemy koncentracji eksploatacji w warunkach zagrożenia tąpaniami i metanem“.
Ustroń-Katowice, Wyd. GIG Katowice, 71-81.
K o l a r c z y k M., 2004. Wybrane właściwości pełnej macierzy wrażliwości wydatków powietrza na zmiany oporów
bocznic w kopalnianej sieci wentylacyjnej. Materiały 3 Szkoły Aerologii Górniczej, Zakopane. Wyd. CEiAG EMAG
Katowice. Art. 6, 82-91
K o l a r c z y k M., 2005. Чувствительности расходов воздуха к изменению сопротивлений ветвей в шахтных
вентиляционных нормальных и диагональных сетях. Горный информационно-аналитический бюллетень.
Аєрология, Wyd. Moskiewski Państwowy Uniwersytet Górniczy. Moskwa, 159-171.
K o l a r c z y k M., 2007. Determination of a matrix column of air volume output sensitivity to changes in side branches
resistance in a mine ventilation network. VSB – TU Ostrava, HGF (w druku)?
R o s z c z y n i a l s k i W., 1988. Wpływ losowych i deterministycznych zmian oporu aerodynamicznego wyrobisk
górniczych na rozpływ powietrza w sieci wentylacyjnej. Zesz. Nauk. AGH, s. Górnictwo, z. 136, Kraków.
S i m o d e E., 1976. Determination de la nature (stable ou instable) des branches d’un reseau. Industr. Miner. Mine 2-76.
Aerage. Document SIM N 3.
Received: 17 June 2007