Journal
of Chemical
Technology
Metallurgy,
1, 2014
Journal
of Chemical
Technology
andand
Metallurgy,
49,49,
1, 2014,
82-89
GEOMETRICAL CHARACTERISTICS
OF THE SOLID BED IN A ROTARY KILN
Rayko Stanev1, Iliyan Mitov2, Eckehard Specht3, Fabian Herz3
Department of Physical Metallurgy
and Thermal Equipment
University of Chemical Technology and Metallurgy
8 Kl. Ohridski, 1756 Sofia, Bulgaria
E-mail: [email protected]
2
“Rua Bulgaria” Ltd., 36 Bogatitsa str.,
1421 Sofia, Bulgaria
3
University “Otto von Guericke”, Universitätsplatz 2,
39106 Magdeburg, Germany
1
Received 30 July 2013
Accepted 25 November 2013
ABSTRACT
Rotary kilns are aggregates for high temperature thermal treatment of a wide range of materials in a continuous
technological process. This type of furnaces is used
��������������������������������������������������������������������������
intensively in many industrial branches such as chemical, metallurgical, silicate, pharmaceutical, etc. In the metallurgy these units find application for heat treatment of bulk materials (e.g.
oxide ores reduction, limestone calcination, sulphide copper stock drying, cleaning of metal swarfs from machine oil, etc).
The present study enlarges the opportunities for a fast and reliable thickness determination of the separate zones of the
solid disperse bed at rolling motion as the most widespread regime of its transverse transport. Data published in a previous work of the authors are used. They are obtained by a mathematical model established on the basis of regularities in
the particles movement in a cylindrical rotary kiln, allowing determination of the total thickness of the layer of processed
material and the active part of it, occupying the area immediately below the free surface of the bed.
Approximation equations for prediction of the maximal values of the enumerated indicators depending on the inner
diameter of the rotary kiln, its rotation speed, the filling degree of the drum with material, the dynamic angle of its repose
and the particles diameter are proposed.
Keywords: rotary kiln, rolling motion, active layer.
INTRODUCTION
Rotary kilns serve for a high-temperature heat treatment of different materials in a continuous technological
process. Their sphere of application gradually expands
and develops [1���������������������������������������
-�������������������������������������
��������������������������������������
3].
������������������������������������
These units are used in many industrial branches like chemical, metallurgical, silicate,
pharmaceutical, etc. [4], as well as for burning the
residuals from different factory installations or the municipal wastes. Drying processes, incineration, mixing,
heating, roasting, cooling, humidification, calcination,
reduction, sintering, melting, gasification, dehydration,
as well as reactions between gas and solid phase are
implemented [5].
82
The application of the considered furnaces in the
metallurgy is for heat treatment of bulk materials [6,
7] such as oxide ores reduction, limestone calcination,
drying of sulphide copper stock, cleaning of swarfs from
machine oil, etc. [8].
At many real production situations the rotary kilns
prove to be the best and often the only solution for fulfillment of a number of processes. Depending on their
function, the temperature of the gases in them can be
higher than 1820 K, as is the situation at the treatment
of the clinker in the cement production and the upper
boundary of this parameter reaches to 2270 K [9]. The
necessary heat for this purpose most often through
combustion of primary fuel is supplied. This required
Rayko Stanev, Iliyan Mitov, Eckehard Specht, Fabian Herz
energy amount to the bed of processed material and to
the inner surface of the furnace wall by convection [10,
11] and by radiation is transferred. For that reason the
characteristics of the burners and the created by them jets
and torches influence very essentially on the complete
work of the equipment.
The considered units allow a treatment of various materials with changing heat-physical properties.
In addition, at appropriate control of their operating
parameters, the load of these installations also permits
considerable deviations from their nominal regime.
Often their universality is a reason to use them for incineration of hazardous wastes, which in relatively deep
beds is performed. Then after the rotary kiln a secondary
combustion chamber is envisaged, which improves the
heterogeneous burning of the corresponding material [3,
12]. Other application of these units except the typical
fire treatment of different media is the gasification of the
rubber waste products (for example of old car tyres) or
wood particles [13, 14].
The analysis of the technological operations pointed
out that the rotary kilns can be used for three main
purposes. These processes are heating, fulfillment of
chemical reactions and drying of solid materials. They
usually run combined, which complicates their consideration from the positions of the individual scientific
directions and imposes a complex approach to the corresponding unit [14].
Several important aspects of engineering and technical standpoint at the design of the rotary kilns should
be had in mind:
 the heat transfer;

the material motion through the cylindrical
workspace;
 the mass transfer between gas and solid phase;
 the chemical reactions.
The experimental data collected from different
rotary kilns by many parameters are influenced. Such
are the diameter of the set, its rotation speed, the feeling degree with a processed material, as well as the
own characteristics of the stuff like the coarseness of
its particles, their shape and state of the surface [15].
An essential significance has also the transverse motion
mode of the bed. The orientation in this variety of factors conditioning the transport phenomena in the rotary
kilns is a difficult task even for skilled researchers and
specialists in their exploitation. For that reason one of the
disposable approaches is the creation and provision of
opportunity for using of mathematical models, predicting
the above mentioned parameters individually for each
of the materials without physical experiments or with a
minimum of the empirical determining quantities [16].
Among the most substantial transport phenomena
running in a given rotary kiln is the particles motion of
the processed solid material. It can be important for its
exploitation [2], because this factor limits the bed ability
to absorb the heat flux evolved by the flame. Furthermore, the outflow of the disperse stuff from the concrete
equipment, which also is an element of the problems
connected with the media motion in it, has an elaborate
and decisive influence on the complete situation in the
workshop. On the one hand, it designs the position of the
following apparatuses in the technological flow. On the
other hand, the parameters at the running out from the
furnace define the initial conditions for modeling of the
bed axial height and its heat exchange surface area. In
the literature this altitude simplified as zero is accepted,
which does not correspond to the physical reality [9].
That is why the mixing and the transportation of the
granular materials treated in the unit have essentially
influence on all other processes and it is an object of
continuously extending researches.
In view of the considerations expressed above, the
aim of the present work is the opportunities for rapid and
reliable thickness determination of the separate zones
of the disperse material bed in a rotary kiln at rolling
motion as the most widespread regime of its transverse
transport to be extended.
BED MOTION IN ROTARY KILNS
The operation knowing of a given rotary kiln requires exhaustive investigations to be conducted with
the aim to determine the key parameters like particles
residence time in the furnace, the temperature field in
it, the filling degree of the cylinder with material, etc.
The big diversity of the processed products in these
installations leads to the necessity for considerable resources to determine the mentioned above parameters
individually for each material. So, the creating and the
possibilities for using of mathematical models, which to
allow their prediction without physical experiments to
be imposed or at reducing to a minimum the number of
the empirical specified quantities is important.
83
Journal of Chemical Technology and Metallurgy, 49, 1, 2014
A. Motion modes
The motions of the solid bed in a rotary kiln can be
differentiated in six basic groups [1]. The possible ways
for transverse particles transportation with the rotational
Froude number, Fr, –, are characterized, which represents a measure of the ratio between the forces of inertia
and weight [1, 17]:
Fr =
n2 R
g
(1)
where n , s–1, is the rotation speed of the kiln, R , m,
is its inner radius, and g , m/s2, - the gravitational acceleration.
The boundary values are shown in Table 1, within
the frames of which can be confidently asserted that the
regime is the pointed out type. There is an impression
that intermediate areas exist, where the transition from
one mode to another occurs.
Except on the cylinder rotation speed, the bed motion in the kiln depends also on the kind of material, the
feeling degree of the drum and the friction forces between the particles and the wall of the vessel. As shown
in Table 1, with the increase of the rotation speed, and
thence – of the rotational Froude number, the transportation modes from slipping motion to centrifuging one
are changed [18]. These regimes are presented in Fig. 1.
The slipping motion begins to run at weak friction
between the bed and the cylinder wall, i. e. at smooth
surfaces of the vessel and the particles. When the kiln
wall has a low roughness, the sliding of the material with
a constant initial point is characterized. Then it turns out
that the cylinder rotates under the treated material, and
the bed remains with a small deviation angle from its
original state, as certain “slipping” is observed.
With the increase of the friction between the wall and
the material, their mutual translation from sliding turns
into a rocking and it reaches to a slumping motion of
Fig. 1. Material motion modes on the cross section of a
rotary kiln.
the bed. It is characterized with a periodical alternation
of an immovable state and of a particles sliding on the
cylinder wall. Such motion should be avoided, because
it is not material mixing, and this leads to an aggravation of the quality. This regime is especially undesired
in industrial conditions, but unfortunately not at all cases
avoidable and it often springs up locally in separated
zones at some technological processes.
The rolling motion is preferable to all other modes,
because it ensures a best mixing of the particles, and
thence also a maximal intensive heat transfer between
the wall of the kiln and the processed material.
The cascading motion by clustering of many particles in the upper zone under the action of the increased
rotation speed is characterized. As it is still insufficient to
begin a material emitting in the freeboard of the furnace,
a descent like avalanche (cascading drop) of the upper
layer to the lower areas of the cross section in reverse
direction of the rotation is observed.
The cataracting motion is typical with the beginning of a particles throwing away from the bed in the
freeboard of the cylinder. With the increase of the rotation speed, the quantity of the ejected particles and the
Table 1. Limits of the particular motion modes according to the rotational Froude number.
No.
1
2
3
4
5
6
84
Motion
Slipping
Slumping
Rolling
Cascading
Cataracting
Centrifuging
Values of the rotational Froude number
Fr < 1x10−5
1x10−5 < Fr < 0.3x10−3
0.5x10 −3 < Fr < 0.2 x10 −1
0.4 x10 −1 < Fr < 0.8x10 −1
0.9x10−1 < Fr < 1
1 < Fr
Rayko Stanev, Iliyan Mitov, Eckehard Specht, Fabian Herz
length of their trajectory grow, while a fully covering
of the cylinder wall with the treated material is turned
out. This motion mode finds an application at the ball
crushers. In them with an enlargement of the rotation
speed of the cylindrical cage, the used metal spheres and
the processed material start to roll on the cylinder wall
and thence also the breaking of the material is improved.
At Fr > 1 an adhesion of the material to the kiln wall
is observed and the motion from cataracting becomes a
centrifuging one.
B. Rolling motion mechanism
As it was already mentioned, this regime of bed
shifting is preferable to the other ones be-cause it ensures
a best particles mixing and gives optimal conditions for
heat exchange between the processed material and the
hot gas, as well as from the kiln wall to the particles
contacting with it.
The rolling motion mechanism is presented in Fig. 2.
The inclination angle of the material surface (the dynamic angle of repose) is approximately constant. The
whole volume occupied with particles with two layers is
characterized: a thin active and a deep passive one. Their
geometrical characteristics in Fig. 2 are also shown. The
boundary line between the two beds is through the turning point W. The particle flux on the surface of the active
layer is with relatively high speed. In its lower area, at
the boundary line WB, the particles pass from the active
to the passive layer and there they start to raise upwards
with the rotation speed of the cylinder, until they reach
the upper section of the boundary line WA, where they
return back in the active bed [19, 20].
In the rotary kilns with a direct burning of fuel above
Fig. 2. Kinematic scheme of a rolling motion (1 - active
layer; 2 - boundary line; 3 - passive layer;
t A - thickness of the active layer, m; t B - vertical depth
of the bed, m; t BA - vertical depth of the active layer, m).
the material, the active layer plays an important role at
the heat transfer from the gaseous to the solid phase. For
this reason big efforts are made to predict the active bed
thickness, the particles speed in it and their residence
time on its surface. The study of the rolling motion at
the rotary kilns with an indirect firing is also important,
as on its complete fulfillment depends on the transport
of the heat from the inner cylinder wall at the contact
of the material with it.
EXPERIMENTAL
A previous work of the authors [21] presents an
analytical model created on the ground of the regularities at the particles motion in a cylindrical rotary kiln.
It allows the calculation of the general thickness of the
bed moving rollingly on its cross section, by reason of
what a “total model” is named, as well as this one of the
active layer, occupying the zone immediately below the
free surface of the material, through its vertical depth.
On the basis of this mathematical apparatus the particles distribution at different rotation speeds of the unit,
feeling degrees of it, f , –, and materials is determined.
Moreover the influence of the diameters of the cylindrical kilns, D , m, and of the processed in them particles,
d , m, is investigated. It is proved that the assumed preconditions at the development of the model correspond
to the nature of the described physical phenomena.
RESULTS AND DISCUSSION
In the referred literature source [21] graphical
dependences are given, which reflect the behaviour
of the enumerated above dimensionless geometrical
characteristics of the whole bed and the active layer
of material on the kiln cross section in a function of
different constructive and operating parameters of the
unit chosen for modeling. Since their assessment with
the spending of considerable resources is connected, a
form for describing of the most important results of the
offered researches with simple and convenient for engineering application relations is sought. The analysis of
the mentioned graphs shows that all they evince a clearly
expressed maximum. Because of that in the present work
their ordinates are statistically processed and presented
in Figs. 3��������������������������������������������
���������������������������������������������
-�������������������������������������������
10�����������������������������������������
. On the highest values of the dimensionless thicknesses and vertical depths of the layers can be
85
Journal of Chemical Technology and Metallurgy, 49, 1, 2014
Fig. 3. Dependence of the maximum dimensionless vertical depth of the bed on the feeling degree of the furnace
with material.
Fig. �������������������������������������������������
7. ����������������������������������������������
Dependence������������������������������������
�����������������������������������
of the maximum dimensionless thickness of the active layer on the particles diameter.
Fig. 4.
�������������������������������������������������
Dependence������������������������������������
����������������������������������������������
of
�����������������������������������
the maximum dimensionless vertical depth of the bed on the dynamic angle of repose of the
material.
Fig. 8. Dependence of the maximum dimensionless vertical
depth of the active layer on the inner diameter of the kiln.
Fig. �������������������������������������������������
5. ����������������������������������������������
Dependence������������������������������������
�����������������������������������
of the maximum dimensionless thickness of the active layer on the inner diameter of the kiln.
Fig. 9. Dependence of the maximum dimensionless vertical
depth of the active layer on the dynamic angle of repose
of the material.
Fig. 6.
�������������������������������������������������
Dependence������������������������������������
����������������������������������������������
of
�����������������������������������
the maximum dimensionless thickness of the active layer on the rotation speed of the kiln.
Fig. 10. Dependence of the maximum dimensionless
vertical depth of the active layer on the feeling degree
of the furnace with material.
obtained an initial notion for their distributions on the
cross sections of the considered rotary kilns.
As seen from the results presented in Figs. 3-10, all
data are mathematically well describable. At this process approximation equations with different structures
are feasible. That is why in the diagrams with a solid
86
Rayko Stanev, Iliyan Mitov, Eckehard Specht, Fabian Herz
Table 2. Dependences derived on the basis of the presented data.
To
Fig.
Тtype of the
dependence
Polynomial
3
Linear
Logarithmic
Polynomial
4
Exponential
Linear
Polynomial
5
Linear
Exponential
Polynomial
6
Exponential
Linear
Polynomial
7
Linear
Exponential
Polynomial
8
Exponential
Linear
Exponential
Polynomial
9
Linear
Polynomial
10
Linear
Logarithmic
Analytical expression
R*
t B / R = −3.0598 f 2 + 3.2293 f + 0.0639 0.9996
0.9966
t B / R = 2.4175 f + 0.0975
0.9709
t B / R = 0.2217 ln f + 0.9236
2
t B / R = 0.0001θ − 0.0028θ + 0.4466 0.9986
0.9954
t B / R = 0.3339 exp(0.0123θ )
0.9914
t B / R = 0.0064θ + 0.2922
Boundaries of validity
0.02 ≤ f ≤ 0.25
0.4 m ≤ D ≤ 6 m
θ = 30°
25° ≤ θ ≤ 45°
0.4 m ≤ D ≤ 6 m
f = 0.15 ; ε = 54.2°
0.4 m ≤ D ≤ 6 m
t A / R = 3x10−5 D 2 + 0.0006 D + 0.1023 0.9778
0.9765
f = 0.15 ; ε = 54.2°
t A / R = 0.0008 D + 0.1021
0.9763
t A / R = 0.1021exp(0.0074 D )
d = 2.2x10−3 m
2
0.2 min −1 ≤ n ≤ 3 min −1
t A / R = 0.0005n + 0.0001n + 0.0985 0.9947
0.9735
t A / R = 0.0977 exp(0.018n )
D = 2 m; d = 2.2 x10−3 m
0.9729 f = 0.15 ; ε = 54.2° ; θ = 30°
t A / R = 0.0018n + 0.0977
−6 2
t A / R = −7 x10 d + 0.001d + 0.0998 0.9944 2x10 −3 ≤ d ≤ 60x10 −3 m
0.9795
t A / R = 0.0006d + 0.1033
D = 2 m; n = 3 min −1
0.9686 f = 0.15 ; ε = 54.2° ; θ = 30°
t A / R = 0.1037 exp(0.0051d )
−5
2
0.4 m ≤ D ≤ 6 m
t BA / R = 7 x10 D + 0.0005 D + 0.125 0.9993
0.9934
f
=
0
.
15 ; ε = 54.2° ; θ = 30°
t BA / R = 0.1246 exp(0.0074 D )
0.9929
t BA / R = 0.0009 D + 0.1246
d = 2.2 x10−3 m
0.9999
25° ≤ θ ≤ 45°
t BA / R = 0.0262 exp(0.045θ )
2
D = 2m
t BA / R = 0.0001θ − 0.003θ + 0.0775 0.9998
0.9917
f = 0.15 ; ε = 54.2°
t BA / R = 0.0059θ − 0.0719
2
0.02 ≤ f ≤ 0.35
t BA / R = −0.5575 f + 0.6283 f + 0.0746 0.9996
t BA / R = 0.4053 f + 0.09
t BA / R = 0.0614 ln f + 0.2835
line the curves obtained by the dependences with the
highest adequacy degrees are displayed. The corresponding to them analytical expressions in Table 2 as a first
proposal for each set of points are systematized. Then
in a descending series of their statistical indicators by
further two variants for mathematical description of the
discrete values are arranged. All polynomial functions
are chosen to be second-degree because their complicating with more terms does not lead to an improvement
of the precision. At the indication of their boundaries
of validity with ε , °, the half of the filling angle of the
kiln cross section is marked.
From Table 2 makes an impression that the correlation coefficients, R * , –, of each of the presented
equation are enough high. This fact shows that the
corresponding dependence describes adequately the
data on the basis of which it is derived. The differences
0.9922
0.9896
D = 2 m; n = 3 min −1
d = 2.2x10−3 m; θ = 40°
between the accuracy, with which they by each of the
three chosen as most suitable formulae can be approximated are often practical negligible. Nevertheless, at a
more careful analysis it is found that at seven from the
whole eight figures best results offers the polynomial
second-degree dependence and only once such priorities
by the exponential correlation are ensured. The linear
equation guarantees the second highest precision in four
of the cases compared to three times on behalf of the
exponential function and one – of the polynomial relation. The logarithmic dependence can be recommended
solely as a third successive option, besides only regarding the data from two of the graphs, namely these ones
shown in Fig. 3 and 10.
The presented graphical and tabular data confirm the
conclusions made in [21] that on the thicknesses and the
vertical depths of the whole bed and the active part of
87
Journal of Chemical Technology and Metallurgy, 49, 1, 2014
it exert influence the inner diameter of the cylindrical
cage, its rotation speed, the filling degree of the drum
with stuff, the particles diameter, the dynamic angle of
repose of the material, its motion regime through the
cylindrical workspace determining where along the
kiln a certain feeling degree would be obtained. The
significance of the specified parameters for the forming
of the separate zones of the bed at its rolling motion
from the presented in [21] graphs can be more precise
evaluated. Here is properly to be generalized that for the
obtaining of a right idea for the transverse motion and
the material mixing in a given rotary kiln the impact of
each of the enumerated factors on it is necessary to be
rendered an account. The overestimation of one of the
listed indicators or conditions for the process running at
the�������������������������������������������������������
expense�����������������������������������������������
������������������������������������������������������
of
����������������������������������������������
another, as well as the still more inadmissible absolute disregard of some of them can lead to an
essential distortion of the results.
CONCLUSIONS
The data obtained by specially created mathematical
model and published in our previous work are analyzed.
Their interpretation form a point of view of the maximum values manifesting in the individual functional
dependences allows a rapid and convenient determination of the influence of the most important factors on the
thicknesses and the vertical depths of the whole bed and
the active part of it at rolling motion, which has a practical significance through its wide currency in the rotary
kilns and the priorities of it over the remaining regimes.
For each set of discrete values characterizing the
height of the maximum in the concrete curve from the
mentioned research by three equations are proposed.
They are ranged in a descending series according to their
adequacy degree, with which they describe the data, on
the basis of what the separate formulae are derived. The
lines corresponding to the dependences, which in best
way the points from the figures in this work approximate
in them are plotted.
The implemented previous and present investigations allow to be generalized that on the thicknesses and
the vertical depths of the whole bed and the active part
of it exert influence the inner diameter of the kiln, its
rotation speed, the degree of its feeling with material,
its dynamic angle of repose and the particles diameter.
The statistical analysis of all derived equations
88
shows that they can be successfully used for prediction of the height of the corresponding maximum. It is
recommended this calculation to occur by the proposed
in the lead correlation, which in more than 87 % of the
cases is a second-degree.
REFERENCES
1.A.A. Boateng, Rotary Kilns., Transport Phenomena
and Transport Processes, Elsevier Inc., Amsterdam,
Boston e. a., 2008.
2.P.V. Barr, Heat Transfer Processes in Rotary Kilns,
Ph.D. thesis, The University of British Columbia,
Vancouver, BC, Canada, 1986.
3.M. Rovaglio, D. Manca, G. Biardi, Dynamic
Modeling of Waste Incineration Plants with Rotary
Kilns: Comparisons between Experimental and
Simulation Data, Chemical Engineering Science, 53,
15, 1998, 2727-2742.
4.R.J. Spurling, Granular Flow in an Inclined Rotating
Cylinder: Steady State and Transients, Ph. D. thesis,
Cambridge University, England, 2000.
5.R. Jauhari, M.R. Gray, J.H. Masliyah, Gas-Solid Mass
Transfer in a Rotating Drum, Canadian J. of Chem.
Eng., 76, 2, 1998, 224-232.
6.M. Chmielowski, E. Specht, Modelling of Heat
Transfer by the Transport Rollers in Kilns, Appl.
Thermal Eng., 26, 7, 2006, 736-744.
7.K.-U. Holzapfel, E. Specht, Heat Transfer between a Rotating Cylinder and a Transported Plate,
Experimental Heat Transfer, 19, 1, 2006, 39-51.
8.Pushnov A.S., Calculation of Average Bed Porosity,
Chem. and Petroleum Eng., 42, 1–2, 2006, 14-17.
9. Y. Shi, The Out-Flowing Behaviour of Particles at the
Discharge End of Rotary Kilns, Ph. D. thesis, University
“Otto von Guericke”, Magdeburg, Germany, 2009.
10. D. Choshnova, B. Stefanov, Structure of a Torch
Formed During Melting of Copper Charge in a
Furnace OUTOKUMPU, Russian J. of Non-Ferrous
Metals, 52, 1, 2011, 62-68.
11. R.D. Stanev, Perfection of Mixing between Gaseous
Fuel and Air in Industrial Burners – Technical and
Terminological Problems, J. Univ. Chem. Technol.
Met. (Sofia), 44, 4, 2009, 395- 402.
12. I. Abe, T. Fukuhara, S. Iwasaki, K. Yasuda, K.
Nakagawa, Y. Iwata, H. Kominami, Y. Kera,
Development of a High Density Carbonaceous
Rayko Stanev, Iliyan Mitov, Eckehard Specht, Fabian Herz
Adsorbent from Compressed Wood, Carbon, 39,
10, 2001, 1485-1490.
13. H.F.M. Elattar, Flame Simulation in Rotary Kilns
Using Computational Fluid Dynamics, Ph. D. thesis, University “Otto von Guericke”, Magdeburg,
Germany, 2011.
14. I.S. Mitov, Investigation of Transport Phenomena in
Rotary Kilns, Ph. D. thesis, UCTM, Sofia, Bulgaria,
2012, (in Bulgarian).
15. X.Y. Liu, Experimental and Theoretical Study of
Transverse Solids Motion in Rotary Kilns, Verlag
Dr. Hut, Münich, 2005.
16. I.S. Mitov, R.D. Stanev, Mathematical Description
of the Falling Particles of Disperse Material
Deviations from a Rotary Kiln, Eng. Sci. (Bulg.),
45, 3, 2008, 42-49, (in Bulgarian).
17. J. Mellmann, E. Specht, Mathematical Modeling
of the Transition Behavior between the Various
Forms of Transverse Motion of Bulk Materials in
Rotating Cylinders (Part 2). Cement, Lime, Gypsum
International, 54, 7, 2001, 380-402.
18. J. Mellmann, E. Specht, Mathematical Modeling
of the Transition Behavior between the Various
Forms of Transverse Motion of Bulk Materials in
Rotating Cylinders (Part 1). Cement, Lime, Gypsum
International, 54, 6, 2001, 281-296.
19. X.Y. Liu, E. Specht, О.G. Gonzales, P. Walzel,
Analytical Solution of the Rolling-Mode Granular
Motion in Rotary Kilns, Chem. Eng. and Proc., 45,
2006, 515-521.
20. E. Specht, A. Queck, S. Augustini, Modeling of
the Regenerative Heat Flow of the Wall in Direct
Fired Rotary Kilns, Heat Transfer Eng., 29, 1, 2008,
57- 66.
21. I.S. Mitov, R.D. Stanev, Application of an Analytical
Model for Determination of the Thickness of the
Layers from Material in a Rotary Kiln, Eng. Sci.,
45, 4, 2008, 58-67, (in Bulgarian).
89