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materials
Article
Application of 3-D Drucker–Prager Material Model to
Determine Optimal Operating Parameters of
Centrifugal Regeneration Device
Waldemar Łatas 1, Rafał Da´nko 2, * and Przemysław Czapla 3
1Institute of Applied Mechanics, Faculty of Mechanical Engineering, Cracow University of Technology,
Al. Jana Pawła II 37, 31-864 Kraków, Poland; waldemar.latas@pk.edu.pl
2Faculty of Foundry Engineering, AGH University of Science and Technology, Al. A. Mickiewicza 30,
30-059 Kraków, Poland
3Eurocast Industries, ul. Grabowa 19, 30-227 Kraków, Poland; pczapla@eurocast-ind.com
*Correspondence: rd@agh.edu.pl
Received: 21 March 2020; Accepted: 30 April 2020; Published: 5 May 2020
Abstract:
The process of metal casting indisposable sand molds is associated with the generation
of large amounts of waste, mainly used molding and core sands, from which the molds and cores
reproducing the external and internal shapes of the castings were made. It is estimated that about
600 kg of waste can come from the production of 1 ton of casting. The main component of the waste
is quartz matrix, which after undergoing appropriate reclamation treatments can be recovered and
reused in the production process. This article presents the theoretical foundations regarding the
existing methods of quartz matrix recovery and an experimentally justified model of the regeneration
process occurring in one of the varieties used in the practice of mechanical regenerators. The goal is
to improve the quality of regenerated molding sand by means of liberating the sand grain’s surface
from the layer of the used binding component. The elastic-plastic material model characterized by
the Drucker–Prager yield criterion was used to describe the deformation of the sand layer during
treatment performed in a centrifugal regenerator. Conclusions based on the results of numerical
calculations, obtained with the use of the software adopting the material point method, enable us
to find out how to control the device in a way that ensures a permanent reclamation effect which is
independent of the working components that wear out over time.
Keywords:
molding sand mechanical reclamation; centrifugal regeneration unit; circular economy;
Drucker-Prager yield criterion; Material Point Method
1. Introduction
In the process of manufacturing cast metals (mainly of iron and non-ferrous alloys) in single-use
sand molds made of chemically bonded molding sands, the external and internal shapes of the casting
are reproduced by the mold and core. The material that creates these elements is quartz sand with
appropriate properties, the individual grains of which are joined together with a binder to form a
mold or casting core [
1
,
2
]. The quartz sand used in the mold and core production process must have
adequate grain size, main fraction content, homogeneity, proper shape and heat resistance [
3
]. Molds
and casting cores are made in such a way that their shapes ensure the correct filling of the interior
of the mold with liquid metal and the creation of a casting free of internal and external defects with
an appropriate, previously established microstructure. It is assumed that in the case of traditional
technologies, the amount of material used to make the mold and cores (quartz sand +binder) is 3.5–5
times greater than the amount of metal poured into the mold for casting [
2
,
4
,
5
]. After the casting
is made, it is separated from the casting mold in the stamping out or knocking out processes and
Materials 2020,13, 2134; doi:10.3390/ma13092134 www.mdpi.com/journal/materials
Materials 2020,13, 2134 2 of 29
subjected to further processing. The molding and core sands formed after separation constitute waste
that can undergo the regeneration process [2,6].
The most important issues for the considered topic of regeneration of used molding and core
sands were analyzed in [
7
–
19
]. To date, these items bring the most complete overview on many aspects
of this process, its variations and applications, which have been ordered and classified into specific
topics and specific issues. The general issues related to the regeneration of used sands in the context of
the process variants applied are highlighted below.
Publications [
7
–
15
] treated the mechanical regeneration processes as specific variations of the
method of releasing sand matrix from the coating of used binding material that occur regardless of
the type of regenerative treatment. Practically to date, no model has been developed that allows a
quantitative description of the phenomena in the process of releasing matrix grains from the coating
of binding material. In publications [
16
,
17
], attempts were made to determine the mechanisms of
destruction of binder bonds during the thermal impact on the regenerated matrix. However, their
application in the regeneration process does not allow a quantitative description. The analytical
model of this process [
18
] based on the crushing theory is also known, but it was not a numerical
solution to the problem, which is demonstrated in the presented work. The distinction in the process
of regenerative treatment of two of its ranges (pre-regeneration and specific regeneration) emphasized
in the literature [
19
] has a conventional nature useful for the certain systematics of the analyzed
phenomena, both at the boundary between the binding material and the matrix grain, as well as in the
contact area of individual grains in spatial terms.
The analysis of the available literature indicates that no attempt has been made so far to develop
the numerical model of the mechanical regeneration process, which is presented in this publication.
The model calculations and verification tests presented in the publication concern one of the
varieties of dry mechanical (proper) regeneration device, namely the centrifugal regenerator. The goal
was to determine a control method of the device’s work which, by proper selection of the roller pressure
on the layer of sand located on the internal side surface of the regenerator batch, will ensure a constant
regenerative effect of used mass grains, independent of its working elements wearing out in time.
The elastic-plastic material model characterized by the Drucker–Prager yield criterion [
20
] was
used to describe the deformation of the sand layer. The regeneration quality indicator related to shear
strain was defined, and, on its basis, the value of the optimal regenerator supply pressure was obtained
depending on the depth of the groove of the device batch’s side surface. In the numerical calculations,
a program based on the material point method [21] was used.
2. Regeneration of Used Molding and Core Sands
Regeneration of used molding and core sands is defined as a treatment that allows the recovery of
at least one of the components of these sands with properties similar to the fresh state of this component
and its re-use for the production of molds and cores. Currently, the main component recovered from
used molding and core sands is quartz sand, constituting the matrix and the main component of
these sands.
Used molding and core sands can be used alone or in conjunction with other components in the
construction industry, as a sealing material in landfills, as a material for the production of cement
aggregate, or asphalt filler [
3
–
5
,
22
]. In the foundry, the sands can be pre-prepared for non-foundry
applications by subjecting them to a pre-regeneration process. However, the optimal way to use the
sands is to treat them so that they can be reused for molds and cores. Possible ways of dealing with
used sand are shown schematically in Figure A1 (Appendix A).
Used sand, regardless of the adopted method of matrix recovery from it, is subjected to treatment
carried out in appropriate systems, most often with the following sequence of actions:
•preliminary separation of mechanical impurities from the used sand, mainly metal ones
•grinding the compacted sand after knocking it out of the mold
•sieving the sand and isolating the grain size range of the material for regeneration
Materials 2020,13, 2134 3 of 29
•re-separation of metal impurities from the sand
•
proper (secondary) regeneration, releasing the matrix from the remains of the used binding
material by using methods that allow removal of a coating of binding material from the surface of
the grains
•removing unwanted regeneration products from the matrix by dedusting it
•isolation of a matrix with a specific grain size and homogeneity (classification by grain size)
The treatment of the used sand consisting of the combination of the first four mentioned activities
is referred to as the primary regeneration process [
2
,
3
]. Its full implementation is a prerequisite for
subjecting the obtained matrix to subsequent, final treatment of the matrix grain purification from the
coatings of binding material (proper regeneration—named secondary regeneration). The process of
secondary regeneration should lead to obtaining a product (quartz sand, matrix) which is characterized
by properties similar to the fresh state of this component and can be its substitute in the process of
manufacturing molds and cores.
Figure A2 (Appendix A) presents a schematic view of the proper (secondary) regeneration methods
used, with the basic classification criterion being the environment in which the regeneration process is
carried out. In this aspect, two basic wet and dry methods can be distinguished.
The dry regeneration process by mechanical means has today found the widest application in
foundries, which results from the following possibilities:
•
matrix recovery from virtually any used sand, assuming a limited degree of matrix cleaning from
residual binding material (regeneration rate)
•
applications for the regeneration of relatively simple devices, often for other original purposes
(for example mixers)
•lower, compared to other methods, costs of the regeneration process
It seems that the variety of regenerative interactions used in mechanical regeneration devices
should be considered through the prism of the so-called elementary operations that share common
characteristics regardless of the treatment method. This approach was presented in [2,3].
The purpose of the mechanical regeneration process is to affect the grain of used sand (sand matrix
surrounded by a layer of used, inactive binder) in such a way as to separate the used binder from the
matrix (grain) and thus clean the matrix grain to a degree similar to that of the matrix when fresh.
This process is implemented through the so-called elementary operations that are carried out in devices
called regenerators [
11
,
23
]. The elementary operations occurring in mechanical regeneration—rubbing,
abrasion and crushing—are presented in Figures A3–A5 and described in more detail in Appendix A.
Model calculations and verification tests presented in the publication concern one of the varieties
of devices for mechanical dry regeneration.
3. Reclamation Device Preliminary Tests Results
The USR-I reclamation device was used in model, preliminary and verification tests (Figure A6,
Appendix B). The device is a centrifugal regenerator with a periodic work cycle and of a relatively
universal purpose, equipped with additional grinding and compressing components placed in the
batch. The device batch and other internal structural components exposed to intense abrasion are
made of abrasion-resistant ceramic materials.
The portion of used sand that can be at one time subjected to reclamation treatment in the device
is 25–30 kg. The regenerator is equipped with a timer which makes it possible to adjust the duration
of reclamation treatment to the requirements (regenerability) of a given used sand. It is generally
assumed that the abovementioned portion of mass is subjected to treatment for a period of about 30 s.
During the treatment process, the batch is put into rotation (400 rpm), forcing centrifugal movement of
the regenerated sand onto the outer peripheral side surface of the batch made of an abrasion-resistant
ceramic material. The dominant regenerative operations carried out in the device are abrasion, crushing
Materials 2020,13, 2134 4 of 29
of sand aggregates, and rubbing of the used coating from the surface of the grains. These operations
are caused by obliquely arranged rollers, and the intensity of the regenerative effect can be changed by
adjusting the pressure of the rollers against the batch.
As stated earlier, the side surface of the regenerator batch is made of an abrasion-resistant material.
However, long-term tests of the device’s workings have shown that this material is in fact subject to
abrasion. The results which demonstrate this are shown in Figure 1. The wear of the side surface
material lowers the effectiveness of the reclamation process. This unfavorable effect can be partially
decreased by reducing the distance between the pressing roller and the side of the device batch.
Materials 2020, 13, x FOR PEER REVIEW 4 of 29
As stated earlier, the side surface of the regenerator batch is made of an abrasion-resistant
material. However, long-term tests of the device’s workings have shown that this material is in fact
subject to abrasion. The results which demonstrate this are shown in Figure 1. The wear of the side
surface material lowers the effectiveness of the reclamation process. This unfavorable effect can be
partially decreased by reducing the distance between the pressing roller and the side of the device
batch.
(a) (b)
(c)
Figure 1. Wear of regenerator batch lining as a result of long-term operation, measurement at constant
supply pressure set as 2.4 barp=: (a) maximum depth of side surface groove (marked by h in
drawing (b)) as a function of regenerator work time; (b) scheme of regenerator batch lining wear; (c)
photo showing regenerator batch lining wear.
The confirmation of the impact of side surface material wear on the quality of the regenerate are
the results of long-term tests of the device presented below. During the 2000h measuring cycle, the
device settings and the binding materials used for making molds did not change. The sand supplier
was Sand Mine Szczakowa (Jaworzno, Poland), the producer of high quality sands for casting
production.
In the long-term tests, together with the measurement of the maximum depth of side surface
groove, shown in Figure 1, each time, the ignition loss of the used sand and the regenerate recovered
from it were determined. The tests were carried out during the foundry’s production activity, where
ALPHASET (Phenolic No-Bake Resin System) resin with the trade name Momentive TPA 70
0.0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
4.0
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Maximum depth of the groove [mm]
Operation time [h]
h
Figure 1.
Wear of regenerator batch lining as a result of long-term operation, measurement at constant
supply pressure set as p=2.4 bar: (
a
) maximum depth of side surface groove (marked by hin drawing
(
b
)) as a function of regenerator work time; (
b
) scheme of regenerator batch lining wear; (
c
) photo
showing regenerator batch lining wear.
The confirmation of the impact of side surface material wear on the quality of the regenerate are the
results of long-term tests of the device presented below. During the 2000 h measuring cycle, the device
settings and the binding materials used for making molds did not change. The sand supplier was Sand
Mine Szczakowa (Jaworzno, Poland), the producer of high quality sands for casting production.
In the long-term tests, together with the measurement of the maximum depth of side surface
groove, shown in Figure 1, each time, the ignition loss of the used sand and the regenerate recovered
from it were determined. The tests were carried out during the foundry’s production activity,
Materials 2020,13, 2134 5 of 29
where ALPHASET (Phenolic No-Bake Resin System) resin with the trade name Momentive TPA 70
(Momentive Performance Materials Italy S.r.l, Varese, Italy), hardened with ACE 1035 (Momentive
Performance Materials Italy S.r.l, Varese, Italy), was used as the molding sand binding material. During
the preparation of the molding sand, the resin addition was 1.1% in relation to the matrix, while the
hardener addition was 0.35%. A mixture of 50% quartz sand with a grain size d
50
=0.32 mm and
50% regenerate obtained from the tested device was used as the matrix. The technology used in the
foundry was designed in such a way that for most components, the castings are made in molds with
an average ratio of molding sand to metal mass of 3.9:1. The foundry produces steel castings, usually
the temperature of pouring molds with liquid metal is in the range of 1550–1600
◦
C. In the tests, as an
indicator of the degree of regeneration of the molding sand, the ignition loss was used, which in the
case of molding sand with an organic binder allows the determination of the residue of the binder on
quartz sand grains. Ignition loss tests were carried out at a temperature of 950
◦
C, and five samples
were tested in parallel, based on which the average value of the measurement was determined. At the
start of the study, the regenerate’s ignition loss was 1.24%. The obtained results recording changes in
ignition loss of the used sand and regenerate are presented in Figure 2.
Materials 2020, 13, x FOR PEER REVIEW 5 of 29
(Momentive Performance Materials Italy S.r.l, Varese, Italy), hardened with ACE 1035 (Momentive
Performance Materials Italy S.r.l, Varese, Italy), was used as the molding sand binding material.
During the preparation of the molding sand, the resin addition was 1.1% in relation to the matrix,
while the hardener addition was 0.35%. A mixture of 50% quartz sand with a grain size d50 = 0.32 mm
and 50% regenerate obtained from the tested device was used as the matrix. The technology used in
the foundry was designed in such a way that for most components, the castings are made in molds
with an average ratio of molding sand to metal mass of 3.9:1. The foundry produces steel castings,
usually the temperature of pouring molds with liquid metal is in the range of 1550–1600 °C. In the
tests, as an indicator of the degree of regeneration of the molding sand, the ignition loss was used,
which in the case of molding sand with an organic binder allows the determination of the residue of
the binder on quartz sand grains. Ignition loss tests were carried out at a temperature of 950 °C, and
five samples were tested in parallel, based on which the average value of the measurement was
determined. At the start of the study, the regenerate’s ignition loss was 1.24%. The obtained results
recording changes in ignition loss of the used sand and regenerate are presented in Figure 2.
Figure 2. Ignition loss of used sand and regenerate as a function of USR-I device operation time,
measurement at constant supply pressure set as p = 2.4 bar.
Based on the data presented in Figure 2, it can be seen that the achieved values of ignition loss
of the used sand in the analyzed period of time were in the range of 2.43–2.55%; a similar analysis
carried out for the regenerate indicates values of ignition loss in the range of 1.24–1.51%. The achieved
values of material ignition loss after the regeneration process indicate a very high efficiency of the
regeneration device operation. However, the average values of this parameter increase as a function
of time. Assuming unchanged parameters of the process of making the mold and its overheating after
the process of pouring with a liquid casting alloy, it can be assumed that the reason for the increase
in the average value of the ignition loss is the decrease in the regeneration efficiency resulting from
the change in the shape of the side surface (Figure 1).
This fact is confirmed by the data presented in Figure 3, which summarizes the average
regeneration efficiency values expressed by Equation (1) for periods of regeneration device operation
time: period 1 (0–500 h), period 2 (500–1000 h), period 3 (1000–1500 h), period 4 (1500–2000 h):
()
1100%
()
LOI R
ELOI US
=− ×
(1)
where:
E—regenerator’s work efficiency (%)
1.00
1.20
1.40
1.60
1.80
2.00
2.20
2.40
2.60
2.80
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Ignition loss [%]
Operation time [h]
Used sand
Regenerate
Figure 2.
Ignition loss of used sand and regenerate as a function of USR-I device operation time,
measurement at constant supply pressure set as p=2.4 bar.
Based on the data presented in Figure 2, it can be seen that the achieved values of ignition loss
of the used sand in the analyzed period of time were in the range of 2.43–2.55%; a similar analysis
carried out for the regenerate indicates values of ignition loss in the range of 1.24–1.51%. The achieved
values of material ignition loss after the regeneration process indicate a very high efficiency of the
regeneration device operation. However, the average values of this parameter increase as a function of
time. Assuming unchanged parameters of the process of making the mold and its overheating after
the process of pouring with a liquid casting alloy, it can be assumed that the reason for the increase in
the average value of the ignition loss is the decrease in the regeneration efficiency resulting from the
change in the shape of the side surface (Figure 1).
This fact is confirmed by the data presented in Figure 3, which summarizes the average regeneration
efficiency values expressed by Equation (1) for periods of regeneration device operation time: period 1
(0–500 h), period 2 (500–1000 h), period 3 (1000–1500 h), period 4 (1500–2000 h):
E= 1−LOI(R)
LOI(US)!×100% (1)
Materials 2020,13, 2134 6 of 29
where:
E—regenerator’s work efficiency (%)
LOI(US)—ignition loss of used sand (%)
LOI(R)—ignition loss of regenerate (%).
Materials 2020, 13, x FOR PEER REVIEW 6 of 29
LOI(US)—ignition loss of used sand (%)
LOI(R)—ignition loss of regenerate (%).
Figure 3. Average efficiency of the USR-I regenerator demonstrated in long-term studies.
Figure 4 presents an example photo of the morphology of the surface of used sand grains and
the regenerate obtained after 2000 h of operation time of the regeneration device. The image shows a
thick layer of used binding material covering the grain of the used sand and a much smaller amount
covering the grain of quartz sand after the process of mechanical regeneration.
(a) (b)
Figure 4. Surface morphology of used sand grains (a) and regenerate (b) obtained after 2000 h of
operation time of the USR-I device; 1000× magnification.
The conclusions resulting from the preliminary and industrial tests presented above indicate the
need to develop a method enabling such control and setting of working elements of the USR-I device
so that it is possible to eliminate the unfavorable effect of wear of these elements, causing a decrease
in the quality of the regenerate.
4. Defining Problem and Purpose of Model Tests
The model tests were aimed at obtaining practical conclusions for determining the pressing force
of the regenerator working elements on the processed grain of used sand, depending on the degree
of wear of the regenerator batch.
49,76
48,46
44,42
41,75
36
38
40
42
44
46
48
50
52
0-500 500-1000 1000-1500 1500-2000
Average regeneration efficiency [%]
Operation time [h]
Figure 3. Average efficiency of the USR-I regenerator demonstrated in long-term studies.
Figure 4presents an example photo of the morphology of the surface of used sand grains and
the regenerate obtained after 2000 h of operation time of the regeneration device. The image shows a
thick layer of used binding material covering the grain of the used sand and a much smaller amount
covering the grain of quartz sand after the process of mechanical regeneration.
Materials 2020, 13, x FOR PEER REVIEW 6 of 29
LOI(US)—ignition loss of used sand (%)
LOI(R)—ignition loss of regenerate (%).
Figure 3. Average efficiency of the USR-I regenerator demonstrated in long-term studies.
Figure 4 presents an example photo of the morphology of the surface of used sand grains and
the regenerate obtained after 2000 h of operation time of the regeneration device. The image shows a
thick layer of used binding material covering the grain of the used sand and a much smaller amount
covering the grain of quartz sand after the process of mechanical regeneration.
(a) (b)
Figure 4. Surface morphology of used sand grains (a) and regenerate (b) obtained after 2000 h of
operation time of the USR-I device; 1000× magnification.
The conclusions resulting from the preliminary and industrial tests presented above indicate the
need to develop a method enabling such control and setting of working elements of the USR-I device
so that it is possible to eliminate the unfavorable effect of wear of these elements, causing a decrease
in the quality of the regenerate.
4. Defining Problem and Purpose of Model Tests
The model tests were aimed at obtaining practical conclusions for determining the pressing force
of the regenerator working elements on the processed grain of used sand, depending on the degree
of wear of the regenerator batch.
49,76
48,46
44,42
41,75
36
38
40
42
44
46
48
50
52
0-500 500-1000 1000-1500 1500-2000
Average regeneration efficiency [%]
Operation time [h]
Figure 4.
Surface morphology of used sand grains (
a
) and regenerate (
b
) obtained after 2000 h of
operation time of the USR-I device; 1000×magnification.
The conclusions resulting from the preliminary and industrial tests presented above indicate the
need to develop a method enabling such control and setting of working elements of the USR-I device
so that it is possible to eliminate the unfavorable effect of wear of these elements, causing a decrease in
the quality of the regenerate.
Materials 2020,13, 2134 7 of 29
4. Defining Problem and Purpose of Model Tests
The model tests were aimed at obtaining practical conclusions for determining the pressing force
of the regenerator working elements on the processed grain of used sand, depending on the degree of
wear of the regenerator batch.
In the steady state of device operation, the rollers rotate around the stationary axes, but relative
to the side surface they roll over a layer formed of grains of the regenerated sand. This causes the
displacement of sand under the rollers and the grains slipping on the side of the regenerator batch,
which contributes to the formation of a gradually deepening groove of a circular cross-section (Figure 1).
The wear of the side surface makes movement of the sand grains relative to each other difficult. For the
regeneration process to take place with sufficient efficiency, it is necessary to gradually increase the
pressing force of the rollers against the sand layer along with the increasing depth of the groove. If the
force is too low, the grains will not be cleaned from the binding material envelopes, while if it is too
high, sand grains may be ground.
The main goal was to build a calculation model that allows determining the optimal, from the
point of view of the quality of the regeneration process, supply pressure in the pipe depending on the
depth of the groove describing the wear of the side surface of the regenerator batch.
5. Regenerator Working Parameters—Preliminary Assumptions
Figure 5is a diagram of the regeneration device from Omega Foundry Machinery Ltd.
(Peterborough, UK), whereas Figure 6shows a cross-section of the device with the marked dimensions
of its components.
Materials 2020, 13, x FOR PEER REVIEW 7 of 29
In the steady state of device operation, the rollers rotate around the stationary axes, but relative
to the side surface they roll over a layer formed of grains of the regenerated sand. This causes the
displacement of sand under the rollers and the grains slipping on the side of the regenerator batch,
which contributes to the formation of a gradually deepening groove of a circular cross-section (Figure
1). The wear of the side surface makes movement of the sand grains relative to each other difficult.
For the regeneration process to take place with sufficient efficiency, it is necessary to gradually
increase the pressing force of the rollers against the sand layer along with the increasing depth of the
groove. If the force is too low, the grains will not be cleaned from the binding material envelopes,
while if it is too high, sand grains may be ground.
The main goal was to build a calculation model that allows determining the optimal, from the
point of view of the quality of the regeneration process, supply pressure in the pipe depending on
the depth of the groove describing the wear of the side surface of the regenerator batch.
5. Regenerator Working Parameters. Preliminary Assumptions
Figure 5 is a diagram of the regeneration device from Omega Foundry Machinery Ltd.
(Peterborough, UK), whereas Figure 6 shows a cross-section of the device with the marked
dimensions of its components.
Figure 5. Diagram of regeneration device from Omega Foundry Machinery Ltd.
Figure 6. Cross-section of regeneration device.
240
65
l =162
1
240
R
=283
r=267 l =639
2
Figure 5. Diagram of regeneration device from Omega Foundry Machinery Ltd.
The basic geometrical data (Figure 6) and working parameters of the regeneration device are
listed below:
–roller diameter: Z=240 mm
–roller height: H=65 mm
–maximum radius of the side surface curvature: R=283 mm
–minimum radius of the side surface curvature: r=267 mm
–average radius of the side surface curvature: R* =275 mm
–gap thickness between the roller and the side surface of the regenerator batch: d=8 mm
Materials 2020,13, 2134 8 of 29
–rotational speed: n=400 rev/min
–regeneration device efficiency: W=4700 kg/h=78 kg/min =0.195 kg/rev
–regenerated sand density: ρ=2300 kg/m3
It can be estimated that under steady state operation with the above parameters, each grain passes
about 10–11 times under each roller. Based on the dimensions presented in Figure 6, the relationship
between the force
N
pressing the roller to the side surface through a layer of the regenerated sand
and the force
F
acting on the lever arm and resulting from the air pressure in the supply line can be
determined. The leverage ratio is: e=l2/l1=639/162 =3.94, therefore: N=e·F=3.94·F.
Materials 2020, 13, x FOR PEER REVIEW 7 of 29
In the steady state of device operation, the rollers rotate around the stationary axes, but relative
to the side surface they roll over a layer formed of grains of the regenerated sand. This causes the
displacement of sand under the rollers and the grains slipping on the side of the regenerator batch,
which contributes to the formation of a gradually deepening groove of a circular cross-section (Figure
1). The wear of the side surface makes movement of the sand grains relative to each other difficult.
For the regeneration process to take place with sufficient efficiency, it is necessary to gradually
increase the pressing force of the rollers against the sand layer along with the increasing depth of the
groove. If the force is too low, the grains will not be cleaned from the binding material envelopes,
while if it is too high, sand grains may be ground.
The main goal was to build a calculation model that allows determining the optimal, from the
point of view of the quality of the regeneration process, supply pressure in the pipe depending on
the depth of the groove describing the wear of the side surface of the regenerator batch.
5. Regenerator Working Parameters. Preliminary Assumptions
Figure 5 is a diagram of the regeneration device from Omega Foundry Machinery Ltd.
(Peterborough, UK), whereas Figure 6 shows a cross-section of the device with the marked
dimensions of its components.
Figure 5. Diagram of regeneration device from Omega Foundry Machinery Ltd.
Figure 6. Cross-section of regeneration device.
240
65
l =162
1
240
R
=283
r=267 l =639
2
Figure 6. Cross-section of regeneration device.
For the known diameter of the pneumatic hose (equal to 40 mm), the relationship between the
regenerator supply pressure pand the pressing force
N
was determined: the pressure pin the hose equal
to 1 bar corresponds to a pressing force
N
equal to 495 N (in further calculations, the rounded value
500 N was used). For the considered regenerative device, the minimum (initial) value of the supply
pressure is equal to p
min
=2.4 bar, whereas the maximum value equals p
max
=4.2 bar. The minimum
value of the roller pressure on the regenerated sand layer is Nmin =1188 N.
It should be emphasized that the purpose of the work is not to accurately simulate the phenomena
occurring during the regeneration process of the material obtained from used molding sands, but to
simplify the analysis of the sand layer deformation process, which would allow the definition of a
simple, phenomenologically justified indicator that can be used to assess the quality of regeneration.
The first assumption is the omission of inertia forces due to the fact that it can be estimated and
justified that these forces have a negligible influence on the stresses in the sand layer in the domain
close to the area of contact with the roller. When building the deformation calculation model, it was
also assumed that despite a relatively high graininess, the layer of regenerated molding sand would be
treated as a continuum.
From the modeling point of view, the problem is analogous (despite the small scale) to the
problems of soil mechanics. It is necessary to adopt a constitutive model describing the elastic-plastic
behavior of sand. The simplest model is described by the Mohr–Coulomb condition on which many
other, more complex phenomenological models are based. In the discussed issue, the material model
defined by the Drucker–Prager yield surface [
24
–
26
], which is often used for geological materials such
as soils, rocks and sands, was adopted to describe the deformation process.
In loose materials, such as soil and sand, it is very difficult to experimentally determine physical
constants (Young’s modulus, Poisson’s ratio), which, additionally, often prove to be dependent on
Materials 2020,13, 2134 9 of 29
pressure. In the simulations carried out, it was assumed that the physical parameters describing the
sand are constant and do not depend on pressure.
6. Drucker–Prager Elastic–Plastic Model
The three-dimensional Drucker–Prager criterion allows the determination of the stress state at
which material failure or plasticization occurs. This criterion is based on the assumption that at the
moment of failure (plasticization), octahedral tangential stress is linearly dependent, through material
constants, on the octahedral normal stress [
6
]. The D–P criterion can be presented as a yield function of
the stress tensor e
σin the following pressure-dependent form:
f(e
σ,κ) = pJ2+αI1−κ=0 (2)
where I
1
is the first invariant of the stress tensor, J
2
is the second invariant of the stress deviator tensor
given by:
I1=σ1+σ2+σ3(3)
J2=1
6h(σ1−σ2)2+(σ2−σ3)2+(σ3−σ1)2i(4)
In the above formulas,
σ1
,
σ2
,
σ3
, denote principal stresses, and
α
and
κ
are material constants (
κ
is the yield stress under pure shear). The Drucker–Prager criterion can be considered as a special case
of the Nadai criterion, or it can also be treated as an extension of the Huber–Mises criterion.
The D–P criterion describes a cone (for
α
>0) or a cylinder (for
α
=0) in the 3D space of
principal stresses. Material constants are usually determined based on the results of standard three-axis
compression and can be expressed in terms of the angle of internal friction ϕand cohesion c.
The Drucker-Prager yield surface can be considered as a smooth version of the Mohr–Coulomb
yield surface. It can be assumed that the Drucker–Prager yield surface circumscribes, inscribes or
middle circumscribes the Mohr–Coulomb yield surface. After adopting the last of these cases, that the
D–P failure cone passes through the interior vertices of the M–C hexagonal pyramid, the following
expressions (taken in the numerical calculations) are obtained:
α=2·sin ϕ
√3(3+sin ϕ),κ=6c·cosϕ
√3(3+sin ϕ). (5)
The main advantages of the D–P criterion are its simplicity and smooth, symmetrical yield surface
in a stress space facilitating numerical implementation.
In the case of deformation of the regenerated sand layer presented in the paper, the plasticity D–P
model described by the associated flow rule without hardening was adopted [
26
]. The constitutive
equations of elastic-plastic deformation process are given in Appendix C.
7. MPM—Material Point Method
The finite element method (FEM), most commonly used in engineering analyses, is not effective
enough for large deformation problems formulated in Lagrange’s description. Large mesh distortions
progressing with body deformation are the cause of large errors of the approximated solution.
Rebuilding the mesh is time consuming and introduces additional errors resulting from the need to
project a solution from a distorted mesh onto a regenerated mesh. In the issue considered in this paper,
an additional problem is the changing common contact surface of the roller and the penetrated sand
layer (contact issue).
One of the methods used in cases of large deformation problems, for example in soil mechanics
and flow problems of loose materials, is the material point method (MPM) [
27
,
28
]. This method, known
in fluid mechanics as the particle-in-cell method (PIC), has been adapted to the problems of solid
Materials 2020,13, 2134 10 of 29
mechanics. One of the generalizations of the MPM method is the generalized interpolation material
point (GIMP) method [29].
In the MPM and GIMP methods, two types of discretization are used. The first discretization type
(related to the description in Lagrange coordinates) consists of dividing the region occupied at the initial
moment by the body in question into a set of sub-regions, each represented by one particle (material
point). The second type of space discretization (referring to the description in Euler coordinates) is
associated with the so-called computational mesh covering possible positions of the analyzed body.
MPM and GIMP owe their effectiveness to the solving equations describing a given problem using
a computational mesh that can be modified arbitrarily during analysis or remain unchanged, while
state variables are assigned to the moving Lagrange particles.
The standard MPM method treats each particle as a concentrated mass, and the GIMP method
replaces calculation of integrals in MPM using the
δ
-Dirac function with weighted average integrals over
a finite domain in the vicinity of the particle using a weight function, also called a characteristic function.
The basic difference between MPM and FEM is that in the former, state variables (i.e., mass,
stress, strain, velocity, etc.) are assigned to material points that are defined independently of the
computational mesh. In the latter, state variables are specified at integration points connected with the
elements. More mathematical details of the MPM are presented in Appendix D.
8. Material Constants—Experimental Research Results
The coefficients describing the constitutive model of sand adopted in numerical calculations were
determined based on the results of experimental research conducted on sand used in the foundry.
Standard triaxial compression, direct shear and odometer tests were carried out, and the elastic-plastic
model parameters obtained are given in Table 1.
Table 1. Physical parameters (elastic-plastic model) of regenerated sand.
Angle of Internal Friction Cohesion Poisson’s Ratio Young’s Modulus
ϕ=32◦−37◦c=0 kPa −4 kPa ν=0.25 E=86.9 ±16.1 MPa
It should be emphasized that for loose materials, the results of experimental tests describing
the elastic behavior are always associated with large errors, hence the significant variability of the
Young’s modulus.
The following values were adopted for numerical simulations:
ϕ
=35
◦
,c=3 kPa. Despite the fact
that for sand, the Young’s modulus depends on pressure, a constant value was used in the simulations:
ES=87.0 MPa.
Details on experimental testing of the properties of regenerated molding sand are presented in
Appendix E, where Figures A7–A10 show a sample after test carried out in the triaxial compression
apparatus and charts with the exemplary results obtained.
9. Geometric Model of System
The built-up geometric model consists of three solids representing a roller, a layer of sand and a
part with the side surface of the batch of the regeneration device (Figure 7).
As the pressure exerted by the roller is located on a relatively small area, there is no need to
consider a layer of sand spread over the entire side of the regenerator batch. It was assumed that the
layer of sand is spread over a side segment with an angular width of 40 degrees. Instead of the entire
roller, its cross-section was formed the upper part of which is a rectangle measuring 212 mm
×
65 mm.
The built-up CAD model is parametric, allowing, among others, continuous change in the sand
layer thickness and the sidewall groove depth, thus enabling a description of its progressive wear.
The thickness of the layer describing the side surface before wear is 20 mm, while its width and the
width of the regenerated sand layer are assumed to be equal to 110 mm.
Materials 2020,13, 2134 11 of 29
Materials 2020, 13, x FOR PEER REVIEW 10 of 29
over a finite domain in the vicinity of the particle using a weight function, also called a characteristic
function.
The basic difference between MPM and FEM is that in the former, state variables (i.e., mass,
stress, strain, velocity, etc.) are assigned to material points that are defined independently of the
computational mesh. In the latter, state variables are specified at integration points connected with
the elements. More mathematical details of the MPM are presented in Appendix D.
8. Material Constants. Experimental Research Results
The coefficients describing the constitutive model of sand adopted in numerical calculations
were determined based on the results of experimental research conducted on sand used in the
foundry. Standard triaxial compression, direct shear and odometer tests were carried out, and the
elastic-plastic model parameters obtained are given in Table 1.
Table 1. Physical parameters (elastic-plastic model) of regenerated sand.
Angle of Internal Friction Cohesion Poisson’s Ratio Young’s Modulus
32 37
ϕ
°=−° 0 kPa 4 kPac=− 0.25
ν
= 86.9 16.1 MPaE=±
It should be emphasized that for loose materials, the results of experimental tests describing the
elastic behavior are always associated with large errors, hence the significant variability of the
Young’s modulus.
The following values were adopted for numerical simulations: φ = 35°, c = 3 kPa. Despite the fact
that for sand, the Young’s modulus depends on pressure, a constant value was used in the
simulations: ES = 87.0 MPa.
Details on experimental testing of the properties of regenerated molding sand are presented in
Appendix E, where Figures A7–A10 show a sample after test carried out in the triaxial compression
apparatus and charts with the exemplary results obtained.
9. Geometric Model of System
The built-up geometric model consists of three solids representing a roller, a layer of sand and a
part with the side surface of the batch of the regeneration device (Figure 7).
Figure 7. System CAD model—side surface of regenerator batch not worn.
As the pressure exerted by the roller is located on a relatively small area, there is no need to
consider a layer of sand spread over the entire side of the regenerator batch. It was assumed that the
layer of sand is spread over a side segment with an angular width of 40 degrees. Instead of the entire
Figure 7. System CAD model—side surface of regenerator batch not worn.
Figure 7shows the model without side surface wear. In Figure 8, the side surface has a groove
with a maximum depth of 5 mm. The cross-section of the groove has a circular shape with rounded
edges. The groove shape was modeled on the basis of the observation of the worn surface of the
reclamation device (Figure 1c).
Materials 2020, 13, x FOR PEER REVIEW 11 of 29
roller, its cross-section was formed the upper part of which is a rectangle measuring 212 mm × 65
mm.
The built-up CAD model is parametric, allowing, among others, continuous change in the sand
layer thickness and the sidewall groove depth, thus enabling a description of its progressive wear.
The thickness of the layer describing the side surface before wear is 20 mm, while its width and the
width of the regenerated sand layer are assumed to be equal to 110 mm.
Figure 7 shows the model without side surface wear. In Figure 8, the side surface has a groove
with a maximum depth of 5 mm. The cross-section of the groove has a circular shape with rounded
edges. The groove shape was modeled on the basis of the observation of the worn surface of the
reclamation device (Figure 1c).
Figure 8. System CAD model—side surface of regenerator batch worn to depth of 5 mm.
10. Assumptions of Computational Model Description in MPMsim
In the case of deformation of the regenerated sand layer presented in the paper, the Drucker–
Prager plasticity model together with the associated flow law without hardening was adopted
(Appendix C). In the numerical calculations, it was assumed that the deformation of the sand layer
under the pressure exerted by the roller is a static process, and the inertial forces resulting from the
movement of the roller in the sand layer were omitted. In addition, the lack of friction between the
sand and the roller and between the sand and the side surface of the regeneration device was
assumed.
Numerical simulations were made using the MPMsim program which employs the material
point method (www.mpmsim.com). The built CAD model was imported into the MPMsim program
in which displacement constraints were imposed on the lower part of the side surface fragment. An
evenly distributed normal load was applied to the pressure roller cross-section area. This load
generates a force pressing the roller against the sand layer, depending on the air pressure in the
supply line. Figure 9 shows the diagram in the MPMsim program of the considered system with the
applied load.
Figure 8. System CAD model—side surface of regenerator batch worn to depth of 5 mm.
10. Assumptions of Computational Model Description in MPMsim
In the case of deformation of the regenerated sand layer presented in the paper, the Drucker–Prager
plasticity model together with the associated flow law without hardening was adopted (Appendix C).
In the numerical calculations, it was assumed that the deformation of the sand layer under the pressure
exerted by the roller is a static process, and the inertial forces resulting from the movement of the roller
in the sand layer were omitted. In addition, the lack of friction between the sand and the roller and
between the sand and the side surface of the regeneration device was assumed.
Numerical simulations were made using the MPMsim program which employs the material point
method (www.mpmsim.com). The built CAD model was imported into the MPMsim program in which
displacement constraints were imposed on the lower part of the side surface fragment. An evenly
distributed normal load was applied to the pressure roller cross-section area. This load generates
Materials 2020,13, 2134 12 of 29
a force pressing the roller against the sand layer, depending on the air pressure in the supply line.
Figure 9shows the diagram in the MPMsim program of the considered system with the applied load.
Materials 2020, 13, x FOR PEER REVIEW 12 of 29
Figure 9. Computational model in MPMsim—normal load of roller cross-section.
In the computational model, it was assumed that the roller and the side surface are made of an
elastic material with a Young’s modulus 11
3.6 10 PaE=× (and with a Poisson’s ratio of 0.27), i.e.,
over 4000 times larger than the Young’s modulus determined experimentally for the tested sand.
With such a high ratio of Young’s modulus, the roller and side surface of the regenerator batch
deformations are negligible compared to the deformations of the sand layer.
For the Drucker–Prager plasticity model, additional parameters describing the behavior of the
material can be given: the angle of dilation and tensile strength.
The angle of dilation
ψ
is responsible for the change in volume occurring during plastic shear.
It is usually assumed that its size is constant. The condition 0
ψ
= means that the volume does not
change during plastic shear deformation. Clays have very low expansion ( 0
ψ
≈), while for sands
the angle of dilation depends on the angle of internal friction [30].
For sands and gravels with the internal friction angle 30
ϕ
>°, the angle of dilation can be
estimated using the expression 30
ψϕ
=− °. A negative value of the angle of dilation is only allowed
for very loose sands. In most cases, the condition assumed is 0
ψ
=.
The physical phenomenon of coherence responsible for the non-zero value of cohesion c is also
responsible for the finite (usually negligible) strength of the geological material (soil, sand) to tensile
stress. The tensile strength is usually a fraction of the cohesion value c.
Numerical calculations assume that the angle of dilation is 5 degrees, while the tensile strength
is assumed to be equal to 1 kPa. In the calculation algorithm used by the MPMsim program, this
parameter is associated with the possibility of reaching a limit state by exceeding the allowable
tension value (the so-called ‘tension cut-off’).
11. Quality Indicator of Sand Regeneration
The purpose of the numerical simulations was to investigate the course of the deformation
process of the sand layer located between the roller and the side surface of the regeneration batch
and, on this basis, to define the simplest possible indicator allowing comparison of the degrees of
regeneration of the used molding sand achieved for different values of air pressure in the supply line.
Figure 9. Computational model in MPMsim—normal load of roller cross-section.
In the computational model, it was assumed that the roller and the side surface are made of an
elastic material with a Young’s modulus
E=
3.6
×
10
11 Pa
(and with a Poisson’s ratio of 0.27), i.e., over
4000 times larger than the Young’s modulus determined experimentally for the tested sand. With such
a high ratio of Young’s modulus, the roller and side surface of the regenerator batch deformations are
negligible compared to the deformations of the sand layer.
For the Drucker–Prager plasticity model, additional parameters describing the behavior of the
material can be given: the angle of dilation and tensile strength.
The angle of dilation
ψ
is responsible for the change in volume occurring during plastic shear.
It is usually assumed that its size is constant. The condition
ψ=
0 means that the volume does not
change during plastic shear deformation. Clays have very low expansion (
ψ≈
0), while for sands the
angle of dilation depends on the angle of internal friction [30].
For sands and gravels with the internal friction angle
ϕ >
30
◦
, the angle of dilation can be
estimated using the expression
ψ=ϕ−
30
◦
. A negative value of the angle of dilation is only allowed
for very loose sands. In most cases, the condition assumed is ψ=0.
The physical phenomenon of coherence responsible for the non-zero value of cohesion cis also
responsible for the finite (usually negligible) strength of the geological material (soil, sand) to tensile
stress. The tensile strength is usually a fraction of the cohesion value c.
Numerical calculations assume that the angle of dilation is 5 degrees, while the tensile strength
is assumed to be equal to 1 kPa. In the calculation algorithm used by the MPMsim program, this
parameter is associated with the possibility of reaching a limit state by exceeding the allowable tension
value (the so-called ‘tension cut-off’).
11. Quality Indicator of Sand Regeneration
The purpose of the numerical simulations was to investigate the course of the deformation process
of the sand layer located between the roller and the side surface of the regeneration batch and, on this
Materials 2020,13, 2134 13 of 29
basis, to define the simplest possible indicator allowing comparison of the degrees of regeneration of
the used molding sand achieved for different values of air pressure in the supply line.
In order for the regeneration process to take place, sufficiently large shear deformations in the
regenerated sand layer, which occur at a sufficiently high stress level, are necessary. Even large stresses
occurring in an area in which, for example, boundary conditions prevent deformation will not cause
particles of sand to move relative to each other and deformation will be small. On the other hand,
large shear strains at low stresses will also not be suitable conditions for sand regeneration.
It is difficult to theoretically determine the optimal conditions under which the regeneration
process should occur and, what is more, it would require a number of experimental studies. On the
basis of the results of the regeneration process taking place in the tested device, however, it is possible
to describe the conditions in which regeneration occurs with optimal efficiency.
It is known that for the device considered in the paper that does not yet have wear of the side
surface (depth of the groove equal to h=0 mm), the optimal regeneration conditions are ensured by
the supply air pressure p=2.4 bar (pressing force
N
=1200 N). It is also known that for a maximum
depth of the groove equal to h=5 mm, optimal regeneration conditions occur for the supply pressure
equal to p=3.6 bar (pressing force
N
=1800 N). This pressure value was determined experimentally in
such a way that the values of ignition losses were comparable in these two cases which are reference
systems, on the basis of which a numerical indicator of regeneration quality will be defined, whose
value for these two systems should be with high accuracy almost the same.
The cross-sections visible in Figures 10 and 11 show the deformation of the sand layer (middle
section) for two reference states intended to define the regeneration quality indicator. The figures show
the displacement in the ydirection, parallel to the axis of the roller.
Materials 2020, 13, x FOR PEER REVIEW 13 of 29
In order for the regeneration process to take place, sufficiently large shear deformations in the
regenerated sand layer, which occur at a sufficiently high stress level, are necessary. Even large
stresses occurring in an area in which, for example, boundary conditions prevent deformation will
not cause particles of sand to move relative to each other and deformation will be small. On the other
hand, large shear strains at low stresses will also not be suitable conditions for sand regeneration.
It is difficult to theoretically determine the optimal conditions under which the regeneration
process should occur and, what is more, it would require a number of experimental studies. On the
basis of the results of the regeneration process taking place in the tested device, however, it is possible
to describe the conditions in which regeneration occurs with optimal efficiency.
It is known that for the device considered in the paper that does not yet have wear of the side
surface (depth of the groove equal to h = 0 mm), the optimal regeneration conditions are ensured by
the supply air pressure p = 2.4 bar (pressing force N = 1200 N). It is also known that for a maximum
depth of the groove equal to h = 5 mm, optimal regeneration conditions occur for the supply pressure
equal to p = 3.6 bar (pressing force N = 1800 N). This pressure value was determined experimentally
in such a way that the values of ignition losses were comparable in these two cases which are
reference systems, on the basis of which a numerical indicator of regeneration quality will be defined,
whose value for these two systems should be with high accuracy almost the same.
The cross-sections visible in Figures 10 and 11 show the deformation of the sand layer (middle
section) for two reference states intended to define the regeneration quality indicator. The figures
show the displacement in the y direction, parallel to the axis of the roller.
Figure 10. Displacements (in meters, middle section) in sand layer in parallel y direction to roller axis:
h = 0 mm, popt = 2.4 bar (N = 1200 N).
Figure 10.
Displacements (in meters, middle section) in sand layer in parallel ydirection to roller axis:
h=0 mm, popt =2.4 bar (N=1200 N).
Materials 2020,13, 2134 14 of 29
Materials 2020, 13, x FOR PEER REVIEW 14 of 29
Figure 11. Displacements (in meters, middle section) in sand layer in parallel y direction to roller axis:
h = 5 mm, popt= 3.6 bar (N = 1800 N).
Based on the observation of the results of the numerical simulations of sand layer deformation,
as the indicator of the quality of regeneration, marked by Q, was taken the maximum displacement
of the middle section of the layer in the y direction, parallel to the axis of the roller, marked as max
y,
divided by the maximum thickness of the sand layer before its deformation, marked by max
δ
:
max
max
y
Q
δ
= (6)
The adopted quality indicator Q, which is in a sense an approximation of the average shear strain
component, may be considered as a function of the pressure p (bar) and the maximum depth of the
groove h (mm): (,)QQph=. For the unused side surface (h = 0 mm) and pressure p = 2.4 bar, the
quality indicator is: 1(2.4,0.0) 0.024985QQ==
, while for the worn side surface with a groove depth
equal to h = 5 mm and pressure p = 3.6 bar is obtained: 2(3.6,5.0) 0.024402QQ==
.
The values
1
Q and 2
Q differ by approximately 2%. As the optimal indicator was arbitrarily
adopted the approximate arithmetic mean of the values
1
Q and 2
Q: 0.024700
OPT
Q=.
It was assumed in the calculations that the thickness of the sand layer for the unused side surface
before the displacement of the roller is 9 mm, i.e., in this case max 9.0 9.0 0.0 9.0 mmh
δ
=+=+= is
obtained, while for the worn side surface with a groove depth of h = 5 mm,
max 9.0 9.0 5.0 14.0 mmh
δ
=+=+= is obtained. For the value of the original layer thickness adopted
in this way, the minimum displacement of the outer surface of the roller (i.e., its part nearest to the
side surface) in a direction perpendicular to its axis at supply pressure p = 2.4 bar is approximately
1.5 mm (exact simulation result: 0.00164 m), i.e., after depressing the roller, the minimum thickness
of the sand layer is approximately 7.5 mm (exact value from the simulation: 0.00736 m). This is the
approximate width of the gap between the roller and the side surface at steady state of the
regeneration device for the unused batch.
12. Results of Numerical Calculations
Based on the defined quality indicator Q, it is possible to estimate the optimal supply pressure
value in the system for side surface groove in the range: 0.0 mm 5.0 mmh<< . After exceeding the
upper value of the range, the ceramic coating of the side surface of the regenerator batch is practically
suitable for replacement.
Figure 11.
Displacements (in meters, middle section) in sand layer in parallel ydirection to roller axis:
h=5 mm, popt=3.6 bar (N=1800 N).
Based on the observation of the results of the numerical simulations of sand layer deformation, as
the indicator of the quality of regeneration, marked by Q, was taken the maximum displacement of the
middle section of the layer in the ydirection, parallel to the axis of the roller, marked as
ymax
, divided
by the maximum thickness of the sand layer before its deformation, marked by δmax:
Q=ymax
δmax (6)
The adopted quality indicator Q, which is in a sense an approximation of the average shear
strain component, may be considered as a function of the pressure p(bar) and the maximum depth
of the groove h(mm):
Q=Q(p
,
h)
. For the unused side surface (h=0 mm) and pressure p=2.4 bar,
the quality indicator is:
Q1=Q(
2.4, 0.0
) =
0.024985, while for the worn side surface with a groove
depth equal to h=5 mm and pressure p=3.6 bar is obtained: Q2=Q(3.6, 5.0) = 0.024402.
The values
Q1
and
Q2
differ by approximately 2%. As the optimal indicator was arbitrarily
adopted the approximate arithmetic mean of the values Q1and Q2:QOPT =0.024700.
It was assumed in the calculations that the thickness of the sand layer for the unused side surface
before the displacement of the roller is 9 mm, i.e., in this case
δmax =
9.0
+h=
9.0
+
0.0
=
9.0
mm
is obtained, while for the worn side surface with a groove depth of h=5 mm,
δmax =
9.0
+h=
9.0
+
5.0
=
14.0
mm
is obtained. For the value of the original layer thickness adopted in this way, the
minimum displacement of the outer surface of the roller (i.e., its part nearest to the side surface) in
a direction perpendicular to its axis at supply pressure p=2.4 bar is approximately 1.5 mm (exact
simulation result: 0.00164 m), i.e., after depressing the roller, the minimum thickness of the sand layer
is approximately 7.5 mm (exact value from the simulation: 0.00736 m). This is the approximate width
of the gap between the roller and the side surface at steady state of the regeneration device for the
unused batch.
12. Results of Numerical Calculations
Based on the defined quality indicator Q, it is possible to estimate the optimal supply pressure
value in the system for side surface groove in the range: 0.0
mm <h<
5.0
mm
. After exceeding the
Materials 2020,13, 2134 15 of 29
upper value of the range, the ceramic coating of the side surface of the regenerator batch is practically
suitable for replacement.
For the groove depth described by the values
h=
1.0, 2.0, 3.0, 4.0
mm
(respectively obtained
at
δmax =
10.0, 11.0, 12.0, 13.0
mm
) such a value of the force
N
(and also the supply pressure p) is
searched for that presses the roller against the sand layer, for which the deformation will be described
by the value of the quality indicator Qnot very different (with an error of not more than 1%) from the
value QOPT =0.024700.
The graph in Figure 12 (and data collected in Table 2) shows the value of the optimal air supply
pressure depending on the depth of the side surface groove. It can be seen that the optimum pressure
value depends approximately linearly on the value of the groove depth.
Materials 2020, 13, x FOR PEER REVIEW 15 of 29
For the groove depth described by the values 1.0, 2.0, 3.0, 4.0 mmh=(respectively obtained at
max 10.0,11.0,12.0,13.0 mm
δ
=) such a value of the force N (and also the supply pressure p) is
searched for that presses the roller against the sand layer, for which the deformation will be described
by the value of the quality indicator Q not very different (with an error of not more than 1%) from
the value 0.024700
OPT
Q=.
The graph in Figure 12 (and data collected in Table 2) shows the value of the optimal air supply
pressure depending on the depth of the side surface groove. It can be seen that the optimum pressure
value depends approximately linearly on the value of the groove depth.
Figures A11–A14 (Appendix F) show the displacements in the sand layer in y direction, parallel
to the axis of the roller, for the different values of the side surface groove and for the optimal air
pressure values in the supply line.
Table 2. Optimal supply pressure (elastic-plastic model) of regenerated sand.
Maximum Depth of the Side Surface
Groove h (mm)
Optimal Supply
Pressure popt (bar)
0.0 2.40
1.0 2.60
2.0 2.85
3.0 3.10
4.0 3.35
5.0 3.60
Figures 13–15 present graphs of, respectively: the maximum displacement in the middle section
layer in the y direction parallel to the axis of the roller; displacement of the roller (its part nearest to
the side surface) in a direction perpendicular to its axis; indicator of the quality of regeneration
depending on the pressure in the supply line for the maximum groove of the side surface (described
by the value h = 5 mm).
Figure 12. Optimal supply pressure popt depending on depth of side surface groove h.
012345
Side surface groove depth [mm]
2.2
2.4
2.6
2.8
3.0
3.2
3.4
3.6
3
.
8
Optimal supply pressure [bar]
Figure 12. Optimal supply pressure popt depending on depth of side surface groove h.
Table 2. Optimal supply pressure (elastic-plastic model) of regenerated sand.
Maximum Depth of the Side Surface Groove h(mm) Optimal Supply Pressure popt (bar)
0.0 2.40
1.0 2.60
2.0 2.85
3.0 3.10
4.0 3.35
5.0 3.60
Figures A11–A14 (Appendix F) show the displacements in the sand layer in ydirection, parallel to
the axis of the roller, for the different values of the side surface groove and for the optimal air pressure
values in the supply line.
Figures 13–15 present graphs of, respectively: the maximum displacement in the middle section
layer in the ydirection parallel to the axis of the roller; displacement of the roller (its part nearest to the
side surface) in a direction perpendicular to its axis; indicator of the quality of regeneration depending
on the pressure in the supply line for the maximum groove of the side surface (described by the value
h=5 mm).
The graphs presented in Figures 13–15 reveal that for the quantities given, their variability is a
non-linear function of the supply pressure, although in the tested pressure range the deviation from
linearity is small. Increasing the pressure would surely give graphs that differ more and more from
linear functions, although above a certain pressure value the grains of sand would start to be destroyed,
which would change the parameters describing it, such as the Young’s modulus and Poisson’s ratio.
Materials 2020,13, 2134 16 of 29
This would require additional experimental testing which would not be justified from a practical point
of view, because the goal of the regeneration process of molding sand is to remove the resin without
destroying its quartz grains.
Materials 2020, 13, x FOR PEER REVIEW 16 of 29
Figure 13. Maximum displacement in sand layer (middle section) in y direction parallel to roller axis
depending on pressure in supply line.
Figure 14. Displacement of roller (its part nearest to side surface) in direction perpendicular to its axis
depending on pressure in supply line for h = 5 mm.
2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
Supply pressure [bar]
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
Maximum displacement in y dir ection [mm]
2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
Supply pressure [bar]
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Minimum displacement in z direct ion [mm]
Figure 13.
Maximum displacement in sand layer (middle section) in ydirection parallel to roller axis
depending on pressure in supply line.
Materials 2020, 13, x FOR PEER REVIEW 16 of 29
Figure 13. Maximum displacement in sand layer (middle section) in y direction parallel to roller axis
depending on pressure in supply line.
Figure 14. Displacement of roller (its part nearest to side surface) in direction perpendicular to its axis
depending on pressure in supply line for h = 5 mm.
2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
Supply pressure [bar]
0.20
0.22
0.24
0.26
0.28
0.30
0.32
0.34
0.36
0.38
Maximum displacement in y dir ection [mm]
2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
Supply pressure [bar]
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
Minimum displacement in z direct ion [mm]
Figure 14.
Displacement of roller (its part nearest to side surface) in direction perpendicular to its axis
depending on pressure in supply line for h=5 mm.
Materials 2020,13, 2134 17 of 29
Materials 2020, 13, x FOR PEER REVIEW 17 of 29
Figure 15. Regeneration quality indicator depending on pressure in supply line for h = 5 mm.
The graphs presented in Figures 13–15 reveal that for the quantities given, their variability is a
non-linear function of the supply pressure, although in the tested pressure range the deviation from
linearity is small. Increasing the pressure would surely give graphs that differ more and more from
linear functions, although above a certain pressure value the grains of sand would start to be
destroyed, which would change the parameters describing it, such as the Young’s modulus and
Poisson’s ratio. This would require additional experimental testing which would not be justified from
a practical point of view, because the goal of the regeneration process of molding sand is to remove
the resin without destroying its quartz grains.
13. Verification Tests Results
In order to confirm model calculations, which showed that with a certain wear level of the
peripheral side surface of the regeneration device, it is possible to increase the regenerative impact
by increasing the supply pressure value of the cylinders pressing the roller to the side surface,
verification tests were carried out.
Verification tests were carried out using the USR-I device, which was excluded from production
for their duration. They consisted of subjecting a portion of the used sand, after the initial
regeneration process, consisting of crushing lumps and clumps of grains, to the regenerative effect in
the USR-I device operating at various settings of pressure supplying the cylinders pressing the rollers
to the side surface.
The tests were carried out after a device operation time of 2000 h, at which the groove depth of
the side surface was 3.8 mm (Figure 1). The following supply pressure values were used: 2.5 bar, 3.0
bar, 3.5 bar, 4.0 bar. Each portion of sand subjected to regeneration was 30 kg, while the regeneration
time was 30 s. Due to the fact that a periodic mode of the device operation was used, it was possible
to classify and dedust regenerates obtained for individual device settings.
During the tests, the following parameters of the used sand and regenerate were determined: ignition
loss, pH, grain size d50, and the amount of dust generated in the regeneration process. The results of
the tests are presented in the set of charts in Figure 16.
2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
Supply pressure [bar]
0.016
0.017
0.018
0.019
0.020
0.021
0.022
0.023
0.024
0.025
0.026
0.027
0.028
Regeneration quality indicator
Figure 15. Regeneration quality indicator depending on pressure in supply line for h=5 mm.
13. Verification Tests Results
In order to confirm model calculations, which showed that with a certain wear level of the
peripheral side surface of the regeneration device, it is possible to increase the regenerative impact by
increasing the supply pressure value of the cylinders pressing the roller to the side surface, verification
tests were carried out.
Verification tests were carried out using the USR-I device, which was excluded from production
for their duration. They consisted of subjecting a portion of the used sand, after the initial regeneration
process, consisting of crushing lumps and clumps of grains, to the regenerative effect in the USR-I
device operating at various settings of pressure supplying the cylinders pressing the rollers to the
side surface.
The tests were carried out after a device operation time of 2000 h, at which the groove depth of
the side surface was 3.8 mm (Figure 1). The following supply pressure values were used: 2.5 bar, 3.0
bar, 3.5 bar, 4.0 bar. Each portion of sand subjected to regeneration was 30 kg, while the regeneration
time was 30 s. Due to the fact that a periodic mode of the device operation was used, it was possible to
classify and dedust regenerates obtained for individual device settings.
During the tests, the following parameters of the used sand and regenerate were determined:
ignition loss, pH, grain size d
50
, and the amount of dust generated in the regeneration process.
The results of the tests are presented in the set of charts in Figure 16.
The results of the ignition loss of regenerates obtained after various impact of the regenerator,
measured by the values of supply pressure, which are presented in Figure 16a, show that as the value
of supply pressure increases, a decrease in the value of the ignition loss of obtained regenerates is
observed. A similar effect is observed by analyzing the results of the pH of obtained regenerates,
which, together with the increase in the intensity of the regenerator’s impact, decrease, aiming at the
reaction obtained for fresh sand (Figure 16b).
The grain size d
50
, for which the results are presented in Figure 16c, also decreases as a result of
the increase in supply pressure. At this point, it should be noted that quartz sand, characterized by its
diameter d
50
=0.32 mm, is used to prepare the molding sand. Therefore, obtained in the verifying tests,
the results of the analysis of regenerate grain size which show a smaller diameter than that specified
for fresh sand may indicate too intensive impact of the regenerator and a progressive, unfavorable
phenomenon of crushing of quartz matrix grains. This effect was obtained for the regeneration process
carried out at a supply pressure of 4 bar.
Materials 2020,13, 2134 18 of 29
Materials 2020, 13, x FOR PEER REVIEW 18 of 29
(a) (b)
(c) (d)
(e)
Figure 16. Results of verification tests: (a) ignition loss of obtained regenerates; (b) pH of the obtained
regenerates; (c) regenerate grain size d50; (d) amount of dust generated in the regeneration process; (e)
ignition loss of dust.
The results of the ignition loss of regenerates obtained after various impact of the regenerator,
measured by the values of supply pressure, which are presented in Figure 16a, show that as the value
of supply pressure increases, a decrease in the value of the ignition loss of obtained regenerates is
observed. A similar effect is observed by analyzing the results of the pH of obtained regenerates,
which, together with the increase in the intensity of the regenerator’s impact, decrease, aiming at the
reaction obtained for fresh sand (Figure 16b).
The grain size d50, for which the results are presented in Figure 16c, also decreases as a result of
the increase in supply pressure. At this point, it should be noted that quartz sand, characterized by
its diameter d50 = 0.32 mm, is used to prepare the molding sand. Therefore, obtained in the verifying
tests, the results of the analysis of regenerate grain size which show a smaller diameter than that
2.50
1.42 1.35 1.25
1.09
0.00
0.50
1.00
1.50
2.00
2.50
3.00
Used sand 2.5 3 3.5 4
Ignition loss [%]
Supply pressure [bar]
11.15
11.09
11.03
10.98
10.73
10.50
10.60
10.70
10.80
10.90
11.00
11.10
11.20
Used sand 2.5 3 3.5 4
pH value
Supply pressure [bar]
0.350
0.343
0.331
0.324
0.309
0.28
0.29
0.30
0.31
0.32
0.33
0.34
0.35
0.36
Used sand 2.5 3 3.5 4
d
50
grain diameter [mm]
Supply pressure [bar]
0.00
2.30
2.80
3.10
4.40
0.00
0.50
1.00
1.50
2.00
2.50
3.00
3.50
4.00
4.50
5.00
Used sand 2.5 3 3.5 4
Dust generated [%]
Supply pressure [bar]
0
38
35
29
22
0
5
10
15
20
25
30
35
40
Used sand 2.5 3 3.5 4
Ignition loss of dust generated [%]
Supply pressure [bar]
Figure 16.
Results of verification tests: (
a
) ignition loss of obtained regenerates; (
b
) pH of the obtained
regenerates; (
c
) regenerate grain size d
50
; (
d
) amount of dust generated in the regeneration process; (
e
)
ignition loss of dust.
Figure 16d presents the results of the amount of dust generated in the regeneration process. It was
possible to conduct this test owing to the use of a periodic device operation mode. Thanks to that,
after the regeneration of 30 kg of used sand, it was possible to carry out a pneumatic classification
and determine the amount of dust generated in the process. These dusts were created as a result of
elementary rubbing, abrasion and crushing operations taking place in the device (Figures A3–A5).
A steady increase in the amount of dust can be seen up to 3.5 bar, after regeneration at a supply pressure
of 4 bar, a step increase in the amount of dust has been observed.
Materials 2020,13, 2134 19 of 29
Comparing this with the value of the dust ignition loss from Figure 16e, showing the amount
of organic substance it contains (binders removed from the surface of the matrix grains as a result of
the regeneration process), it can be stated that the regeneration process carried out at the value of
the supply pressure of the roller cylinders exceeding 3.5 bar leads to the adverse effect of crushing of
quartz matrix grains.
14. Discussion
The results of the experimental tests presented in Section 13 are consistent with the results of
numerical calculations presented in Section 12. The diagram in Figure 12 shows that the optimal
pressure for groove depth h=5 mm is equal to p=3.3 bar. By approximating the given values from
Figure 16a, the ignition loss for pressure p=3.3 bar equals 1.29, i.e., the value is slightly different
from the value obtained for h=0 mm and p=2.4 bar, equal to 1.24. Therefore, the introduced quality
indicator Q, based on the quantities related to shear deformation of the sand layer, can be associated
with the ignition loss of the regenerated sand.
The regeneration quality indicator defined in the work, associated with the shear deformation
in the middle section of the sand layer, justified by the physical course of the regeneration process,
allowed the determination of the optimal air supply pressure of the centrifugal device, depending on
the degree of wear of the side surface of the regenerator batch, described by the depth of the groove
formed in it.
It is worth emphasizing that the presented results of the calculations of the optimal air pressure
relate to a specific regenerative device and are valid for specific parameters, such as the diameter of the
pneumatic hose, leverage ratio and given dimensions of the roller and batch.
Based on the conducted verification tests, correctness of the developed model and compliance
of the test results obtained with the parameters of devices regulated on the basis of the results of
numerical calculations were confirmed.
The developed theoretical model allows the optimization of the industrial device operation while
maintaining the required reclaim quality determined by the range of loss on ignition as well as the size
and shape of quartz basis grains. The model calculations will enable the development of the device’s
automated work system responding to the emerging effect of wear on the side surface by means of
changing the distance (via controlling the supply air pressure) between and mutual arrangement of
the cooperating mechanical system components, the roller and the side surface of the batch.
15. Conclusions
The simulation and experimental studies were carried out for a specific regenerative machine and
a specific molding sand described in the publication. However, the qualitative conclusions that can be
drawn from the results obtained can be used for another regenerative machine of similar type with
different dimensions and for sand with different physical parameters. Resting upon the implemented
regeneration quality index based on the deformation of the sand layer occurring in the regenerator, the
optimal machine process parameters, determined by the supply pressure (i.e., the optimal regenerator
roller pressing force on the layer) and depending on the degree of side surface wear (given by the
maximum groove depth), can be obtained. An approximate linear relationship was obtained for the
case examined in the paper, but the result does not always have to be identical. Therefore, the supply
pressure can be changed depending on side surface wear to ensure adequate quality of the obtained
regenerate, and the method of pressure control can be obtained still prior to starting the regeneration
process by virtue of anticipating a change in geometry of the side surface as it wears.
Author Contributions:
Conceptualization, methodology, writing—original draft preparation, P.C., R.D. and W.Ł.;
formal analysis, writing—review and editing, R.D. and W.Ł.; validation, visualization, R.D. and W.Ł. All authors
have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Materials 2020,13, 2134 20 of 29
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Materials 2020, 13, x FOR PEER REVIEW 20 of 29
was obtained for the case examined in the paper, but the result does not always have to be identical.
Therefore, the supply pressure can be changed depending on side surface wear to ensure adequate
quality of the obtained regenerate, and the method of pressure control can be obtained still prior to
starting the regeneration process by virtue of anticipating a change in geometry of the side surface as
it wears.
Author Contributions: Conceptualization, methodology, writing—original draft preparation, P.C., R.D. and
W.Ł.; formal analysis, writing—review and editing, R.D. and W.Ł.; validation, visualization, R.D. and W.Ł.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Figure A1. Management methods for used molding sands—developed on the basis of [3].
Figure A2. Schematic view of proper (secondary) regeneration methods used [3].
Elementary operations occurring in mechanical regeneration, carried out at ambient
temperature, can be illustrated schematically (Figures A3–A5) and characterized as below.
Rubbing (Figure A3)—an elementary operation of the regeneration process consisting of a
gradual decrease in the thickness of binding material coatings removed from the surface of the grains
under the influence of mutual, frictional interaction. Rubbing occurs in a cluster of loose sand matrix
grains that are in relative motion and in direct contact with one another.
USED SAND
Primary regeneration
Sand after primary regeneration
Sand directed for use
outside foundry
Sand directed for reuse
in foundry
Without additional regenerative
treatment
(as a component of filling sands)
With additional regenerative
treatment (proper regeneration) -
for matrix recovery for
molding and core sands
PROPER REGENERATION
DRY REGENERATION WET REGENERATION
Mechanical Pneumatic
High-temperature
Combined
At ambient
temperature
At reduced
temperature
Low-temperature
Thermal
Mechanical-thermal-
pneumatic
Medium-temperature
Mechanical-
thermal
Mechanical-
pneumatic
Figure A1. Management methods for used molding sands—developed on the basis of [3].
Materials 2020, 13, x FOR PEER REVIEW 20 of 29
was obtained for the case examined in the paper, but the result does not always have to be identical.
Therefore, the supply pressure can be changed depending on side surface wear to ensure adequate
quality of the obtained regenerate, and the method of pressure control can be obtained still prior to
starting the regeneration process by virtue of anticipating a change in geometry of the side surface as
it wears.
Author Contributions: Conceptualization, methodology, writing—original draft preparation, P.C., R.D. and
W.Ł.; formal analysis, writing—review and editing, R.D. and W.Ł.; validation, visualization, R.D. and W.Ł.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
Appendix A
Figure A1. Management methods for used molding sands—developed on the basis of [3].
Figure A2. Schematic view of proper (secondary) regeneration methods used [3].
Elementary operations occurring in mechanical regeneration, carried out at ambient
temperature, can be illustrated schematically (Figures A3–A5) and characterized as below.
Rubbing (Figure A3)—an elementary operation of the regeneration process consisting of a
gradual decrease in the thickness of binding material coatings removed from the surface of the grains
under the influence of mutual, frictional interaction. Rubbing occurs in a cluster of loose sand matrix
grains that are in relative motion and in direct contact with one another.
USED SAND
Primary regeneration
Sand after primary regeneration
Sand directed for use
outside foundry
Sand directed for reuse
in foundry
Without additional regenerative
treatment
(as a component of filling sands)
With additional regenerative
treatment (proper regeneration) -
for matrix recovery for
molding and core sands
PROPER REGENERATION
DRY REGENERATION WET REGENERATION
Mechanical Pneumatic
High-temperature
Combined
At ambient
temperature
At reduced
temperature
Low-temperature
Thermal
Mechanical-thermal-
pneumatic
Medium-temperature
Mechanical-
thermal
Mechanical-
pneumatic
Figure A2. Schematic view of proper (secondary) regeneration methods used [3].
Elementary operations occurring in mechanical regeneration, carried out at ambient temperature,
can be illustrated schematically (Figures A3–A5) and characterized as below.
Rubbing (Figure A3)—an elementary operation of the regeneration process consisting of a gradual
decrease in the thickness of binding material coatings removed from the surface of the grains under
the influence of mutual, frictional interaction. Rubbing occurs in a cluster of loose sand matrix grains
that are in relative motion and in direct contact with one another.
Materials 2020, 13, x FOR PEER REVIEW 21 of 29
Figure A3. Schematic diagram of matrix release from coatings of used binding material—result of
elementary rubbing operation [3].
Abrasion (Figure A4)—an elementary operation of the regeneration process consisting of a
gradual decrease in the thickness of binding material coatings removed from the surface of the grains
under the influence of the mechanical effect of a machine structural component. Coating abrasion
occurs in larger clusters or on individual grains of sand matrix in motion relative to the movable or
permanent working components of the machine with which they are in contact.
Figure A4. Schematic diagram of matrix release from coatings of used binding material—result of
elementary abrasion operation [3].
Crushing (Figure A5)—an elementary regeneration operation consisting of abrupt reduction in
the thickness of binding material coating on grains and reduction of its particles. Crushing is caused
by the pressure of external static or dynamic forces causing an increase in contact loads transmitted
to the coatings by matrix grains. As a result of this interaction, the coatings crack and the bridges
connecting the individual grains break. In a loose medium in motion, crushing occurs at moments of
rapid changes in the momentum of the grains, caused, for example, by their hitting on appropriately
shaped device components or impact plates oriented horizontally or vertically.
Figure A5. Schematic diagram of matrix release from coatings of used binding material–result of
elementary crushing operation [3].
Figure A3.
Schematic diagram of matrix release from coatings of used binding material—result of
elementary rubbing operation [3].
Materials 2020,13, 2134 21 of 29
Abrasion (Figure A4)—an elementary operation of the regeneration process consisting of a gradual
decrease in the thickness of binding material coatings removed from the surface of the grains under
the influence of the mechanical effect of a machine structural component. Coating abrasion occurs in
larger clusters or on individual grains of sand matrix in motion relative to the movable or permanent
working components of the machine with which they are in contact.
Materials 2020, 13, x FOR PEER REVIEW 21 of 29
Figure A3. Schematic diagram of matrix release from coatings of used binding material—result of
elementary rubbing operation [3].
Abrasion (Figure A4)—an elementary operation of the regeneration process consisting of a
gradual decrease in the thickness of binding material coatings removed from the surface of the grains
under the influence of the mechanical effect of a machine structural component. Coating abrasion
occurs in larger clusters or on individual grains of sand matrix in motion relative to the movable or
permanent working components of the machine with which they are in contact.
Figure A4. Schematic diagram of matrix release from coatings of used binding material—result of
elementary abrasion operation [3].
Crushing (Figure A5)—an elementary regeneration operation consisting of abrupt reduction in
the thickness of binding material coating on grains and reduction of its particles. Crushing is caused
by the pressure of external static or dynamic forces causing an increase in contact loads transmitted
to the coatings by matrix grains. As a result of this interaction, the coatings crack and the bridges
connecting the individual grains break. In a loose medium in motion, crushing occurs at moments of
rapid changes in the momentum of the grains, caused, for example, by their hitting on appropriately
shaped device components or impact plates oriented horizontally or vertically.
Figure A5. Schematic diagram of matrix release from coatings of used binding material–result of
elementary crushing operation [3].
Figure A4.
Schematic diagram of matrix release from coatings of used binding material—result of
elementary abrasion operation [3].
Crushing (Figure A5)—an elementary regeneration operation consisting of abrupt reduction in
the thickness of binding material coating on grains and reduction of its particles. Crushing is caused
by the pressure of external static or dynamic forces causing an increase in contact loads transmitted
to the coatings by matrix grains. As a result of this interaction, the coatings crack and the bridges
connecting the individual grains break. In a loose medium in motion, crushing occurs at moments of
rapid changes in the momentum of the grains, caused, for example, by their hitting on appropriately
shaped device components or impact plates oriented horizontally or vertically.
Materials 2020, 13, x FOR PEER REVIEW 21 of 29
Figure A3. Schematic diagram of matrix release from coatings of used binding material—result of
elementary rubbing operation [3].
Abrasion (Figure A4)—an elementary operation of the regeneration process consisting of a
gradual decrease in the thickness of binding material coatings removed from the surface of the grains
under the influence of the mechanical effect of a machine structural component. Coating abrasion
occurs in larger clusters or on individual grains of sand matrix in motion relative to the movable or
permanent working components of the machine with which they are in contact.
Figure A4. Schematic diagram of matrix release from coatings of used binding material—result of
elementary abrasion operation [3].
Crushing (Figure A5)—an elementary regeneration operation consisting of abrupt reduction in
the thickness of binding material coating on grains and reduction of its particles. Crushing is caused
by the pressure of external static or dynamic forces causing an increase in contact loads transmitted
to the coatings by matrix grains. As a result of this interaction, the coatings crack and the bridges
connecting the individual grains break. In a loose medium in motion, crushing occurs at moments of
rapid changes in the momentum of the grains, caused, for example, by their hitting on appropriately
shaped device components or impact plates oriented horizontally or vertically.
Figure A5. Schematic diagram of matrix release from coatings of used binding material–result of
elementary crushing operation [3].
Figure A5.
Schematic diagram of matrix release from coatings of used binding material–result of
elementary crushing operation [3].
Materials 2020,13, 2134 22 of 29
Appendix B
Materials 2020, 13, x FOR PEER REVIEW 22 of 29
Appendix B
(a) (b)
(c)
Figure A6. Diagram of the USR-I centrifugal device with cyclic operation, intended for regeneration
of used sands: (a) regenerator scheme: 1—drive motor, 2—regenerator working unit, 3—used mass
tank, 4—fluidization and dedusting gutter, 5—classifier chamber, 6—regenerate discharge; (b)
regenerator centrifugal batch assembly. Arrows indicate directions of air flow during sand portion
introduction and regeneration; (c) USR-I regenerator working batch [3].
The USR-I device has a compact structure allowing it to be placed in a space with dimensions
given in mm: 1870 (width) × 1563 (depth) × 2890 (height). Sand circulation during regeneration is
supported by air blown from below through a gap around the perimeter of the batch. When emptying
the batch, the air is blown from above and supports the spilling of the regenerated matrix which falls
onto the fluidizing feeder. Passing through the pneumatic classifier, the regenerate is deprived of
dust regeneration products and fine matrix fractions. The classification air is supplied by a centrifugal
fan connected to the device.
Appendix C
The process of plastic deformation of a wide class of materials can be modeled by the general
relationship in the form:
p
ij
ij
g
dd
ελ
σ
∂
=∂
(A1)
In the above,
p
ij
ε
is the plastic strain tensor element,
ij
σ
is the stress tensor element, g is the
scalar function, called the plastic potential, and
λ
is the plastic multiplier to be determined from the
hardening rule. The Equation (A1) describes the so-called non-associated flow rule. After taking as
Figure A6.
Diagram of the USR-I centrifugal device with cyclic operation, intended for regeneration of
used sands: (
a
) regenerator scheme: 1—drive motor, 2—regenerator working unit, 3—used mass tank,
4—fluidization and dedusting gutter, 5—classifier chamber, 6—regenerate discharge; (
b
) regenerator
centrifugal batch assembly. Arrows indicate directions of air flow during sand portion introduction
and regeneration; (c) USR-I regenerator working batch [3].
The USR-I device has a compact structure allowing it to be placed in a space with dimensions
given in mm: 1870 (width)
×
1563 (depth)
×
2890 (height). Sand circulation during regeneration is
supported by air blown from below through a gap around the perimeter of the batch. When emptying
the batch, the air is blown from above and supports the spilling of the regenerated matrix which falls
onto the fluidizing feeder. Passing through the pneumatic classifier, the regenerate is deprived of dust
regeneration products and fine matrix fractions. The classification air is supplied by a centrifugal fan
connected to the device.
Appendix C
The process of plastic deformation of a wide class of materials can be modeled by the general
relationship in the form:
dεp
ij =∂g
∂σij
dλ(A1)
In the above,
εp
ij
is the plastic strain tensor element,
σij
is the stress tensor element, gis the
scalar function, called the plastic potential, and
λ
is the plastic multiplier to be determined from the
hardening rule. The Equation (A1) describes the so-called non-associated flow rule. After taking as the
Materials 2020,13, 2134 23 of 29
plasticity function the yield function, the so-called associated flow rule is given, because the flow rule
is associated with a particular yield criterion.
It should be emphasized that the associated flow rule cannot be regarded as a law of nature. It is a
convenient mathematical approach that enables us to describe the deformation of materials in a way
that is consistent with the results of experimental research.
A wide class of materials (perhaps all realistically behaving) meet the so-called Drucker’s stability
postulate. Two conditions result from it: the plastic strain increment vector is outward normal to
the yield surface (normality rule); the plasticity surface in a stable plastic material must be convex
(convexity rule). The Drucker’s stability postulate is satisfied for a strain-hardening material subject to
the associated flow rule.
The elastic-plastic deformation process of the isotropic material described by the Drucker–Prager
model, subject to the associated flow rule, for the plasticity function given by Equation (2), is described
by the following equations [31]:
e
σ=e
D:(e
ε−e
εp)(A2)
κ=w(β)(A3)
.
e
εp=.
λ∂f
∂e
σ=.
λ αe
δ+e
s
2√J2!(A4)
.
β=r2
3k
.
e
εpk(A5)
.
λ≥0, f(e
σ,κ)≤0, .
λf(e
σ,κ) = 0 (A6)
where:
(A2)—linear elastic law
(A3)—hardening rule
(A4)—evolution law of plastic strain
(A5)—evolution law of hardening variable
(A6)—loading-unloading conditions
The following notations are introduced in Equations (A2)–(A6):
e
D
—elastic stiffness tensor,
e
ε
—strain tensor,
e
εp
—plastic strain tensor,
e
δ
—unit second order tensor,
e
s
—stress tensor deviator,
β—hardening variable, λ—plastic multiplier.
The flow rule Equation (A4) applies at all points of the conical yield surface with the exception of
the apex, and an upper dot marks the derivative with respect to time.
For the described model, the evolution of the hardening variable is directly related with the plastic
multiplier. Substitution of the flow rule Equation (A4) to Equation (A5) leads to the relationship:
.
β=k.
λ(A7)
with a constant parameter
k=p1/3+2α2
. In the simplest case of linear hardening, the hardening
function is a linear function of β:
w(β) = σ0+Kβ(A8)
where σ0is the initial yield stress, Kis the hardening modulus, case K=0 means no hardening.
Appendix D
In the MPM method, the continuous area
G
is represented by a set
p=
1
. . . Np
of material points
(particles). Each material point (particle) is assigned: mass, stress tensor, velocity vector and density,
indicated by:
mp
,
e
σp
,
vp
,
ρp
. The domain assigned to the material point with the index pis marked by
Gp
, and its volume by
Vp
. For example, for three-dimensional problems, domains
Gp
can be cuboids.
Materials 2020,13, 2134 24 of 29
Additionally, a computational grid of nodes is defined
i=
1
. . . Nn
, with the help of which the equations
governing a given problem are solved. Interpolation between material points and the mesh is done by
nodal shape functions Si(x).
In the GIMP method, the characteristic function is usually taken in the form:
χp(x) = (1 for x∈Gp
0 outside domain Gp(A9)
The classical MPM method is obtained after adopting as the characteristic function the
δ
-Dirac
function. For example, mass density field is then described by the expression:
ρ(x) =
Np
X
p=1
mpδ(x−xp)(A10)
where xpdenotes the position vector of the p-th material point.
The basic equation is the law of momentum change written for a continuous material medium:
ρdv
dt =∇·e
σ+ρb(A11)
where:
ρ=ρ(x,t)—spatial density distribution
v=v(x,t)—spatial velocity distribution
e
σ=e
σ(x,t)—Cauchy stress tensor
b=b(x,t)—mass force (force per unit mass)
The law of momentum change Equation (A11) is transformed into a discrete form after substituting
the mass density field Equation (A10) and expressing the fields of
v
,
e
σ
,
b
by the shape functions and
nodal parameters, defined on the computational mesh.
Appendix E
Materials 2020, 13, x FOR PEER REVIEW 24 of 29
Appendix D
In the MPM method, the continuous area G is represented by a set 1
p
pN= of material
points (particles). Each material point (particle) is assigned: mass, stress tensor, velocity vector and
density, indicated by:
p
m,
p
σ
,
p
v,
p
ρ
. The domain assigned to the material point with the index p
is marked by
p
G, and its volume by
p
V. For example, for three-dimensional problems, domains
p
G
can be cuboids. Additionally, a computational grid of nodes is defined 1n
iN=, with the help of
which the equations governing a given problem are solved. Interpolation between material points
and the mesh is done by nodal shape functions ()
i
Sx
.
In the GIMP method, the characteristic function is usually taken in the form:
1for
() 0 outside domain
p
p
p
xG
xG
χ
∈
=
(A9)
The classical MPM method is obtained after adopting as the characteristic function the
δ
-Dirac
function. For example, mass density field is then described by the expression:
1
() ( )
p
N
p
p
p
x
mxx
ρδ
=
=−
(A10)
where
p
x
denotes the position vector of the p-th material point.
The basic equation is the law of momentum change written for a continuous material medium:
dv b
dt
ρρ
=∇⋅ +
σ
(A11)
where:
(,)
x
t
ρρ
=—spatial density distribution
(,)vvxt=—spatial velocity distribution
(,)
x
t=
σσ
—Cauchy stress tensor
(,)bbxt=—mass force (force per unit mass)
The law of momentum change Equation (A11) is transformed into a discrete form after
substituting the mass density field Equation (A10) and expressing the fields of v,
σ
, b by the
shape functions and nodal parameters, defined on the computational mesh.
Appendix E
Figure A7. Sample of molding sand after tests in triaxial compression apparatus.
Figure A7. Sample of molding sand after tests in triaxial compression apparatus.
Laboratory methods of testing the basic static parameters of soils (mainly for civil engineering
purpose) are based on using the equipment and procedures used in accordance with European norms
and ISO standards (ISO/TS 17892 standard, www.iso.org), and they are as follows:
Materials 2020,13, 2134 25 of 29
•direct and residual shear apparatus
•triaxial compression apparatus
•edometer (deformation modules)
Materials 2020, 13, x FOR PEER REVIEW 25 of 29
Laboratory methods of testing the basic static parameters of soils (mainly for civil engineering
purpose) are based on using the equipment and procedures used in accordance with European norms
and ISO standards (ISO/TS 17892 standard, www.iso.org), and they are as follows:
• direct and residual shear apparatus
• triaxial compression apparatus
• edometer (deformation modules)
Figure A8. Deviatoric stress during testing of sample of regenerated sand in triaxial compression
apparatus.
Figure A9. Stress path during tests in triaxial compression apparatus.
Figure A8.
Deviatoric stress during testing of sample of regenerated sand in triaxial compression
apparatus.
Materials 2020, 13, x FOR PEER REVIEW 25 of 29
Laboratory methods of testing the basic static parameters of soils (mainly for civil engineering
purpose) are based on using the equipment and procedures used in accordance with European norms
and ISO standards (ISO/TS 17892 standard, www.iso.org), and they are as follows:
• direct and residual shear apparatus
• triaxial compression apparatus
• edometer (deformation modules)
Figure A8. Deviatoric stress during testing of sample of regenerated sand in triaxial compression
apparatus.
Figure A9. Stress path during tests in triaxial compression apparatus.
Figure A9. Stress path during tests in triaxial compression apparatus.
Materials 2020,13, 2134 26 of 29
Materials 2020, 13, x FOR PEER REVIEW 26 of 29
Figure A10. Mohr circle obtained in study of molding sand in triaxial compression apparatus.
Appendix F
Figure A11. Displacements (in meters, middle section) in sand layer in parallel y direction to roller
axis: h = 1.0 mm, popt=2.6 bar (N = 1300 N).
Figure A10. Mohr circle obtained in study of molding sand in triaxial compression apparatus.
Appendix F
Materials 2020, 13, x FOR PEER REVIEW 26 of 29
Figure A10. Mohr circle obtained in study of molding sand in triaxial compression apparatus.
Appendix F
Figure A11. Displacements (in meters, middle section) in sand layer in parallel y direction to roller
axis: h = 1.0 mm, popt=2.6 bar (N = 1300 N).
Figure A11.
Displacements (in meters, middle section) in sand layer in parallel ydirection to roller
axis: h=1.0 mm, popt=2.6 bar (N=1300 N).
Materials 2020,13, 2134 27 of 29
Materials 2020, 13, x FOR PEER REVIEW 27 of 29
Figure A12. Displacements (in meters, middle section) in sand layer in parallel y direction to roller
axis: h = 2.0 mm, popt = 2.85 bar (N = 1425 N).
Figure A13. Displacements (in meters, middle section) in sand layer in parallel y direction to roller
axis: h = 3.0 mm, popt = 3.1 bar (N = 1550 N).
Figure A12.
Displacements (in meters, middle section) in sand layer in parallel ydirection to roller
axis: h=2.0 mm, popt =2.85 bar (N=1425 N).
Materials 2020, 13, x FOR PEER REVIEW 27 of 29
Figure A12. Displacements (in meters, middle section) in sand layer in parallel y direction to roller
axis: h = 2.0 mm, popt = 2.85 bar (N = 1425 N).
Figure A13. Displacements (in meters, middle section) in sand layer in parallel y direction to roller
axis: h = 3.0 mm, popt = 3.1 bar (N = 1550 N).
Figure A13.
Displacements (in meters, middle section) in sand layer in parallel ydirection to roller
axis: h=3.0 mm, popt =3.1 bar (N=1550 N).
Materials 2020,13, 2134 28 of 29
Materials 2020, 13, x FOR PEER REVIEW 28 of 29
Figure A14. Displacements (in meters, middle section) in sand layer in parallel y direction to roller
axis: h = 4.0 mm, popt = 3.35 bar (N = 1675 N).
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March 2020).
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2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access
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