01 - matus krizan ondruska soos

ANALYSIS OF TOOL GEOMETRY

FOR SCREW EXTRUSION MACHINES

MAT Milo, (SK), KRIAN Peter, (SK), ONDRUKA Juraj,

(SK), OO ubomr, (SK)

Abstract. The contribution deals with the technology of biomass briquetting into the solid high-

grade biofuel by screw extrusion machines. It is focused mainly on the theory of compacting

tools for screw briquetting presses, their analysis, stress conditions and geometry. The main aim

is analyzing of pressing screw geometry and determination process of its design. Analysis of

force conditions on the screw is necessary for designed geometry verification and for stress

analysis. The determination process of the frictional power is instrumental to main power drive

design. Knowledge of these processes is the base of the new tools research for screw presses,

the increase of tools lifetime and the competitiveness of whole technology.

Key words. biomass, briquetting, tool, screw, screw machine, screw profile

Mathematics Subject Classification: Primary 28A75; Secondary 49Q10.

1 Introduction The world trend of the fossil fuel substitution by renewable energy sources affects the research direction of production machines in field of the material agglomeration. Biomass represents the most perspective renewable energy source with the most effective possibilities of the energy storage. It gives rise to a necessity of the research in field of the biomass treatment and transformation into solid high-grade biofuel. The briquetting technology is successfully used for production of solid biofuel destined for industrial sector as well as household. The biomass briquetting is based on the pre-treatment material pressing and its extruding through the pressing die. It is a complicated production process where the mount of conditions affecting the binding mechanism within the briquette has to be fulfilled. The production process and the production quality are influenced by many technological and structural parameters. The briquetting technology of biomass uses three types of machines hydraulic presses, mechanical crank presses and screw extrusion machines. Ever principle of machine with own advantages and disadvantages has its justification in the production of solid high-grade biofuel. In light of the achieved production quality, as a determining factor on the biofuel market, the most suitable is the principle of screw

Aplimat Journal of Applied Mathematics

volume 4 (2011), number2

404

extrusion machine. This principle allows production of different shapes of briquettes (cylindrical, n-angular; with or without hole, etc.). The main disadvantage of this type of press is high operating costs following the low lifetime of very expensive tools - screws. Nowadays, manufactured pressing screws reach lifetime only a few decades of working hours. The biomass briquetting by the screw presses is progressive technology of solid biofuel production. It is high actual and perspective to deal with the research in this area. Increase of the tools lifetime and decrease of the operating costs will be reflected on the lower price of production and may contribute for wider utilization of renewable energy sources. 2 Tools for screw briquetting presses The briquetting process by screw presses is continual without beats. The high compacting pressure is characteristic what causes high density and high strength of briquettes. The surface of briquette is high-class. By screw presses is possible to produce different shapes of briquettes with or without hole. There are put material and geometric requirements on the tools. Material requirements include high wear resistance, toughness and thermal stability. Specification of geometrical requirements in not simple and it is dependent on the pressed material. The main geometrical request is creating the rapid increase of pressure into the pressed material. The tool geometry has also to insure the material axial movement and the fluency of the pressing process. Tools consist of the feeding screw, the pressing screw and the pressing chamber included particulate dies (Fig. 1).

Figure 1. Tools of screw extrusion machines

(1 - feeding screw, 2 - pressing screw, 3 - pressing chamber, 4 - dies) The feeding screw is not stressed by high working load. Its main task is fluent axial movement of material toward the pressing screw and the homogenous filling the whole screw profile cross-section. The pressing screw as the tool is exposing to high workload, abrasion and temperature. Its geometry provides a high degree of compaction of material in the pressing chamber and the material extrusion through a particular dies, thereby achieving a compact pellet of high density, strength and surface quality. This contributes to compact briquette of high density, strength and surface quality. Pressing screw is the most loaded machine part with the highest rate of wear. The most worn part of the pressing screw is the spike (Fig. 2) and the first 1.5 thread, which shows the distribution of the workload on the tool. Material movement, compression, wear rate, stress


volume 4 (2011), number 2

405

distribution depends primarily on the chosen screw geometry. Therefore it is extremely important in the design of the screw to pay great attention to its geometry.

Figure 2. Pressing screw (1 - working screw part, 2 - calibration spike)

Pressing chamber must be sufficiently strong to withstand the internal pressure of compression. Each die in the pressing chamber have to copy the pressing screw, its geometry has to prevent the rotation of the material and provide its axial displacement. Their geometry is chosen so that on the one side of the pressing chamber, they copy the screw and on the other side, they pass fluently to the required shape of briquettes. They are highly stressed by compression pressure, temperature and abrasion especially. Therefore, their material has to be hard and abrasion resistant on the surface, tough in the core. 3 Analysis of pressing screw 3.1 Volume of screw profile Assuming that the thread space is filled with material to 100%, the volume of the screw profile represents the amount of material transported by the screw. Feeding screw is designed to achieve a fill volume of the screw profile. It is set coaxially with the pressing screw and form one unit. The main task of the feeding screw is the homogenization of entering material throughout cross-section of the screw profile.

1 - face angle

2 - backoff angle

s - pitch

e - sliding surface

h - profile depth

D - outside diameter of screw Figure 3. Screw profile (meridian cross-section)

Screw profile volume of i-course screw can be expressed (i - number of courses):



406

+

+

+

+=

)2

180.(30

2cos1

22sin

2sin.10

.

3

2.

6

1)

1180.(

301

cos1

12sin

1sin.10

.

3

1.

6

1

)2

180.(360

2sin

2cos1

.

2

2.

21)

1180.(

3601

sin

1cos1

..

2

1.

21

2sin.

1sin

)21

sin(

.

3

3

22

.2

1

2sin.

1sin

)21

sin(

.2

1.1...1

3.

D

r

D

r

D

r

D

h

D

r

D

h

D

h

D

h

D

h

s

ei

D

s

D

h

D

h

D

V

(1)

The expression can be considerably simplified if we implement some assumptions valid for special

types of structures, such as 02

21

21===

D

r

D

ror

.

( )( )eishDhVs

ei

D

s

D

h

D

h

D

V...11

3=

=

(2)

The volume profile of one thread can be changed following the screw length. Such changes will be achieved by:

-

changing the outer diameter (conical screw), -

changing the profile depth (conical screw core),changing the pitch angle (screw with progressive pitch).Several parameters can be changed also at the same time. Then the ratio

of volumes in the monitored areas on the screw is known as the compression ratio, thus: k = V/Vi 3.2 Speed and force relationships in the screw Material, in our case biomass, is delivered to the screw profile in loose form. The feeding screw profile under the hopper is permanently flooding, thus the filling of the screw profile is kept at the interface constant value during the compression process. Fluent and constant supply of material from the hopper is provided by the filling screw. The advantage of using the filling screw is the possibility of partial pre-compacting of material before entry into the feeding respectively pressing screw area. For describing the movement of material in the screw is introduced following assumptions:

-

thread profile is completely filled with solid,material moves in the direction of the screw pitch,neglecting the influence of curvature on the bottom of screw profile ,friction coefficient between material and monitored surfaces is constant,processed material does not transmit shear stresses.

If the assumptions are valid and in steady conditions, we can monitor the movement of defined particles in Fig 4.



407

- spiral angle

- feeding angle

uo - circumferential speed

vr - relative circumferential speed component

va - absolute circumferential speed component

Figure 4. Speed conditions in screw profile The material is moving at a speed vr relative to screw (in the moving coordinates). The material is moving at a speed va relative to the die (in the fixed coordinates). From Fig 4 we can derive:

sincos.

.

sin.coscos.sin

sin.

)sin(

sin.000

+

=

+

=

+

=

tg

tguuuva

(3)

tg

tguuuv

r

.cossin

.

sin.coscos.sin

sin.

)sin(

sin.000

+

=

+

=

+

= (4)

For the material transport is applied only axial velocity component vax, which is given by:

tgtg

tgtgu

tg

tguvv

aax

+

=

+

==.

.sincos.

sin.sin.

00 (5)

If is valid, that the vax has to be greater than 0, then the numerator in the fraction expression has to be greater than zero. Therefore has to be that the feeding angle belongs to the interval 0< < /2. If = 0 then tg = 0, and so vax = 0. The first boundary state occurs. In other words, we can say that the material will not be transported in the direction of the axis of the screw, but will rotate with it. The second boundary state occurs, if = /2. This value can not be appointed directly to the above relation, because we get an indefinite term. Therefore the original relationship is necessary to modify:

sincos.

sin.sin.

0

+

==

tg

tguvv

aax (6)

Substituting for = /2 we get:

aax

vutgutg

tguv ==

+

=

000.

90sin90cos.

90sin.

(7)

In this case, the material is moving in the direction of the screw axis without circumferential velocity component (such as nut and bolt). It follows that the real value of the angle will lie somewhere between these extremes. Using those relationships and knowledge can be expressed extruded amount. For simplicity, here is mentioned relationship for angular profile with the neglecting radius on the bottom of the screw profile:



408

SHaxs

iehhDvsV .1.)..(.

= (8)

Substituting:

SHaxs

iehhDv

tgtg

tgtgusV

.1.)..(.

..

0

+= (9)

(SV - volume flow of material, SH - bulk density of material).

For circumferential speed uo is valid:

nDu ..0 = (10)

(D screw diameter, n number of screw revolutions). Then we can write:

SH

tgtg

tgtgn

s

iehhDDVs

.

...1.)..(.2

+

= (11)

As conditions will vary in length l, the feeding angle will vary too. When the bulk material is pressed, the feeding angle shrink lengthwise. On the defined particles when the screw profile is completely filling, the forces affect by the following Fig. 5.

Figure 5. Force conditions in screw profile

Three-dimensional system of forces is converted to the planar so that all forces are reduced to the outer diameter of the screw. We respect the same pitch by introduction the pitch angles, see Fig. 6.



409

D

stg

p

.

=

D

htg

hD

stg

p

=

=

1

1.

).(0

D

htg

hD

stg

ps 21

1.

)2.(

=

=

Figure 6. Pitch angles on screw profile From Fig. 6 for forces balance in the direction of relative speed is valid:

)cos(..sin

cos.1...

...cos.1...2

.sin.cos.1....

sin....2

)sin(.sin.cos.1....

0

0

+

=

=

+

+

+

+

=

p

p

p

p

s

s

szz

p

p

pp

dls

iesfp

fD

hDF

D

hD

s

ieshdp

D

hDdl

s

iesfp

D

hDdlhfp

dl

s

iesfp

D

hDF

(12)

Where: fz - friction coefficient between the material and the screw, fp - friction coefficient between the material and the die. Exclusion of F we get:

[ ])sin(.)cos(..

sin

cos.1...

.cos.1...2

.sin

.cos.1....sin

....20

0

++

=

=

+

+

psp

p

p

p

s

szz

fdls

iesfp

D

hD

s

ieshdp

D

hDdl

s

iesfp

D

hDdlhfp

(13)

It is also valid:

( )[ ] dl

D

hD

s

iesf

D

hDhff

s

iesf

p

dp

D

hD

s

iesh

s

s

zzpsp

p

p

p

++

=

=

2.

sin

cos.1...

sin.2sin.)cos(.

sin

cos.1..

..cos.1..

0

0

(14)

After treatment:

( )[ ]

s

z

z

zpz

p

pp

tgD

hhD

hDf

s

ie

D

s

f

ftgftg

D

h

D

h

f

dl

D

p

dp

.cos.

2.

cos.sin.1.

.2

sincos..sin.cos.cos

.

1

1..

cos

1.

000

0

+

=

(15)



410

Assuming that the right-side expression the variables will not depend on the length l nor the pressure p, then integrating and using the boundary condition

00 ppl == , where l is the working

length of the screw with filled profile, we get:

( )[ ]

z

z

z

zpz

p

p

tgD

hhD

hDf

s

ie

D

s

f

ftgf

D

hDtg

D

h

f

p

p

l

D

.cos.

2.

cos.sin.1.

.2

cossin.sin.cos.

.cos..

ln.

000

00

=

(16)

The expression on the right side will denote as the constant of proportionality A.

( )[ ]

z

z

z

zpz

p

p

tgD

hhD

hDf

s

ie

D

s

fftgf

D

hDtg

D

h

fA

0000

cos

2

cossin1

2cossinsincos

cos

= (17)

Then it is a valid:

Dl

A

epp.

0.= (18)

The above relations show that the pressure in the screw profile is exponentially dependent on the length of the screw. Proportionality constant A depends on the geometry of screw profile and the friction coefficient between material and die fp, and the material and screw fz. The condition of steep growth pressure requires that the coefficient of friction fp is the largest and coefficient fz the smallest. Coefficient fz can be greatly affected by the quality of screws surface. The aim is to achieve the least surface roughness. Enlargement of the coefficient fp can be achieved by rougher surface treatment of the die or grooves in the screw axis. Using the grooves will not only increase friction, but form closure prevents rotation of material and causes the so-called block axial flow.

3.3 Frictional power of screw

Movement of material in the filled screw profile has connection with friction power. The friction power is transformed into heat. Elementary friction power on the die is given by expression:

a

p

papv

dl

s

iesfppkdP .

sin.cos.1....

= (19)

(ka is coefficient of pressure anisotropy). Elementary friction power on the screw is given by expression:

( )

+

++

=

sin..

sin.cos.1.....

sin..2.2.cos.1....

00

0

r

p

prrssaZ vdl

s

iesfsfppv

dlhfspv

s

iesfspkdP (20)



411

Using those relationships and integration within the limits from 0 to 1 under the above assumptions we get:

( )

l

p

p

ppu

s

iesfpkPp

l

l

p

p

a .

ln

..sin

cos.1...

0

0

0

+

=

(21)

( )

( )( )

+

+

+

+=

p

p

ss

s

l

l

s

iesfpka

hD

Dh

s

ieskp

p

p

ppfsluPz

sin

cos.1....

sin

1..2

sin

sin.

sin

cos.1...

ln

.sin..sin.

0

0

0

0

0

0

(22)

The resulting friction power with filled profile:

PzPpPv += (23)

Torque on the screw:

n

PvMk

..2= (24)

Friction power Pv and torque on the screw Mk are necessary for drive of machines design. 4 Application of the theory of geometry pressing screw design The usefulness of that theory we demonstrate on three variants of the pressing screw geometry (Fig. 7, 8, 9). For comparison simplicity of variants, the same input parameters are chosen for all variants: outer screw diameter (D = 70 mm), inner screw diameter (d = 40 mm), cross-sectional area of thread (S = 325 mm2), width of the screw guide surface (e = 5 mm), number of threads (z = 3).

Figure 7. Screw cross-section - variant A



412

Figure 8. Screw cross-section - variant B

Figure 9. Screw cross-section - variant C

Figure 10 shows that the work pressure is growing slowest in the screw geometry of variant A. Conversely, in the variant C the pressure grows steepest with the number of threads. The thread area is constant in all three variants. Just screw pitch varies depending on changes in the profile. In terms of condition of steep pressure growth is the most advantageous the geometry of variant C. It can not be definitively held that the mentioned thread profile is optimal in all aspects. Certainly it is necessary to verify that assumption also from other aspects - strength, technological and economic.

Figure 10. Dependence of pressure available by screw and number of screw-threads at constant initial

pressure (p0 = 1.05 MPa)



413

5 Conclusion Compaction of biomass is relatively complicated process. Therefore good design of the pressing tool geometry is a fundamental for the success of the technology. The mentioned analysis of the pressing screw geometry can help in the design of new progressive tools that would eliminate the deficiencies of current instruments and allow the increase in biomass briquetting technology and application in a larger share of utilization of solid high-grade biofuel within the renewable energy sources. Acknowledgement This contribution was created by realization of project Development of progressive biomass

compacting technology and production of prototype and high-productive tools (ITMS Project

code: 26240220017), on base of Operational Programme Research and Development support

financing by European Regional Development Fund. References [1] TOMIS, F.: Zklady gumrensk a plastiksk technologie. SNTL Praha, 1975. [2] MAT, M., KRIAN, P., KOVOV, M.: Analza kontruknch parametrov

vplvajcich na vsledn kvalitu vlisku. In Energie z biomasy X.: Proceedings, VT Brno, 2009.

[3] RVAY, J.: Analza a optimalizcia silovch pomerov pri lisovan na nstroji zvitovke. Psomn prca k dizertanej skke, STU Bratislava, 2006.

Current address Milo MAT, M.Sc. Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Institute of Manufacturing Systems, Environmental Technologies and Quality Management, Nam. Slobody 17, 812 31 Bratislava, Slovak Republic. Phone: +421 257 296 573, e-mail: [email protected] Peter KRIAN, M.Sc., Ph.D. Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Institute of Manufacturing Systems, Environmental Technologies and Quality Management, Nam. Slobody 17, 812 31 Bratislava, Slovak Republic. Phone: +421 257 296 537, e-mail: [email protected] Juraj ONDRUKA, M.Sc., Ph.D. Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Institute of Manufacturing Systems, Environmental Technologies and Quality Management, Nam. Slobody 17, 812 31 Bratislava, Slovak Republic. Phone: +421 257 296 560, e-mail: [email protected]



414

ubomr OO, Prof. M.Sc., Ph.D. Slovak University of Technology in Bratislava, Faculty of Mechanical Engineering, Institute of Manufacturing Systems, Environmental Technologies and Quality Management, Nam. Slobody 17, 812 31 Bratislava, Slovak Republic. Phone: +421 257 296 180, e-mail: [email protected]

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