KLASIFIKACIONA [EMA METODA ZA RJE[AVANJE PROBLEMA...

Ma{instvo 3(8), 143 – 152, (2004) M.[abi}: KLASIFIKACIONA [EMA METODA...

KLASIFIKACIONA [EMA METODA ZA RJE[AVANJE PROBLEMA LAYOUTA

Mr Muharem [abi}, dipl. in`. ma{.; Ministry of Defence of Federation of Bosnia and Herzegovina, Hamdije Kre{evljakovi}a 98, Sarajevo

REZIME Raspodjela prostora u cilju pobo j{anja produktivnosti i efikasnosti ve} dugo je va`na u mnogim slu~ajevima kao {to je raspodjela elemenata fabrika, prostornih struktura odràvanja i skladi{ta. Po{to danas postoje mnoge metode i razvijeni softve i javlja se potreba za njihovom klasifikacijom kako bi layout plane i i inìnjeri mog izabrati odgovaraju}u metodu za traènje optimalnog rje{enja za pojedina~ne i vi{estruke prostorne layout probleme. Postoje razli~iti pristupi za generisanje rje{enja layout problema i za velike probleme koje je te{ko rje{avati egzaktnim kompjuterskim softve ima potrebno je razvijati suboptimalna rje{enja. Zato je va`no postaviti krite ije pomo}u kojih se moè mjeri i kvalitet razli~itih rje{enja. Prezenti an je jedan stvarni problem koji predstavlja pojedina~ni layout problem sa normiranim pravougaon m rastojanjima.

l

rr li

rr

t ri

i rril r

r

Klju~ne rije~i: metode problema layouta, klasifikaciona {ema, globalni minimum, optimalno rje{enje

CLASSIFICATION SCHEME OF METHODS FOR RESOLVING FACILITY LAYOUT PROBLEMS

M.Sc. Muharem [abi}, B.Sc. Mech. Eng.; Ministry of Defence of Federation of Bosnia and Herzegovina, Hamdije Kre{evljakovi}a 98, Sarajevo

SUMMARY The importance of space allocation in contributing to productivity and efficiency has long been recognized in many contexts, such as factories offices, maintenance facilities and warehouses. Because of many existing methods and software, there is a need for their classification so that layout planners and engineers can choose the suitable ones in search ng fo the optimal solution of single and multiple floor facility layout problem. The e are various approaches to generate solutions for the facility layout problem and suboptimal solut ons need to be considered for large layout problems since optimal algorithms are computationa ly infeasible. The efore, it is important to set crite ia to measure the quality of various solutions. An example is presented to illustrate weighted rectangular minisum problem.

Keywords: facility layout methods, classification scheme, global minimum, optimal solutions

1. UVOD Lociranje nove pojedina~ne prosotrne jedinice je problem minimizacije funkcije cilja koja naj~e{}e uklju~uje Euklidsko ili pravougaono rastojanje izme|u nove i skupa ve} postoje}ih prosotornih jedinica. Naj~e{}e kori{tena funkcija cilja je ukupno pre|eni put ili ukupni tro{kovi unutra{njeg transporta. Ovi problemi su uobi~ajeni kod rada na problemima layouta ( npr. lociranje nove ma{ine u nekoj postoje}oj radionici ili lociranje novog elementa unutar skladi{ta).

1. INTRODUCTION Locating a single new facility is the problem of minimizing an objective function involving Euclidian or rectilinear distances between the new facility and collection of existing facilities having known planar locations. The most used objective function is total travel distance or total travel cost. These problems occur on regular basis when working on layout problems (e.g. locating a machine in a shop or items inside warehouse).

- 143 -


Problem lociranja vi{e prostornih jedinica u ve} postoje}u strukturu od pojedina~nog razlikuje se u najmanje dva va`na aspekta:

- lociraju se najmanje dvije nove prostorne jedinice i

- tro{kovi transporta izme|u parova novih prosotrnih jedinica su proporcionalni njihovom rastojanju

U stvarnosti preporu~eni model nije dovoljno uzeti i primjeniti na neki layout problem. Postojanje i najrealisti~nijeg modela je od male vrijednosti ako on nije kompjuterski obradljiv. S druge strane ako je model vi{e realniji moè biti skuplji za izvo|enje i obezbije|enje potrebnih podataka. Ponekad je bolje imati jednostavniji model koji je lak{e pribliìti naru~iocu koji èli njegovo kvalitativno razumijevanje koje je koeizistentno sa njegovim iskustvom. Danas postoji veliki broj razvijenih primjenljivih metoda i softvera i njihova klasicikacija u glavne grupe omogu}ava dizajnerima lak{i izbor odgovaraju}e metode za neki problem. Ravanski modeli lociranja principijelno imaju vrijednost zbog toga {to daju razumijevanje problema i jednostavni su za primjenu, vi{e nego {to daju ta~na rje{enja za nominovane probleme. Vi{e sofisticirani modeli omogu}avaju analizu osjetljivosti {to je od velike va`nosti za razumijevanje problema i sticanje povjerenja u model. Kroz rje{avanje realnog problema (lociranje nove ma{ine u radionici sa pet postoje}ih ma{ina) sa pravougaonim rastojanjima kroz matemati~ku analizu pokazano je egzistiranje optimalnih rje{enja.

Multifacility location problems differ from the previous one in two important respects:

- at least two new facilities are being located and

- involved costs are proportional to distances between some pairs of new facilities.

In reality it is not easy just to take proposed model and apply it to a particular layout problem. Having the most realistic model is of little value if the model is computationally intractable. Further, the more realistic the model is, the more expensive it may be to construct and to obtain the data for it. Sometimes it is good to have a simple model so that a builder can explain the model to a client, who may want to have quality insight into it, consistent with the client’s experience. Today, there are huge numbers of applicable methods and softwares and their classification into major groups allow designers proper choice for particular problem. Planar location models are principally of value for the insight they provide, and the simplicity of their construction and use, rather than accuracy with which they represent the problems of interest. More sophisticated models allow sensitivity analysis, which is of major importance in gaining insight into the problem and gaining confidence in the model. By resolving a real problem (locating a new machine in a shop with five existing machines) with rectangular distances by math analysis the existence of optimal solutions has been shown.

2. KLASIFIKACIONA [EMA PROBLEMA LOCIRANJA

Pozicija 1 / Pozicija 2 / Pozicija 3 / Pozicija 4 / Pozicija 5 Pozicija 1: Broj novih prostornih jedinica koje treba locirati Pozicija 2: Prostor lociranja

P ravanska lokacija, nove prostorne jedinice su ta~ke u ravni

D diskretna lokacija, nove prostorne jedinice se nalaze u specificiranim ta~kama

N mre`ne lokacije, nove prostorne jedinice su ta~ke u nekoj mreì

Pozicija 3: Specijalni slu~ajevi problema lokacije

R nove prostorne jedinice su ograni~ene u definisana podru~ja

Wi normirana rastojanja

2. CLASSIFICATION SCHEME FOR LOCATION PROBLEMS

Position 1 / Position 2 / Position 3 / Position 4 / Position 5 Position 1: The number of new facilities to be located Position 2: The space in which facilities are located

P planar location, new facility locations are points in the plane

D discrete location, new facility locations occur at specified points

N network location, new facilities are points on a network

Position 3: Special Cases of Location problems

R new locations are restricted to occur in specific areas

Wi weighted distances

- 144 -


Pozicija 4. Kori{tena funkcija rastojanja L2 Euklidsko rastojanje L1 Pravougaono rastojanje l∞ ^ebi~ijevo rastojanje lp "lp " rastojanje

Pozicija 5: Specificira vrstu funkcije cilja

S - suma funkcija rastojanja tzv. "median" or "minimum" problem max – maksimum funkcije rastojanja tzv. "centralni" ili "minimax" problem

Uop{teni Fermatov problem je: 1 / P / wi / l2 / �

Position 4. the distance function used L2 Euclidean distance L1 Rectangular distance l∞ Tchebyshev distance lp "lp distance"

Position 5: Specifies the type of objective function

S - sum of functions of distance, often called "median" or "minisum" problems max - maximum of functions of distance, called "center" or "minimax" problems

The generalized Fermat problem is: 1 / P / wi / l2 /

3. POJEDINA^NI RAVANSKI PROBLEMI LOCIRANJA

3.1. Problem minimuma sa normiranim pravougaonim rastojanjem: 1/P/wi/l1/ � Date su razli~ite ta~ke Pi=(ai,bi) u ravni, i pozitivne te`ine wi, for i = 1, . . . n. Problem je na}i ta~ke X = (x, y) koje mimimiziraju sumu normiranih rastojanja od X do datih ta~aka. Defini{emo funkciju f(X) sa

( ) ( )∑=

=n

iii PXlwXf

11 , , (1)

gdje je

( ) iii1 byaxP,Xl −+−= , (2)

pravougaono rastojanje od X do Pi. Cilj je minimizacija f(X). Daljim razvojem dobijamo

3. PLANAR SINGLE-FACILITY LOCATION PROBLEMS

3.1 The weighted rectangular distance minisum problem: 1 / P / wi / l1 / � The distinct points Pi=(ai,bi) in the plane, and positive weights wi, for i = 1, . . . n are given. The problem is to find the points X = (x, y) that minimize the weighted sum of the rectangular distances from X to the given points. The function f(X) is defined by

( ) ( )∑=

=n

iii PXlwXf

11 , , (1)

where

( ) iii1 byaxP,Xl −+−= (2)

is the rectangular distance from X to Pi. The problem is to minimize f(X). By further developing of the equation we get

( ) ( ) ∑ ∑∑= ==

−+−=−+−=n

1i

n

1iiiii

n

1iiii bywaxwbyaxwXf . (3)

Defini{u}i By defining

( ) ∑=

−=n

1iii1 axwxf i ( ) ∑

=

−=n

1iii2 bywyf . (4)

Rje{enja X* problema minimuma f(X) mogu se obezbjediti rje{avanjem odvojenih problema: min f1(x) za x* i min f2(y) za y* i uzimaju}i X* = (x*, y*).

Razmotrimo problem min ( ) ∑=

−=n

1iii1 axwxf .

Then solutions X* to the problem min f(X) may be obtained by solving the independent problems: min f1(x) for x* and min f2(y) for y* and taking X*=(x*, y*). Let us consider the problem min

( ) ∑=

−=n

1iii1 axwxf .

- 145 -


Pretpostavimo da skup {ai, i = 1,. . ,n} prvih koordinata postoje}ih prostornih jedinica sadr`i k ≤ n razli~itih vrijednosti. Ozna~avamo k razli~itih vrijednosti kao ar i poredamo ih tako da je a1 < a2 < . . < ak. Defini{emo s(r) = { i : ai = ar } i

. ( )∑∈

=rsi

ir ww

Tada je

( ) ∑∑==

−=−=k

1rrr

n

1iii1 axwaxwxf (5)

Onda je za bilo koji i bilo koji

, f

[ k1 a,ax ∈∗ ][ ]k1 a,ax ∉∗

1(x) > f1(x*) tako da optimalna

rje{enja le`e u intervalu [a1, ak]. Primjer: Pet postoje}ih ma{ina su locirane u nekoj radionici u sljede}e ta~ke: P1 = (3, 5), P2 = (4, 2), P3= (4, 6), P4 = (6, 3), i P5 = (8, 5) (slika 1). Potrebno je locirati jednu novu ma{inu u odnosu na postoje}e. Putevi unutra{njeg transporta postavljeni su du` strukture pravougaonih rastojanja. Broj komada koje traba transportovati izme|u postoje}ih i nove ma{ine je w1 = 2, w2 = 1, w3 = 3, w4 = 3, i w5 = 3. Gdje treba locirati novu ma{inu da put transportnog sredstva bude minimalan?

Let us assume the set {ai, i = 1,. . ,n} of the existing facility first coordinates contains k ≤ n of distinct values. Let us rename the k distinct values as ar and order them so that a1 < a2 < . . < ak. Let us define s(r) = { i : ai = ar } and

( )∑∈

=rsi

ir ww .

Then

( ) ∑∑==

−=−=k

1rrr

n

1iii1 axwaxwxf . (5)

Observe that for any and any [ k1 a,ax ∈∗ ][ ]k1 a,ax ∉∗

, f1(x) > f1(x*) so that all optimal

solutions lie in the interval [a1, ak]. Example: Five existing machines are located in a shop as follows:P1 = (3, 5), P2 = (4, 2), P3= (4, 6), P4 = (6, 3), and P5 = (8, 5) (Figure 1). It is desired to locate one new machine with respect to the existing machines. Travel between machines is along a rectilinear aisle structure. The amount of item movement between the new facility and each existing machine is given as w1 = 2, w2 = 1, w3 = 3, w4 = 3, and w5 = 3. Where should the new facility be located to minimize the travel distance?

y

Slika 1 Centri postoje}ih ma{ina i optimalno mjesto za novu ma{inu :Figure 1 Centroids of existing machines and optimal solution for a new machine :

3 [2]

4 [4]

6 [3] 8 [3]

[1] 2

[3] 3

[5] 5

[3] 6

P3 (4,6) [3]

P5 (8,5) [3] P1 (3,5)

[2]

Optimalno mjesto za novu mašinu - Optimal solution for a new machine

P4 (6,3) [3] P2 (4,2)

[1]

x

- 146 -


Za f1(x), k = 4, a1 = 2 , a2 = 4 , a3 = 6 , a4 = 7 i w1 = 1, w2 = 4, w3 = 3 and w4 = 2. Tako je f1(x) = |x – 2| + 4|x – 4| + 3|x – 6| + 2|x – 7|. Funkcija f1(x) = |x – 2| + 4|x – 4| + 3|x – 6| + 2|x – 7| je pokazana na slici 2.

For f1(x), k = 4, a1 = 2 , a2 = 4 , a3 = 6 , a4 = 7 and w1 = 1, w2 = 4, w3 = 3 and w4 = 2. Thus f1(x) = |x – 2| + 4|x – 4| + 3|x – 6| + 2|x – 7|. The function f1(x) = |x – 2| + 4|x – 4| + 3|x – 6| + 2|x – 7| is shown in Figure 2.

f(x)

50

28 32

20

x 8

[3] 3

[2] 4

[4] 6

[3]

Slika 2 Funkcija cilja za x osu :Figure 2: Objective function for x axis

Za tra`enje minimuma funkcije f1(x) koriste se metode minimizacije. Kriti~ne ta~ke f1(x) su ta~ke gdje je prvi izvod f1(x) jednak nuli , i ta~ke gdje f1 (x) nema izvoda. Funkcija f1(x) je diferencijabilna za sve x ≠ ar , r= 1, . . . , k. Pretpostavimo da je ap < x < ap+1 i p = 1, . . , k–1. Tada se f1(x) mo`e napisati kao:

( ) ( ) ( )∑∑+==

−+−=k

1prrr

p

1rrr1 xawaxwxf , (6)

koja pokazuje de je f1(x) djelimi~no linearna. Prvi izvod f1(x) je dat izrazom

. (7) ( ) ∑∑+==

−=k

1prr

p

1rr1

' wwxf

x je kriti~na ta~ka ako je ap < x < ap+1 i

f'1(x) = 0, t.j. . ∑∑+==

=k

1pii

p

1ii ww

To find a minimum solution to f1(x), we apply standard minimization techniques. The critical points of f1(x) are the points where the derivative of f1(x) is zero, and the points where f1 (x) has no derivative. Observe that the function f1(x) is differentiable for all x ≠ ar , r = 1, . . . , k. Let us assume that ap < x < ap+1 for some p = 1, . . , k–1. Then f1(x) may be written as:

( ) ( ) ( )∑∑+==

−+−=k

1prrr

p

1rrr1 xawaxwxf , (6)

which shows that f1(x) is piecewise linear. The derivative of f1(x) is given by

. (7) ( ) ∑∑+==

−=k

1prr

p

1rr1

' wwxf

Thus x is a critical point if ap < x < ap+1 and

f'1(x) = 0, i.e. ∑∑+==

=k

1pii

p

1ii ww

- 147 -


Po{to je f1(x) konveksna funkcija, a ta~ka x takva da je f'1(x) = 0 to je istovremeno globalni minimum. Vidljivo je da f1(x) nije diferencijabilna u ta~kama ar, r = 1, . . . , k. Stoga su ta~ke ar, r = 1, . . . , k, kriti~ne ta~ke. Kriti~na ta~ka ap je ta~ka minimuma ako je f1(x) nerastu}a, t.j., f'1(x) ≤ 0, za x < ap, i ako je f1(x) neopadaju}a, t.j., f'1(x)≥0, za x > ap. To je slu~aj, ako je

Since f1(x) is convex, a point x such that f'1(x) = 0 is a global minimum. Observe that f1(x) is not differentiable at the points ar, r = 1, . . . , k. Thus the points ar, r = 1, . . . , k, are critical points. The critical point ap is a minimum point if f1(x) is nonincreasing, i.e., f'1(x) ≤ 0, for x < ap, and if f1(x) is nondecreasing, i.e., f'1(x)≥0, for x > ap. That is, if

( ) ∑∑=

−

=

≤−=k

prr

1p

1rr1

' 0wwxf i . (8) ( ) ∑∑+==

≥−=k

1prr

p

1rr1

' 0wwxf

Koristi se konvencija da je ako je

p=1, i ako je p=k.

∑−

=

=1p

1rr 0w

∑+=

=k

1prr 0w

Zaklju~uje se, ako je za neke ta~ke p = 1, . ., k,

∑ ∑−

= =

<−1p

1r

k

prrr 0ww i

, (9) ∑ ∑= +=

>−p

1r

k

1prrr 0ww

onda je ap globalni minimum, ili

, tada su ta~ke x takve da

su a

∑ ∑= +=

=−p

1r

k

1prrr 0ww

p≤ x ≤ ap+1 ta~ke minimuma. Za f1(x) u primjeru, izaberemo p = 2, i vidimo da

je . ∑ ∑= +=

=−=−p

1r

k

1prrr 055ww

Tako slijedi da je svako x iz intervala 4 ≤ x ≤ 6 optimalno rje{enje. Isti pristup se koristi I za min

( ) ∑=

−=n

1iii2 bywyf .

Pretpostavimo da skup {bi, i = 1,. . ,n} drugih koordinata postoje}ih prostornih jedinica sadr`i l ≤ n razli~itih vrijednosti. Ozna~avamo l razli~itih vrijednosti kao bj i poredamo ih tako da je b1 < b2 < . .< bl.

Defini{emo s(j) = { i : bi = bj } i . ( )∑∈

=lsi

ll ww

Tada je

Here we use the convention that if

p=1, and if p=k.

∑−

=

=1p

1rr 0w

∑+=

=k

1prr 0w

In summary, if for some p = 1, . . . , k,

∑ ∑−

= =

<−1p

1r

k

prrr 0ww and

∑ ∑ (9) = +=

>−p

1r

k

1prrr 0ww

then ap is a unique minimum point, or

, then all x are such that

a

∑ ∑= +=

=−p

1r

k

1prrr 0ww

p≤ x ≤ ap+1 are minimum points. For f1(x) in the expample, choose p = 2, and

observe that . ∑ ∑= +=

=−=−p

1r

k

1prrr 055ww

Thus each x from the interval 4 ≤ x ≤ 6 is optimal solution. The same approach is used for

the subproblem min ( ) ∑=

−=n

1iii2 bywyf .

Let us assume that the set {bi, i = 1,. . ,n} of existing facility second coordinates contains l ≤ n distinct values. We rename the l distinct values as bj and order them so that b1 < b2 < . . < bl . Let us define s(j) = { i : bi = bj } and define

( )∑∈

=lsi

ll ww .

Then

( ) ∑∑==

−=−=l

1jll

n

1iii2 bxwbxwyf . (10)

- 148 -


Za f2(y), l = 4, b1 = 1, b2 = 3, b3 = 5, b4 = 7 i w1 = 1, w2 = 3, w3 = 3 i w4 = 3. Tada je f2(y) = |y – 1| + 3|y – 3| + 3|y – 5| + 3|y – 7|. Funkcija f2(y) = |y – 1| + 3|y – 3| + 3|y – 5| + 3|y – 7| je pokazana na slici 3.

For f2(y), l = 4, b1 = 1, b2 = 3, b3 = 5, b4 = 7 and w1 = 1, w2 = 3, w3 = 3 and w4 = 3. Thus f2(y) = |y – 1| + 3|y – 3| + 3|y – 5| + 3|y – 7|. The function f2(y) = |y – 1| + 3|y – 3| + 3|y – 5| + 3|y – 7| is shown in Figure 3.

Slika 3 Funkcija cilja za y osu :Figure 3: Objective function for y axis

Uslovi optimalnog rje{enja su dati kako slijedi. Za p = 1, . . . , l, ako je

∑ ∑−

= =

<−1p

1j

l

pjll 0ww i (11) ∑ ∑

= +=

>−p

j

l

pjll ww

1 10

Tada je bp jedinstvena ta~ka minimuma. Ako je

∑ ∑= +=

=−p

1j

l

1pjll 0ww (12)

Onda je svako y takvo da je b≤ y≤ bp+1 ta~ka minimuma. Za f2(y) iz primjera, izaberemo p = 3, i vidimo da je

∑ ∑−

= =

<−=−1p

1j

l

pjll 064ww i

. ∑ ∑= +=

>−=−p

1j

l

1pjll 037ww

Tako slijedi da je y = b3 = 5 je globalni minimum i optimalno rje{enje. Skup optimalnih rje{enja za minimum f(X) je { X = (x, y) : x ∈ [4, 6] i y = 5 } i pokazan je boldiranom linijom na slici 1.

The optimality conditions are given as follows. For p = 1, . . . , l: If

∑ ∑−

= =

<−1p

1j

l

pjll 0ww and ∑ ∑ , (11)

= +=

>−p

j

l

pjll ww

1 10

then bp is a unique minimum point. If

∑ ∑= +=

=−p

1j

l

1pjll 0ww , (12)

then all y such that b≤ y≤ bp+1 are minimum points. For f2(y) in the example problem, choose p = 3, and observe that

∑ ∑−

= =

<−=−1p

1j

l

pjll 064ww and

. ∑ ∑= +=

>−=−p

1j

l

1pjll 037ww

Thus y = b3 = 5 is a unique optimal solution. Thus the set of optimal solutions to min f(X) is { X = (x, y) : x ∈ [4, 6] and y = 5 } and is represented by the bold line in Figure 1.

6 [3]

40

30

20

10

f(y)

y 3

[3] 2

[1] 5

[5]

- 149 -


3.2. Problem minimuma sa normiranim Euklidskim rastojanjem: 1/P/wi/l22/

Date su razli~ite ta~ke Pi = (ai,bi) u ravni, i pozitivne teìne wi, for i = 1, . . . n. Problem je traènje ta~ke X = (x, y) koja minimizira normiranu sumu kvadratnih Euklidskih rastojanja od X do datih ta~aka. Defini{emo funkciju f(X) sa

, (13) ( ) ( )[∑=

=n

1iii P,XdwXf ]

gdje je

( )[ ] ( ) ( )212

11 byaxP,Xd −+−= . (14)

Problem je minimizacija funkcije f(X). Ovaj problem ima zatvorenu formu rje{enja koja proizilazi iz uslova optimalnosti prvog reda.

( )∑=

=−=∂∂ n

1iii 0axw2

xf

i

( )∑=

=−=∂∂ n

1iii 0byw2

yf

(15)

Rje{avanjem x i y dobijamo:

∑

∑

=

== n

1ii

n

1iii

w

awx i

∑

∑

=

== n

1ii

n

1iii

w

bwy (16)

3.3. Pojedina~ni problem medijane sa êbi~ijevim rastojanjima: 1/P/wi/l/ � Date su razli~ite ta~ke Pi = (ai,bi) u ravni, i pozitivne teìne wi, for i = 1, . . . n. Problem je traènje ta~ke X = (x, y) koja minimizira normiranu sumu êbi~ijevih rastojanja od X do datih ta~aka. Defini{emo funkciju f(X) sa

( ) ( )∑=

∞=n

iii PXlwXf

1

, , (17)

gdje je

( ) }{ iii byaxPXl −−=∞ ,max, . (18)

Problem je minimizacija funkcije f(X). Prilaz ovom problemu je transformacija êbi~ijevih rastojanja u pravougaona i onda rej{avanje u tom obliku.

3.2. The weighted, squared Euclidean distance minisum problem: 1/P/wi/l22/ Given distinct points Pi = (ai,bi) in the plane, and positive weights wi, for i = 1, . . . n. The problem is to find a point X = (x, y) that minimizes the weighted sum of the squared Euclidean distances from X to the given points. Let us define the function f(X) by

, (13) ( ) ( )[∑=

=n

iii PXdwXf

1

, ]where

( )[ ] ( ) ( )212

11 byaxP,Xd −+−= . (14)

The problem is to minimize f(X). This problem has a closed form solution that is obtained from the the first order conditions for optimality.

( )∑=

=−=∂∂ n

1iii 0axw2

xf

and

( )∑=

=−=∂∂ n

1iii 0byw2

yf

(15)

Solving for x and y we get:

∑

∑

=

== n

1ii

n

1iii

w

awx and

∑

∑

=

== n

1ii

n

1iii

w

bwy (16)

3.3 The one facility median problem with Tschebychev distance: 1/P/wi/l/ � The distinct points Pi = (ai, bi) in the plane, and positive weights wi, for i = 1, . . . n are given. The problem is to find a point X = (x, y) that minimizes the weighted sum of the Tschebychev distances from X to the given points. Let us define the function f(X) by

( ) ( )∑=

∞=n

iii PXlwXf

1, , (17)

where

( ) }{ iii byaxPXl −−=∞ ,max, . (18)

The problem is to minimize f(X). The approach to this problem is to transform the Tschebychev distance into the rectilinear distance.

- 150 -


3.4. Pojedina~ni problem minimuma sa “lp” rastojanjima: 1 / P / wi / lp / � Date su razli~ite ta~ke Pi = (ai,bi) u ravni, i pozitivne teìne wi, za i = 1, . . . n. Problem je traènje ta~ke X = (x, y) koja minimizira normiranu sumu lp rastojanja od X do datih ta~aka. Defini{emo funkciju f(X) sa

( ) ( )∑=

=n

iipi PXlwXf

1, , (19)

gdje je

( ) [ ] ppi

piip byaxPXl

1, −+−= (20)

Problem je minimizacija funkcije f(X). Ako izaberemo p=2, lp je kvadratno Euklidsko rastojanje, i za p=1 l1 je pravougaono rastojanje. Tako|e,

daje êbi~ijevo rastojanje. ∞∞→ = ll pplim

3.4. The one facility minisum problem with lp distance: 1 / P / wi / lp / � Given distinct points Pi = (ai, bi) in the plane, and positive weights wi, for i = 1, . . . n. The problem is to find a point X = (x, y) that minimizes the weighted sum of the lp distances from X to the given points. Define the function f(X) by

( ) ( )∑=

=n

iipi PXlwXf

1, , (19)

where

( ) [ ] ppi

piip byaxPXl

1, −+−= . (20)

The problem is to minimize f(X). Observe that for p=2, lp is the Euclidean distance, and for p=1 l1

is rectengular distance. Also, ∞∞→ = ll pplim

gives the Tschebbychev distance.

4. VI[ESTRUKE LOKACIJE SA PRAVOUGAONIM RASTOJANJIMA: M / P / wi / l1 / � Problem je lociranje nekoliko novih prostornih jedinica u odnosu na dati skup postoje}ih prostornih jedinica istovremeno uvaàvaju}i njihov me|usobni odnos, tako da se minimizira ukupno normirano rastojanje izme|u njih. Izaberemo Pi = (ai, bi) i = 1, . . , n ta~aka u prostoru Rn. Neka Xj = (xj, yj), j = 1, . . , m ozna~avaju m novih prostornih jedinica koje treba locirati. Neka su wji nenegativne vrijednosti teìna vezanih sa rastojanjima svakog Xj i Pi za i = 1, . . , n i j = 1, . . , m. Neka su vjk nenegativne vrijednosti teìna vezanih sa rastojanjima svakog Xj i Xk za 1≤ j< k≤ m. Tada se problem lociranja vi{estrukih prostornih jedinica sa pravougaonim rastojanjima moè napisati u obliku:

4. MULTIFACILITY LOCATION WITH RECTANGULAR DISTANCE: M / P / wi / l1 / �

The problem is to locate several new facilities with respect to a given set of existing facilities and with respect to other new facilities, so as to minimize the total weighted distance between pairs of new facilities and between pairs of new and the existing facilities. Let Pi = (ai, bi) i = 1, . . , n be given points in Rn. Let Xj = (xj, yj), j = 1, . . , m denote the m new facilities to be located. Let wji be a nonnegative weight associated with the distance between each Xj and Pi for i = 1, . . , n and j = 1, . . , m. Let vjk be a nonnegative weight associated with the distance between each Xj and Xk for 1≤ j< k≤ m. Then the multifacility medium location problem with rectangular distance can be stated as:

( ) ( ) ( )∑ ∑∑≤<≤ ==

−+−+−+−=mkj

m

jijijji

n

ikjkjjk byaxwyyxxvXf

1 11

. (21)

Tada svaka od m novih prostornih jedinica koja treba biti locirana u odnosu na n postoje}ih i tako|e u odnosu jedne na drugu. Lociranje neke Xj moè zavisiti od lokacije od nekih Xk zbog njihovog odnosa definisanog kroz vjk. Nove prostone jedinice Xj i Xk su povezane ako je vjk je pozitivna vrijednost i nisu povezane ako je vrijednost vjk jednaka nuli. Pretpostavlja se da je svaka nova prostorna lokacija Xj povezana sa najmanje jednom novom, u suprotnom lokacija Xj moè se razmatrati nezavisno kao zaseban problem.

Thus each of the m new facilities is to be located with respect to the n existing facilities and also with respect to the other new facilities. The location of Xj may depend on the location of some point Xk because of the terms involving vjk. New facility locations Xj and Xk are said to be linked if vjk is positive and not linked if vjk is zero. It is assumed that each new facility location Xj is linked with at least one other new facility location, otherwise the location of Xj could be determined independently of the other new facility locations by considering a separate problem.

- 151 -


Nove prostone jedinice Xj i postoje}e Xk su povezane ako je wji je pozitivna vrijednost i nisu povezane ako je vrijednost wji jednaka nuli. Ako neka nova prostorna jedinica Xj nije povezana sa bilo kojom postoje}om, tada mora biti povezana sa nekom novom koja je povezana sa nekom od postoje}ih prostornih jedinica. U suprotnom, skup svih novih prostornih jedinica koje nisu povezane sa bilo kojom od postoje}ih moè se locirati u zajedni~ku ta~ku bilo gdje. Nadalje, pretpostavlja se da je vi{estruki problem lociranja dobro formulisan ako su prostorne jedinice povezane. Ove pretpostavke impliciraju da postoji optimalno rje{enje. Pretpostavlja se da su sve vridnosti wji i vjk pozitivne. Problem vi{estrukog lociranja sa pravougaonim rastojanjima moè se razdvojiti na zasebne subprobleme sa varijablama xj i yj. Ti subproblemi su:

New facility location Xj and the existing facility location Pi are said to be linked if wji is positive and not linked if wji is zero. If a new facility Xj is not linked to any existing facility, then it must be linked to some new facility that is linked to some existing facility. Otherwise, the set of all new facilities that are not linked to any existing facility can be located at a common point anywhere. Henceforth, we assume the multifacility location problem is well formulated with respect to facilities being linked to one another. These assumptions imply that an optimal solution exists. For convenience of presentation, we assume all the wji and all the vjk are positive. The multifacility medium location problem with rectangular distance can be separated into independent subproblems in the variables xj and in the variables yj. These subproblems are:

( ) ( ) ( )∑ ∑∑≤<≤ ==

−+−=mkj

m

jijji

n

ikjjk axwxxvXf

1 111 (22)

( ) ( ) ( )∑ ∑∑≤<≤ ==

−+−=mkj

m

jijji

n

ikjjk bywyyvYf

1 112 (23)

5. ZAKLJUÂK Dobro izabran layout koji odgovara proizvodnoj filozofiji je fundamentalna polazna ta~ka za neki proizvodni sistem. Sa klasificiranim softverskim alatima i metodama koje su dostupne danas, layout dizajn moè se kreirati znatno brè sa u{tedama u tokovima materijala i kra}im vremenima proizvodnje. Dobijanje optimalnog layouta nekog problema podrazumijeva da je za njegovo rje{avanje izabrana odgovaraju}a metoda i da je ona dobro adaptabilna za isti problem.

5. CONCLUSION A good layout well suited to the manufacturing philosophy is the fundamental starting point for total production system design. With classified software tools and methods now available, layout designs can be created in much less time with greater reductions in material flow and shorter production times. Achievement of the optimal layout of a problem means that suitable method is chosen and its adaptability to particular layout problem is high.

6. LITERATURA – REFERENCES [1] Bozer Y.A., Meller R.D., Erlebacher S.J.: “An

Improvement-type Layout Algorithm for Single and Multiple-floor Facilities”, Management Science 40 (1994), 918-932;

[2] Chien-Wen C., D.Y. Sha: “A new approach to

the multiple objective facility layout problem”, Integrated Manufacturing Systems 12/1 (2001), 59-66;

[4] Sly D, Grajo E, Montrenil B.: “Layout Design and Analysis Software”, Industrial Engineering Solutions 28 (1996), 18-25.

[3] Kochar J.S, Heragu S.S: “Facility Layout in a Changing Enviroment”, International Journal of Production Research 37 (1999), 2429-2446;

- 152 -

Ma{instvo 3(8), 153 – 166, (2004) S.Bali},...: PRORAÛN I DIZAJN AUTOMOBILSKIH...

PRORAÛN I DIZAJN AUTOMOBILSKIH TURBOPUMPI ZA VODU PRIMJENOM NUMERI^KIH ANALIZA

CIJELOG PROTO^NOG TRAKTA

Doc. dr. Senad Bali}, dipl. ing., Ma{inski fakultet u Zenici, Univerzitet u Zenici, Zenica, Bosna i Hercegovina Prof. dr. Joè Duhovnik, dipl. ing., Fakulteta za strojni{tvo, Univerza v Ljubljani , Ljubljana, Slovenia

REZIME U radu je definisana metoda prora~una i dizajna centrifugalnih turbopumpi u cirkulacionim sistemima za hla|enje automobilskih motora. Bazirana je na primjeni CAD tehnologija za oblikovanje elemenata turbopumpi i za generiranje numeri~kih mreà, te CFD tehnologija (baziranih na metodi kona~nih volumena) za nume i~ke analize strujanja radnog fluida u elementima ovih pumpi. Namijenjena je za inènjerske i razvojne poslove prora~una i dizajna pomenutih centrifugalnih turbopumpi u cirkulacionim sistemima hla|enja automobilskih motora, u fazi prije ispitivanja prototipa na opitnim postrojenjima.

r

r

r

tr i

t

Kod nume i~kih prora~una prvi put je na modelima automobilskih turbopumpi primijenjen egzaktan postupak prora~una, s rotiranjem radnog kola, kori{tenjem pokretnih mreà. Finalni prora~uni realizovani su na modelu cijelog proto~nog trakta izabrane automobilske turbopumpe. Metoda je prakti~no primijenjena i potv |ena rezultatima eksperimentalnih ispitivanja izabranih modela automobilskih turbopumpi.

Klju~ne rije~i: automobilska turbopumpa za vodu, numeri~ka simulacija toka, metod kona~nih volumena, pokretne mreè.

DESIGN OF AUTOMOTIVE WATER TURBOPUMPS BY NUMERICAL ANALYSIS OF ENTIRE FLOWING TRACT

Ass. Prof. Dr. Senad Bali}, B. Sc. Mech. Eng, Faculty of Mechanical Engineering in Zenica, University of Zenica, Zenica, Bosnia and Herzegovina Prof. Dr. Joè Duhovnik, B. Sc. Mech. Eng, Faculty of Mechanical Engineering, University of Ljubljana, Ljubljana, Slovenia

SUMMARY Method of calculation and water turbopump design in the automotive engine circulation cooling system is defined in this work. It is based on the CAD technologys applica ion for turbopump elements forming and for nume ical grid generation, as well as on CFD technology (based on f nite volume method) for numerical analyses of the fluid circulation in the elements of these pumps. It is intended for engineering and developing works of calculation and water turbopump design in the automotive engine circulation cooling system, in the phase before examina ion of the prototypes on the experimental equipments. In the numerical calculations, exact calculation has been applied for the first time on the automotive turbopumps, with rotation of impeller, using of moving grids. The final calculations are realized on the whole flowing tract model of chosen automotive turbopump. Method was practically applied and confirmed by results of experiments on chosen models of automotive turbopumps.

Key words: automotive water turbopump, numerical flow simulation, finite volume method, moving grids.

- 153 -


1. UVOD Kod centrifugalnih turbopumpi, ili op}enito kod turboma{ina, radi se o sloènim fenomenima strujanja i interakcije fluida i radnih elemenata pumpi, s velikim brojem uticajnih faktora razli~itog karaktera i ~itavim spletom njihovih me|uzavisnih djelovanja. Stalno je prisutno nastojanje da se, u svjetlu mogu}nosti savremenih metoda dizajna, prora~una i simulacija podrànih ra~unarom (CAD, CFD, FVM, FEM, ...), defini{u efikasni postupci kojima se cijelom problemu dizajna turboma{ina moè pri}i uz primjenu efikasnih metoda virtualnih analiza, s ciljem da se omogu}i ostvarenje upotrebljivih rje{enja za {to kra}e vrijeme i uz najniè tro{kove. Pri tome se od ovih metoda o~ekuje da odstupanja dobivenih rezulatata od realnih stanja ne}e prelaziti nivo prihvatljive gre{ke. Unato~ pomenutim nastojanjima, uobi~ajena je praksa da se CFD analize strujanja fluida u centrifugalnim pumpama i kompresorima provode primjenom pribli`nih postupaka prora~una, u kojima se ima nedopustivo visok stepen razli~itih aproksimacija i pojednostavljenja. To je ili postupak prora~una s rotiranjem koordinatnih sistema (The Multiple Rotating Reference Frames (MRF) model) [8], ili se prora~un provodi na modelu jednog sprovodnog me|ulopati~nog kanala (ili jednog segmenta toka od ulaza do izlaza pumpe), uz odgovaraju}e dodavanje parametara rotiraju}eg kretanja radnog kola [3, 6, 9, 11]. U ovom radu je pokazano da se CFD analize strujanja u turbopumpi mogu i trebaju provoditi tako da se obuhvati cijeli tok fluida kroz istu, i to uz primjenu egzaktnog postupka prora~una, s rotiranjem radnog kola. Na ovaj na~in se u najve}oj mogu}oj mjeri pribliàva realnom strujanju u turbopumpi. Ovaj postupak prora~una na punom modelu toka fluida jedne turbopumpe posebno je vaàn u finalnim analizama kod dizajna istih [7], kada se, pored sagledavanja strujanja fluida, mogu egzaktno izra~unati sve radne karakteristike modela turbopumpe: protok Q, napor H, op}i napor ∆p, stepen iskori{tenja η, odrediti rezultuju}a radijalna sila Fr na fiktivnom radnom kolu, te konstruisati odgovaraju}e Q-∆p krive. Kao opitni model, na osnovu kojeg su generirani virtualni modeli i realizovani numeri~ki prora~uni, izabrana je jedna automobilska turbopumpa s otvorenim radijalnim radnim kolom s cilindri~nim lopaticama. Ova pumpa se ugra|uje u sistem hla|enja motora nekih tipova teretnih vozila poznatog svjetskog proizvo|a~a automobila. Na slici 1. prikazan je meridijalni presjek, a na slici 2. crte` radnog kola izabrane pumpe.

1. INTRODUCTION Centrifugal turbopumps, or generally turbomachines, have complex phenomena of flow and interaction between fluid and pump working elements, with a great number of influence factors with different character and number of their mutual dependent influences. In respect to modern possibilities of computer aided design, calculations and simulation (CAD, CFD, FVM, FEM, …) there are constant attempts to define effective procedures by which it would be possible to look at the whole problem of turbo machines design by application of effective methods of virtual analysis, with aim to provide useful solutions as fast and as cheap as possible. Besides that, it is expected from the methods that deviations between obtain results and real states will not excide the level of acceptable error.

Despite the mentioned attempts, it’s a common practice to perform CFD (Computational Fluid Dynamics) analysis of fluid flow in the centrifugal pumps and compressors using approximate calculations procedures with unacceptable high level of different approximations and simplifications. It is either procedure to perform the calculation with rotation of coordinate systems (The multiple Rotating Reference Frames (MRF) model) [8], or the calculation is being performed on the model with one internal flow impellers channel (or on one segment of entire flow), with the appropriate addition of impeller’s rotational movement parameters [3, 6, 9, 11]. This paper presents that CFD analysis of flow in turbopump can and should be performed in such manner to take into consideration entire flow of fluid trough it, and that it should be done using exact calculation, with rotation of impeller. This procedure gives closest approximation of real state in turbopump. This procedure of calculation, on full model of fluid flow in one turbopump, has special importance in their final design analysis [7], when, beside getting overview of fluid flow, all work characteristics of turbopump model can be exactly calculated: flow Q, effort H, general effort ∆p, efficiency factor η, resulting radial force Fr and torque Mz on fictive impeller, and also appropriate Q-∆p curves can be constructed. As an experimental model, based on which virtual models were generated and numerical calculations realized, an automotive turbo pump with an open radial impeller and cylindrical vanes was chosen. The pump is built into motor cooling systems of some types of cargo vehicles from a well-known automotive manufacturer. The meridian section is shown in Figure 1, and a drawing of the impeller from the chosen pump is shown in Figure 2.

-154 -


2. MATEMATI^KI MODEL I NUMERI^KI METOD

Za realizaciju CFD analiza u ovom radu je primijenjen softver Comet (CD adapco Group). U [4] je predsta-vljen matematski model transportnih procesa koji mogu biti simulirani primjenom pomenutog softvera. Ovaj model obuhvata jedna~ine bilansa mase, momenta, energije i entropije u integralnoj formi, jedna~inu o~uvanja prostora, koja mora biti zadovoljena kod rje{avanja problema s pokretnom mre`om, konstitutivne relacije potrebne za formiranje zatvorenog sistema jedna~ina, modele turbulentnog toka i grani~ne uvjete. Elementi numeri~ke metode: principi diskretizacije, formiranje sistema algebarskih jedna~ina, algoritmi rje{avanja, implementacija grani~nih uvjeta su tako|e dati u [4].

2.1 Osnovne jedna~ine Tok radnog fluida u me|ulopati~nim kanalima turbopumpe se posmatra kao visozan i turbulentan. Potpuno je opisan s dvije vremenski usrednjene jedna~ine bilansa: jedna~inom kontinuiteta i jedna~inom momenta, koje su dopunjene s jedna~inama modela turbulentnog toka.

Slika 1. Meridijalni presjek izabrane automobilske turbopumpe

1. kolektor-uvodnik pumpe, 2. ulaz u radno kolo, 3. međulopatični kanal radnog kola

Figure 1. Meridian section of the chosen autonotive turbopump

1. pump collector-feeder, 2. inlet port, 3. internal flow impeller channel

1

2

3

Slika 2. Oblik lopatica radijalnog radnog kola analizirane turbopumpe (D1m = 55,5 mm, D2 = 107 mm, β1 = 36°, β2 = 15°) Figure 2. Radial impeller vane shape of the analyzed turbopump

2. MATHEMATICAL MODEL AND NUMERICAL METHOD

In [4] the mathematical model of transport processes that can be simulated with software Comet is presented. It includes the mass, momentum and energy balance equations in integral form, a space conservation law, which has to be satisfied if the problem is solved using a moving grid, constitutive relations required for the problem closure, models of turbulence in fluid flow, and boundary conditions. Elements of the numerical method: diskretization principles, derivation of algebraic equation systems, solution procedure, implementation of boundary conditions are also given in [4]. 2.1 Governing equations The water flow in a turbopump flow passage is considered viscous and turbulent. It is fully described by the time-averaged equations of continuity and momentum conservation, which are accompanied by the turbulence model equations.

- 155 -


Jedna~ine su date za kontrolni volumen KV, ograni~en povr{inom S (koja se moè i kretati), u integralnoj formi [4]. Jedna~ina kontiniteta je:

,0)( dddd

s =⋅−+ ∫∫SV

ρVρt

svv (1)

gdje je ρ gustina kontinuuma, v brzina fluida, vs brzina pokretne numeri~ke mreè, a s je spoljnji vektor povr{ine u proizvoljnoj ta~ki iste. Jedna~ina momenta je:

Equations are given for the control volume CV bounded by (possibly moving) surface S in the integral form [4]. The continuity equation is:

,0)( dddd

s =⋅−+ ∫∫SV

ρVρt

svv (1)

where ρ is the density of continuum, v is the fluid velocity, vs is the computational grid velocity, and s is the outward pointing surface vector. The momentum equation is:

,d dd)(ddd

bs ∫∫∫∫ +⋅=⋅−+VSSV

VVt

fsTsvvvv ρρ (2)

gdje je T Cauchyiev tenzor napona i fb rezultuju}a zapreminska sila po jedinici volumena. Kako je kod CFD analiza modela turbopumpi prisutna i pokretna mreà, to se gornjim jedna~inama dodaje i jedna~ina o~uvanja prostora:

.0dddd

=⋅− ∫∫S

sV

Vt

sv (3)

koja povezuje brzinu promjene zapremine V i brzinu vs. Zbog toga {to je strujanje fluida u turbopumpi turbulentno, nije mogu}e rje{avanje jedna~ina (1) i (2) za skalu s kratkim vremenskim intervalima. Umjesto toga, primijenjeno je vremensko usrednjavanje varijabli toka, koje zamjenjuje

varijablu φ sa zbirom njene srednje vrijednosti φ

i fluktuiraju}e komponente φ':

).,(')(),( tt rrr φφφ += (4)

Primjena takvog usrednjavanja veli~ina u jedna~ini (2) daje Reynoldsove usrednjene Navie -Stokesove (RANS) jedna~ine:

r

where T is the Cauchy stress tensor, and fb is the resultant body force per unit volume. Since turbopump CFD calculations involve moving grid, the equation of space conservation must be solved:

.0dddd

=⋅− ∫∫S

sV

Vt

sv (3)

which links the rate of change of volume V and velocity vs. Because the flow in the turbopump is turbulent, it is difficult to resolve the equations (1) and (2) on a small time scale. Instead, time averaging of flow variables is used, which replaces the flow variable

φ by its mean value φ and fluctuation φ':

).,(')(),( tt rrr φφφ += (4)

Using such averaged quantities in equation (2) results in Reynolds averaged Navier-Stokes (RANS) equations:

.dd)(d)(ddd

∫∫∫∫ +⋅′′−=⋅−+V

bSS

sV

VVt

fsvvTsvvvv ρρρ (5)

RANS jedna~ine sadrè ~lan '' vvρ , poznat kao

komponenta tzv. tenzora Reynoldsovih napona. Ovaj ~lan zahtijeva poseban tretman, po{to ne moè biti izraèn u ~lanovima osnovnih varijabli toka. Isti moè biti izra~unat primjenom nekog od modela turbulencije, koji predvi|a odgovaraju}a inènjerska pojednostavljenja, odnosno prilago|avanja. U ovom slu~aju kori{ten je standardni k-ε model [4], koji je poznat i testiran i najvi{e ra{iren u inènjerskoj praksi. Time se uvode dvije dodatne parcijalne diferencijalne jedna~ine, koje zatvaraju sistem RANS jedna~ina [4]:

The RANS equations contain term '' vvρ , known

as Reynolds stresses. This term needs special treatment, since it may not be expressed in terms of basic flow variables. It may be calculated by using a turbulence model, which relays on adequate engineering assumptions. The standard k-ε model [4] was used in our case, which is well known and tested, and widely used in engineering practice. It introduces two additional partial differential equations, which close the system of RANS equations [4]:

,d)(dgrad)(d)(ddd

∫∫∫∫ −++⋅+=⋅−+V

BSS

sV

VPPkkVkt k

t ρεσµ

µρρ ssvv (6)

- 156 -


,ddivCCC

dgrad)(d)(ddd

3

2

21∫

∫∫∫

⎟⎟⎠

⎞⎜⎜⎝

⎛−−+

+⋅+=⋅−+

V

SSs

V

Vkk

P

Vt

t

v

ssvv

ρεερε

εσµ

µερερε

(7)

gdje je k kineti~ka energija turbulencije, ε je njena brzina disipacije, a µt turbulentni viskozitet.

Produkcija turbulentne kineti~ke energije usljed smicanja P je modelirana kao:

.div)div(32:2 vvDD kP tt ρµµ +−= && (8)

Veli~ine: C1, C2, C3, σk i σε su empirijski

koeficijenti.

where k is the kinetic energy of turbulence, ε is its dissipation rate and µt is the turbulent viscosity.

The production of turbulent kinetic energy by shear P is modeled as:

.div)div(32:2 vvDD kP tt ρµµ +−= && (8)

The quantities: C1, C2, C3, σk and σε are

empirical coefficients.

2.2. Grani~ni uvjeti U prikazanom radu, date jedna~ine (1, 2 i 3) su rje-{avane za specijalni slu~aj strujanja u punom modelu cijelog proto~nog trakta automobilske turbopumpe. Domena izra~unavanja, koja obuhvata proto~ne kanale kroz pumpu, ograni~ena je ve}im dijelom zidovima, gdje su primijenjeni grani~ni uslovi bez klizanja. Brzina strujanja fluida u blizini zida aproksimirana je primjenom zidne funkcije [4], kojom se profil brzine u ovoj zoni predstavlja logaritamskom funkcijom. Neki od zidova (kontaktne povr{ine izme|u impelera i fluida – boundary regions [4]) se kre}u (rotiraju) tokom tranzientnog strujanja odre|enom brzinom. Ulaz u model pumpe je kanal 2, prikazan na slici 3, dok izlaz iz modela predstavlja izlaz iz spirale, prikazan na slici 7. Na ulazu i na izlazu modela specificirane su odgovaraju}e vrijednosti stati~kog pritiska.

2.2 Boundary conditions In the present study, the given equations (1, 2, and 3) are solved for a special case of flow in full model of the automotive turbopump's entire flowing tract. The pump flow passages calculation domain is bounded mainly by walls, where no-slip boundary condition was applied. The fluid velocity in the vicinity of the wall was approximated by using wall function [4], which assume logarithmic region in the velocity profile. Some of the walls (contact surfaces of impeller and fluid – boundary regions [4]) move (rotate) during the transient flow, with noted velocity. Model pump's inlet is the passage 2 shown in Figure 3, and the model's outlet is the spiral's passage shown in Figure 7. At the inlet and at the outlet the total and static pressure were prescribed, respectively.

3. MODEL CIJELOG PROTO^NOG TRAKTA TURBOPUMPE

Kod dizajna modela centrifugalnih turbopumpi moraju se ispuniti odre|eni preduvjeti, koji u prvom redu proisti~u iz karaktera primijenjene numeri~ke metode. Jedan od osnovnih preduvjeta je da se kod modeliranja u startu mora definirati zajedni~ka geometrijska osnova (na bazi rezultata konvencionalnih prora~una i preliminarnog dizajna), da bi se kasnije, prilikom kreiranja numeri~kih modela, svi dijelovi analiziranog domena mogli povezati u cjelinu. Na bazi ovako dobivene geometrijske osnove, potrebno je aktivnosti kreiranja geometrijskih modelâ i njihove podjele na sekcije i segmente, te generiranja mre`a, provoditi uskla|eno i simultano za sve dijelove turbopumpe, odnosno analiziranog domena.

3. MODEL OF TURBOPUMP'S ENTIRE FLOWING TRACT

In the design of turbopumps models certain prerequisites, mainly resulting from the character of applied numerical method, have to be met. One of the main prerequisites is that common geometric base (based on the results from conventional calculations and preliminary design) has to be defined from the moment of the start of modeling, so that later, during the creation of numerical model, all the parts of an analyzed domain could be linked together. Based on the obtained geometrical base, it is necessary that all the activity of creating the geometric models and their division into sections and segments are coordinated and simultaneous for all the parts of the turbopump, i.e. the analyzed domain.

- 157 -


Jedna od specifi~nosti automobilskih turbopumpi je postojanje dva ulazna otvora. Kroz prvi od njih, u periodu normalnog rada zagrijanog motora, uvodi se pothla|ena voda iz hladnjaka, dok je drugi u funkciji u periodu zagrijavanja motora, kada se voda s temperaturom niòm od radne usmjerava pored hladnjaka direktno u pumpu. Kod kamionskih turbopumpi se primjenjuju dvije varijante rje{enja za obezbje|enje dva ulazna otvora. Kod prve varijante (primijenjene kod analizirane pumpe), na vratilu pumpe ugra|en je ventilator (u nizu: hladnjak-ventilator-pumpa-blok motora) i time sprije~ena mogu}nost dotoka vode u pravcu ose kola. Zbog toga je funkcija uvo|enja vode u pumpu iz dva "izvora" ostvarena inkorporiranjem u ku}i{tu pumpe dodatnog prostora ispred radnog kola, koji okruùje vratilo pumpe i na koji se s bo~nih strana nadovezuju dva ulazna otvora (dio 1 na slici 1). Model ovog dodatnog prostora, nazvanog u internoj terminologiji kolek or uvodnik, na kome se jasno vide dva kraka za ulaz vode, prikazan je na slici 3.

t -

One of the automotive turbopumps specifics is the existence of the two inlet ports. Through the first, during the period of normal operation of a warm motor, cooled water from a radiator is drawn, while the other is operational during the period of the motor heating, when water with the temperature lower than the working is directed around the radiator directly into the pump. There are two turbopumps solution variants in use that provide the two inlet ports. In the first variant (used in the analyzed pump), a fan is mounted on the pump shaft (in line: radiator-fan-pump-motor block), preventing the possibility of water flow in direction of the impeller axis. Because of that the function of feeding the pump with water from two sources is realized by incorporation of additional space in front of the impeller in the pump housing. The space is surrounding the pump shaft, and the two inlet ports are added from the lateral sides (the part 1 in Figure 1). Model of the additional space, called a collector-feeder in internal terminology, with the two inlet ports clearly noticeable, is shown in Figure 3.

Kod realizacije prostornih numeri~kih analiza primijenjen je egzaktan postupak prora~una, s rotiranjem radnog kola. Kod ovog postupka potrebno je formirati klizne interfejse izme|u vode u ulazu u radno kolo i u spirali, i vode u me|ulopati~nim kanalima, koja rotira zajedno s radnim kolom. Pri tome dodirne povr{ine koje tvore klizni interfejs (ravne, cilindri~ne, ili koni~ne) u numeri~kom modelu, moraju biti iste, odnosno moraju se poklapati i nisu dozvoljene praznine ni na jednoj od njih. Kra}e kazano, jedan region ne moè formirati klizni interfejs s dva ili vi{e regiona [4].

In a realization of 3D numerical analysis exact calculation procedure is used with rotations of the impeller. In this procedure it is necessary to form sliding interfaces between the water in the entrance in the impeller and in the spiral, and the water in internal flow impeller channels which rotates together with the impeller. In addition, contact areas creating the sliding interface (flat, cylindrical, or conical) in the numerical model, have to be the same, i.e. they have to be aligned, and no gaps on them are allowed. In brief, one region cannot form sliding interface with two ore more regions [4].

Slika 3. Model kolektora-uvodnika s dva ulazna otvora 1) ulaz od hladnjaka, 2) ulaz od motora Figure 3. Collector-feeder model with two inlet ports 1) inlet from a radiator, 2) inlet from a motor

2

1

- 158 -


a) b)

Ulazna i izlazna ivica lopatica radijalnih radnih kola mogu biti razli~ito oblikovane (slika 4.). Kod radnih kola sa zaobljenom ulaznom ivicom i izlaznom lopaticom koja se zavr{ava o{tro u jednoj ta~ki (kao na slici 4a), klizni interfejsi se, prema pomenutom zahtjevu, mogu bez problema formirati. Me|utim, kod radnih kola kod kojih lopatice imaju odre|enu {irinu na ulaznom i izlaznom pre~niku kola (kao na slici 4b), za formiranje kliznih interfejsâ potrebno je kreirati dodatne prstenove, koji }e rotirati zajedno s fiktivnim radnim kolom. Tako je kod modela analizirane pumpe na 3D modelu vode ulaza u radno kolo odsje~en tanki koni~ni prsten (slika 5.) koji je preko proizvoljnog interfejsa vezan za model vode u me|ulopati~nim kanalima (slika 6.), i s kojima rotira istom ugaonom brzinom. Unutra{nja koni~na povr{ina prstena tvori klizni interfejs sa susjednom povr{inom preostalog dijela vode ulaza u radno kolo. Drugi klizni interfejs se formira na izlaznom pre~niku radnog kola, gdje voda iz me|ulopati~nih kanala ulazi u spiralu. Na ovom mjestu se formira prsten, debljine jednake zazoru izme|u kola i spirale, koji rotira (brzinom radnog kola) zajedno s vodom u me|ulopati~nim kanalima. Vanjska cilindri~na povr{ina ovog prstena tvori klizni interfejs s povr{inom spirale koju dodiruje. Kod analizirane automobilske turbopumpe, ovaj prsten je formiran na ra~un me|ulopati~nih kanala, prema slici 6., na kojoj je prikazan jedan njegov segment. Spoljnji pre~nik prestena u ovom slu~aju je jednak vanjskom pre~niku radnog kola D2.

Inlet and outlet vane edges of radial impellers can be differently shaped (Figure 4). For impellers with a chamfered inlet edge vane and an outlet vane ending sharply in one point (as in Figure 4a), the sliding interfaces can be easily created accordingly to the mentioned condition. However, for the impellers with vanes with the certain width on the inlet and the outlet circumference (as in Figure 4b) additional rings need to be created, rotating together with the impeller, in order to form the sliding interfaces. In such way, in the model of the analyzed pump, a thin conical ring was cut off from the 3D model (Figure 5) of the water in the impeller inlet. The ring is through the arbitrary interface connected to the water model of the internal flow impeller channels (Figure 6) and rotates with it at the same angular velocity. The inside conical area of the ring makes the sliding interface with the adjacent area of the remaining part of the water in the impeller inlet.

The second sliding interface is formed on the outlet impeller circumference, where the water from the internal impeller tracts enters into the spiral. Ring is formed in this spot with thickness equal to the clearance between the impeller and the spiral, rotating (with the speed of the impeller) together with the water in the internal impeller’s tracts. The external area of the ring makes a sliding interface with the spiral area it contacts. In the analyzed automotive turbopump, the ring was formed using internal flow impeller channels. One segment of the ring is shown in Figure 6. External diameter of the ring in this case is equal to the impeller external diameter D2.

Slika 4. Radijalna radna kola centrifugalne turbopumpe a) sa zaobljenim (na ulazu) i o{trim krajevima lopatica (na izlazu),

b) s odre|enom {irinom lopatica na ulaznom i izlaznom pre~niku kola

Figure 4. Radial impellers of centrifugal turbopump a) with chamfered (on inlet) and sharp edges (on outlet),

b) with certain vane width on impeller inlet and outlet

- 159 -


Slika 5. Formiranje koni~nog prstena na modelu Slika 6. Modifikovan model me|ulopati~nog kanala, ulaza u radno kolo, u svrhu obezbje|enja za analize s rotiranjem fiktivnog radnog kola kliznog interfejsa Figure 6. Modified model of internal flow impeller Figure 5. Forming of conical ring on impeller channel, for analysis with rotation of fictive impeller inlet model, in order to provide sliding interface

4. GENERIRANJE MRE@E I OBEZBJE\ENJE ROTIRANJA NJENIH POKRETNIH DIJELOVA

Zbog sloène geometrije modelâ dijelova toka fluida kroz turbopumpu i potrebe formiranja interfejsâ izme|u rotiraju}ih i nerotiraju}ih dijelova fluida, primijenjen je specifi~an na~in generiranja mreà pomenutih modela. Na slici 7. prikazana je mreà modelâ cijelog proto~nog trakta analizirane automobilske turbopumpe, za analize s rotiranjem fiktivnog radnog kola. Numeri~ka mreà je kreirana od heksaedralnih (H-mreà) kontrolnih volumena (CV) i mjestimi~no od prizmati~nih KVa. Veza izme|u rotiraju}ih me|ulopati~nih kanala i fiksnih dijelova proto~nog trakta pumpe ostvarena je preko dva klizna interfejsa. Prvi od njih, koji predstavlja vezu izme|u ulaznog dijela proto~nog trakta i rotiraju}ih me|ulopati~nih kanala, prikazan je na slici 8. s odvojenim rotiraju}im prstenom. Drugi klizni interfejs, koji povezuje me|ulopati~ne kanale i spiralu, ostvaren preko dodatog prstena na izlazu iz radnog kola, moè se vidjeti na slici 9. Kod generiranja mreè proto~nog trakta analizirane turbopumpe, na vi{e mjesta je primijenjen proizvoljni interfejs [4]. Time se dobila kvalitenija mreà, kao {to je slu~aj sa spiralom, a na nekim dijelovima domena to je bio i jedini na~in da se kreira H-mreà.

4. GRID GENERATION AND GRID MOTION

Because of complex geometry of flow parts models trough turbopump, and need for creation of interface between rotating and non-rotating parts of fluid, the specific type of grid generation mentioned models was used. The grid of the analyzed automotive model's entire flowing tract, for the analyses with rotation of the fictive impeller, is shown in Figure 7. Numerical grid is created using hexahedron control volumes (CVs), and locally using prism CVs. The connection between rotating impeller flowing tracts and fixed parts of pump flowing tract is realized by two sliding interfaces. The first, representing the connection between inlet part of the flowing tract and the rotating impellers flowing tracts, is shown in Figure 8 with a separated rotating ring. The second sliding interface, connecting the impeller flowing tracts and the spiral, realized by an additional ring on the impeller outlet is shown in Figure 9. When the flowing tract grid of the analyzed pump was generated, arbitrary interface [4] was applied in several cases. Higher quality was obtained by it, as in case of the spiral, and in the certain parts of the domain that was the only way to generate H-type grid.

- 160 -


Slika 7. Mreža modelâ cijelog protočnog trakta analizirane analizirane turbopumpe

1) ulaz od motora, 2) ulaz od hladnjaka, 3) izlaz pumpe Figure 7. The mesh of analysed automotive turbopump model's entire flowing tract

1) motor inlet, 2) radiator inlet, 3) pump outlet

1

2 3

Slika 8. Mreža dva podvolumena ulaza u radno kolo 1) fiksni dio, 2) rotirajući prsten Figure 8. Grids of two impeller inlet subvolumes

1) fixed part, 2) rotating ring

2 1

Slika 9. Mreža međulopatičnih kanala (s rotirajućim prstenom) i spirale analizirane pumpe Figure 9. Internal flow impeller channel (with rotating ring) and spiral grid of analysed model

- 161 -


5. REZULTATI I DISKUSIJA

5.1. 3D-prora~uni toka fluida Za provedbu CFD analiza primijenjen je egzaktan tranzientni postupak prora~una, s rotiranjem radnog kola turbopumpe, baziran na metodi kona~nih volumena, uz kori{tenje pokretnih mre`a s }elijama proizvoljne topologije [5, 10]. Prora~un je ra|en s deset iteracija po jednom koraku, jer se poslije desete iteracije dostizala zadovoljavaju}a ta~nost, bliska zadanoj. Strujanje s ustaljenim promjenama pra}enih veli~ina postizalo se nakon 2 do 5 obrtaja. Rezultati provedenih tranzientnih analiza s rotiranjem radnog kola, na modelima cijelog proto~nog trakta centrifugalne turbopumpe, pokazuju da se mogu dobiti simulacije strujanja radnog fluida u svim dijelovima radnih elemenata ovih pumpi, sa svim detaljima potrebnim kod ovakvih analiza, i s ta~no{}u kod koje rezultati simulacija ne}e odstupati vi{e od ± 5 % od stvarnog stanja. Na bazi rezultata numeri~kih prora~una mogu se dobiti prikazi rasporeda pritiska i/ili brzine (kao i relativne brzine) toka fluida, te veli~ina svih relevantnih parametara (protok Q, napor H, op}i napor ∆p, stepen iskori{tenja η, rezultuju}a radijalna sila Fr na fiktivnom radnom kolu), a tako|e i Q-∆p krive, na osnovu kojih se mogu dobiti podaci o uspje{nosti realizovanih dizajna pumpi, kao i donijeti odluke o potrebnom redizajnu njenih dijelova. Na slici 10. i 11. prikazani su rasporedi pritiska i brzine na dijelovima modela cijelog proto~nog trakta analizirane automobilske turbopumpe.

5. RESULTS AND DISCUSSION

5.1. 3D-calculation of fluid flow For realization of CFD analysis, the exact transient calculation procedure was used, with turbopump's impeller rotation, based on finite-volume numerical method, using moving grids with cells of arbitrary topology [5, 10]. The calculation was performed with ten iterations per step because after the tenth iteration satisfactory accuracy was achieved, close to assigned. Flow with settled changes of monitored values was achieved after 2 to 5 revolutions. Results of performed transient analysis with impeller rotation, on entire flow tract of centrifugal turbopump, show that it's possible to get simulations of working fluid's flow in all parts of working elements of these pumps, with all details needed in such analysis, and with precision of simulation's results will not deviate more than ± 5 % from real condition. Based on numerical calculation results it’s possible to obtain a presentation of fluid flow’s pressure and velocity (as well as relative velocity) distribution, values of all relevant parameters (flow Q, effort H, general effort ∆p, efficiency factor η, resulting radial force Fr and torque Mz on fictive impeller), and Q-∆p diagrams. The Q-∆p diagrams can be used to obtain data about realized pumps design efficiency, as well for making decisions about possible needed redesign of pump parts. Distribution of pressure and velocity in model's parts of analyzed automotive turbopump's entire flowing tract are shown in Figure 10. and Figure 11.

a) b) Slika. 10. Raspored pritiska u modelu cijelog proto~nog trakta analizirane automobilske turbopumpe a) kolektor-uvodnik pumpe, b) me|ulopati~ni kanali i spirala pumpe Figure 10. Distribution of pressure in model of entire turbopump's flowing tract a) pump collector-feeder b) internal flow impeller channels and spiral of pump

- 162 -


a) b) Slika. 11. Raspored brzine u modelu cijelog proto~nog trakta analizirane automobilske turbopumpe a) kolektor-uvodnik pumpe, b) me|ulopati~ni kanali i spirala pumpe Figure 11. Distribution of velocity in model of entire turbopump's flowing tract a) pump collector-feeder b) internal flow impeller channels and spiral of pump

5.2. Uporedba numeri~kih i eksperimentalnih rezultata

Za prakti~no potvr|ivanje rezultata numeri~kih analiza poslu`ili su rezultati na opitnim postrojenjima realizovanih eksperimentalnih ispitivanja. Konstruisane su uporedne Q-∆p krive analizirane turbopumpe i njenog numeri~kog modela (slika 12.), koje pokazuju zadovoljavaju}i stepen podudaranja. Kriva a) dobivena je na osnovu rezultata eksperimentalnih ispitivanja na opitnom postrojenju., pri n = 3000 o/min. Upotrijebljene su srednje vrijednosti ve}eg broja mjerenja na istoj pumpi. Kriva b) rezultat je numeri~kih prora~una, realizovanih na modelu analizirane turbopumpe primjenom egzaktnog postupka prora~una s rotiranjem fiktivnog radnog kola, pri istoj brzini obrtanja. Kod numeri~kih modela su dobivene veli~ine protoka ve}e za oko 10 % u odnosu na eksperimentalno dobivene rezultate na opitnom postrojenju, za iste nivoe op{teg napora �p. Razlog ovome je {to nisu uzeti u obzir gubici na priklju~nim stezaljkama opitnog postrojenja. S ura~unavanjem ovih gubitaka, pomenuta razlika dobivenih vrijednosti protoka svodi se na oko 5 %.

5.2. Comparison of the numerical and experimental results

To confirm results of numerical analysis in practice, the results of performed experimental tests on experimental assembly have been used. Comparative Q-∆p curves of analyzed turbopump and its numerical model have been constructed (Figure 12), and they show satisfactory level of accordance. Curve a) was obtained by experiment results (for experiment at n = 3000 min-1). Average values from numerous measurements on the same pump were used. Curve b) is result of numerical calculations, carried out on the analyzed pump model, using exact calculation procedure with fictive impeller rotation, at the same angular velocity. The values obtained by numerical models are around 10% larger that the experimental one for the same general effort levels �p. A cause of this is that losses on connection nipples were not taken into account. When these losses are included, the mentioned difference comes to about 5%.

- 163 -


0

0,2

0,4

0,6

0,8

1

1,2

1,4

0 0,001 0,002 0,003 0,004 0,005 0,006 0,007

Q m3 /s

p b

ar

b)a)

Slika 12. Q-∆p krive analizirane automobilske turbopumpe i njenog numeri~kog modela, pri n=3000 o/min a) based on the experiment results, b) based on results from the numerical calculations

Figure 12. Q-∆p curves of the analyzed automotive turbopump at n = 3000 min-1

a) based on the experiment results, b) based on results from the numerical calculations

6. ZAKLJUÂK U ovom radu je kod numeri~kih prora~una prvi put na modelima automobilskih turbopumpi primijenjen egzaktan postupak prora~una, s rotiranjem radnog kola, kori{tenjem pokretnih mreà. Prora~uni su realizovani na modelu cijelog proto~nog trakta izabrane automobilske turbopumpe. Primjena CFD analiza za prora~une toka cijelog proto~nog trakta centrifugalnih turbopumpi, u ovom slu~aju posebno automobilskih turbopumpi za vodu, na na~in kako je to predloèno u ovom radu, moè se uspje{no uklju~iti u postupak prora~una i dizajna istih, jer omogu}ava simulacije toka visokog stepena podudarnosti s realnim stanjem, s ta~no{}u kod koje rezultati simulacija ne}e odstupati vi{e od ± 5 % od stvarnih vrijednosti.

6. CONCLUSION In the numerical calculations of this paper, exact calculation has been applied for the first time on the automotive turbopumps, with rotation of impeller, using of moving grids. The calculations are realized on the whole flowing tract model of chosen automotive turbopump. Using CFD analysis in flow calculation of entire flowing tract of centrifugal turbopumps, speciality of automotive water turbopumps in this case, in a manner suggested in this paper, can be successfully included in their calculation procedure and design, because they enable flow simulations of high degree of coincidence with real condition, and with precision of simulation's results will not deviate more than ± 5 % from real values.

6. LITERATURA - REFERENCES

[1] Bali}, S., "Numerical analysis of flow process and stress condition in centrifugal pumps for automotive engine cooling systems", Ph. D. Thesis, University of Sarajevo, Faculty of Mechanical Engineering in Zenica, Zenica, 2002.

[2] Bali}, S. in Duhovnik, J., "Analiza proto~nega trakta centrifugalne ~rpalke DN 200 za agresivne medije", Dr. Duhovnik d.o.o., Ljubljana, 2003.

- 164 -


[3] Carlson, J. and Pipkorn, N., "CFX helps design more efficient diesel engine water pumps", CFX Update, No 22, Autumn 2002

[4] Comet User Manual and Tutorials, Version

2.00, ICCM Institute of Computational Continuum Mechanics GmbH, Hamburg, 2001.

[5] Demird`i}, I. and Muzaferija, S., Numerical

method for coupled fluid flow, heat transfer and stress analysis using unstructured moving grids with cells of arbitrary topology", Computer methods in applied mechanics and engineering, 125, 1995, pp 235-255.

[6] Didandeh H., Toksoy, C., Cutler, F. and the

other, "Computer-Aided Design of a Water Pump Impeller for the Chrysler 4.0 Liter 6 Cylinder Engine", SAE International Congress & Exposition, Detroit, Michigan, February 24-27, 1997, pp 82-93.

[7] Duhovnik, J. and Bali}, S., "Detail Functionality

Analysis Using the Design Golden Loop", Fourth International Seminar and Workshop on Engineering Design in Integrated Product

Development, Management of Design Complexity, EDIProD’2004, Zielona Góra, Poland, 2004.

[8] FLUENT web site for Multiple Rotating

Reference Frames Method http://www.fluent.com/solutions/pumps/ex164.pdf

[9] Minemura, K., Uchiyama, T. and the other,

"Prediction of Air-Water Two-Phase Flow Performance of a Centrifugal Pump Based on One-Dimensional Two-Fluid Model", Journal of Fluids Engineering, Vol 120, June 1998, pp 237-334.

[10] Muzaferija, S. and Gosman, D., "Finite-Volume

CFD Procedure and Adaptive Error Control Strategy for Grids of Arbitrary Topology", Journal of Computational Physics, 138, 1997, pp 766-787.

[11] Pak, E. T. and Lee, J. C., "Performance and

pressure distribution changes in a centrifugal pump under two-phase flow", Proc Instn Mech Engrs, Vol 213, Part A, 1998, pp 165-171.

- 165 -


- 166 -

Ma{instvo 3(8), 167 – 176, (2004) A.[uta,...: UTICAJ GEOMETRIJSKIH MODIFIKACIJA...

UTICAJ GEMOETRIJSKIH MODIFIKACIJA NA KOEFICIJENT PRETVORBE PRITISKA DIFUZORA SA KOLJENOM

[uta Alem, dipl.ing.ma{., JP Elektroprivreda BiH Sarajevo, Hidroelektrane na Neretvi, Jablanica Doc. dr. Ejub Dàferovi}, Ma{inski fakultet u Sarajevu, Univerzitet u Sarajevu

REZIME U radu je prikazana trodimenzionalna numeri~ka simulacija turbulentnog strujanja fluida u difuzoru hidroturbine sa koljenom. Izabrana geometrija difuzora je naj~e{}e kori{teni dizajn kod Kaplanovih turbina u 40 i 50 godinama pro{log vjeka. Ima pravilan dizajn i o{tre ivice te predstavlja potencijal za unapre|enje u pore|enju sa difuzorima novije generacije. Istraìvanje je vr{eno pomo}u softvera COMET (COntinuum Mechanic Engineering Tool) koji je baziran na metodi kona~nih volumena i koji ima implementiran tzv. RANS (Reynolds Ave aged Navier Stokes) pristup. Turbulencija je modelirana pomo}u standardnog �-ε modela turbulencije. Generisanje numeri~ke mreè vr{eno je kombinacijom tri razli~ita pristupa: metoda uvoza podataka, metode nivoa }elija i multi-blok metode. Grani~ni uslovi specificirani su prema rezultatima eksperimentalnih mjerenja. Simulacije su ra|ene za dva razli~ita reìma rada turbine. Prvo su analizirane osnovne karakteristike strujanja u originalnom difuzoru. Validacija rezultata tako|e je prezentovana; mjerenja su upore|ivana sa simulacijama. Kao standardna mjera performansi difuzora kori{ten je koeficijent pretvorbe pritiska. Ovaj koeficijent predstavlja bilans enegije izme|u dva presjeka difuzora; ve}a vrijednost zna~i bolje performanse difuzora i bolju iskori{tenost hidroelektrane u cjelini. U drugom dijelu istraìvanje je fokusirano na uticaj geometrije na koeficijent pretvorbe pritiska. Ura|ene su tri modifikacije originalne geometrije difuzora. Modifikovan je izlazni divergiraju}i dio difuzora (ulazni konus i koljeno difuzora nisu bili predmet redizajna). Simulacije su pokazale pove}anje koeficijenta pretvorbe pritiska u pore|enju sa originalnom konfiguracijom difuzora.

r

r

Klju~ne rije~i: metod kona~nih volumena, difuzor, turbulencija, koeficijent pretvorbe pritiska

INFLUENCE OF GEOMETRICAL MODIFICATIONS ON PRESSURE RECOVERY FACTOR OF AN ELBOW DIFFUSER

[uta Alem, B.Sc.Mech.Eng., JP Elektroprivreda BiH Sarajevo, Hydro Power Plants on Neretva, Jablanica Ph.D. Ejub Dàferovi}, asiss. prof. Mechanical Engineering Faculty, University in Sarajevo

SUMMARY In this paper numerical simulation of the turbulent flow in a three-dimensional elbow diffuser is performed. The diffuser geometry chosen was a commonly used design for Kaplan turbine during the 1940's and 1950's. It was a straight forward design with sharp corner and it is a potential for improvements compared to modern diffusers. The investigation is carried out with finite volume solver COMET (COntinuum Mechanic Engineering Tool) implementing RANS (Reynolds Averaged Navie Stokes) equation. Turbulence was modeled using standard �-ε model. Numerical grid is generated using a combination of three different approaches: data import approach, cell-layer approach and multi-block generation. Boundary conditions were specified according to the experimental data. The flow is analyzed for two different modes of operation. Firstly, basic flow patterns are analyzed in the original configuration of the diffuser. Validation of results is also presented; measurements are compared with computations. As a standard measure of the performance of diffuser the static pressure recovery factor is used. This coefficient represents the energy balance between the sections of the diffuser; a higher value means a higher performance and corresponding power plant efficiency. In the second part investigation is focused on the influence of the geometry on the pressure recovery factor. Three modification of original configuration is performed. Straight diverging part of the diffuser is modified (inlet cone and elbow was not subject of redesign). Simulations showed increasing of the pressure recovery factor compared to the original configuration.

Key words: finite volume method, diffuser, turbulence, pressure recovery factor.

- 167 -


1. UVOD

Veliki broj hidroelektrana i njenih komponenti stari, {to zna~i da predstavljaju potencijal za modifikaciju i implementaciju promjena u dizajnu da bi se pove}ala iskoristivost i izlazna snaga turbine kao i ve}a radna stabilnost. Uobi~ajeno je da su radno kolo i sprovodni aparat u fokusu u projektima modifikacija. Zbog velikih gra|evinskih tro{kova spiralno ku}i{te i difuzor rijetko predmet redizajna. Moderne hidrauli~ke turbine su dizajnirane tako da imaju visok stepen iskori{tenja (do 95%). Bez obzira na to poznato je da pove}anje koeficijenta iskori{tenja od nekoliko desetih dijelova procenta moè zna~ajno pove}ati profit. Difuzor je jedna od najva`nijih komponenti hidroelektrane koja ima zadatak da pretvori maksimum dinami~kog pritiska u stati~ki pritisak i da smanji brzinu strujanja vode koja napu{ta radno kolo turbine a da se pri tome izbjegne odvajanje strujanja. Ugao {irenja konusa rezultat je tog kompromisa. Divergiraju}a geometrija difuzora prouzrokujue nepovoljan gradijent pritisaka koji uve}ava rizik od nastanka odvajanja strujanja. Vrtlog ima suprotan efekat i protivi se odvajanju grani~nih slojeva. Kao pokazatalj efikasnosti difuzora naj~e{}e se koristi koeficijent pretvorbe pritiska . pC

2. MATEMATSKI MODEL Pona{anje kontinuuma je opisano sa tzv. transportnim jedna~inama baziranim na osnovnim zakonima fizike koji izraàvaju bilans (konzervaciju) slijede}ih veli~ina: • mase i • koli~ine kretanja (drugi Newtonov zakon).

2.1 Jedna~ina kontinuiteta Jedna~ina koja izraàva zakon o konzervaciji mase, poznata kao jedna~ina kontinuiteta, moè biti napisana kao:

0dddd

=⋅+∫ ∫ svV S

ρVρt

(1)

gdje je ρ gusto}a kontinuuma, v je brzina kontinuuma a s je povr{inski vektor usmjeren na vanjsku stranu povr{ine.

2.2 Jedna~ina koli~ine kretanja Primjena drugog Newtonovog zakona na proizvoljni kontrolni volumen vodi do jedna~ine koli~ine kretanja poznate kao Cauchyjev prvi zakon kretanja:

∫∫ ∫∫ +⋅=⋅+V

bS SV

VρVρt

dddddd fsTsvvv (2)

1. INTRODUCTION

A large number of hydro power plants and its components are ageing which means that they represent a potential for the refurbishment and implementing changes in the design for improved efficiency and power output as well as greater operating stability. Usually the runner and guide vanes are focused upon in the refurbishment process. Due to capital constructional costs the spiral casing and the diffuser are seldom redesigned. Modern hydraulic turbo machines are designed with high efficiency (up to 95%). Nevertheless it must recognize that efficiency improvements of only a few tenths of a percent can significantly increase profit. Diffuser is one of the most important part of hydroelectric power plant which act to convert maximum of dynamic pressure into static and to reduce the velocity of fluid that leave turbine impeller avoiding flow separation. The opening angle of the cone results from this compromise. The divergent geometry introduces an adverse pressure gradient that raises the risk of flow separation. The swirl has the opposite effect and prevents boundary layer separation. As a standard measure of efficiency pressure recovery factor is usually used. pC

2. MATHEMATICAL MODEL The behavior of continuum is governed by the so-called transport equations based on the basic laws of physics expressing balance (conservation) of: • mass and • momentum (Newton's second law).

2.1. Mass balance equation The equation expressing the mass conservation law, known as the continuity equation, can be written as:

0dddd

=⋅+∫ ∫ svV S

ρVρt

(1)

where ρ is the density of continuum, v is continuum velocity and s is the outward pointing surface vector.

2.2. Momentum equation The Newton's second law when applied to an arbitrary control volume leads to the equation of momentum balance known as Cauchy's first law of motion:

∫∫ ∫∫ +⋅=⋅+V

bS SV

VρVρt

dddddd fsTsvvv (2)

- 168 -


gdje je T Cauchyjev tenzor naprezanja, a je

rezultiraju}a zapreminska sila po jedinici volumena. bf

2.3. Modeliranje turbulencije

Numeri~ko rje{avanje turbulentnog strujanja (Direct Numerical Simulation DNS) zahtijeva mreù sa razmacima manjim od najmanjeg turbulentnog vrtloga i vremenski korak koji }e biti manji od najmanje vremenske promjene turbulentnih fluktuacija. Ovaj pristup zahtijeva takve kompjuterske resurse koji nisu dostupni sa dana{njim razvojem kompjuterskih tehnologija. Alternative su LES (Large Eddy Simulation rije{eni su samo najve}i vrtlozi a ostali su modelirani) i rje{avanje RANS jedna~ina. LES se tek po~inje koristiti u analizi kompleksnih strujanja, tako da je rje{avanje inènjerskih problema uglavnom bazirano na RANS pristupu.

2.4. RANS jedna~ine

RANS jedna~ine su dobijene koriste}i statisti~ki opis turbulentnog strujanja. Jedan takav opis koristi se u Reynoldsovom usrednjavanju gdje je svaka zavisna varijabla izraèna kao suma njene srednje

vrijednosti φ i fluktuacijske komponente φ ′′ :

φφφ ′′+= (3)

gdje je:

∫−

+=2

2

)d(1)(

τ

τ

ξξ,tτ

,t rr φφ (4)

i vremenski interval τ dovoljno velik u odnosu na vremensku veli~inu turbulentnih fluktuacija a mali u odnosu na vremensku veli~inu svih ostalih efekata koji zavise od vremena.

where T is the Cauchy stress tensor, and is

the resultant body force per unit volume. bf

2.3. Turbulence modeling

Numerical solution of turbulent flow (Direct Numerical Simulation DNS) requires a mesh with spacing smaller than the length scale of the smallest turbulent eddies, and time steps smaller than the smallest time scale of turbulent fluctuations. This approach requires computer resources which are not available at the present state computer technology. Alternatives are LES (Large Eddy Simulation only the largest unsteady motions are resolved and the rest is modeled) and the solution of RANS equations. LES is just beginning to be used for complex flows and the solution of engineering problems is mostly based on RANS approach.

2.4. RANS equations

The RANS equations are obtained by using a statistical description of turbulent motion. One such description uses the Reynolds averaging where each dependent variable is expressed as the sum of its

mean value φ and a fluctuating component φ ′′ :

φφφ ′′+= (3)

where:

∫−

+=2

2

)d(1)(

τ

τ

ξξ,tτ

,t rr φφ (4)

and the time interval τ is large enough with respect to the time scale of the turbulent fluctuations, but small with respect to the scale of other time dependent effects.

2.5. Standardni κ-εεmodel turbulencije

Standardni�κ-εεmodel turbulencije je naj~e{}e kori{teni eddy-viscosity model. Kineti~ka energija turbulencije κ i njena dispiacija ε definisane su sa:

T)(grad:grad

21

vv

vv

′′=

′⋅′=

ρµε

κ (5)

i dobijene su rje{avanjem njihovih odgovaraju}ih transportnih jedna~ina:

2.5. Standard κ-εεmodel

The standard κ-εεmodel is the most widely used eddy-viscosity model. The kinetic energy of turbulence κ and its dissipation rate ε are defined as:

T)(grad:grad

21

vv

vv

′′=

′⋅′=

ρµε

κ (5)

and are obtained by solving their respective transport equations:

(∫ ∫∫∫ −++⋅=⋅+SS

dddddd

VB

V

VρεPPρκVρκt

sqsv κ ) (6)

2

1 2 3 4S S V

d d d d max 0 divd ε B

V

ε ε ε dρε V ρε C P C ρ C (P , ) C ρε Vt κ κ κ

⎛ ⎞+ ⋅ = ⋅ + − + −⎜ ⎟

⎝ ⎠∫ ∫ ∫ ∫v s q s v (7)

- 169 -


Veli~ine C1=1.44, C2=1.92 , C3=1.44 i C4=–0.33 u izrazu (7) su empirijski koeficijenti, P i su

udjeli kineti~ke energije turbulencije zbog smicanja i uzgona.

BP

The quantities C1=1.44, C2=1.92, C3=1.44 and C4=-0.33 in equations (7) are empirical coefficients,

P and are contributions of kinetic energy of turbulence due to the shear and buoyancy.

BP

3. NUMERI^KI METOD Sve konzervativne jedna~ine (osim jedna~ine kontinuiteta) mogu se napisati u obliku slijede}e op{te transportne jedna~ine, [2]:

∫∫

∫ ∫∫+⋅

+⋅=⋅+

VV

SS

S SV

Vq

ρVρBt

dd

dgradΓdddd

φφ

φφ φφ

sq

ssv

(8)

dok je jedna~ina kontinuiteta kombinovana sa jedna~inom koli~ine kretanja da bi se dobila jedna~ina za pritisak ili korekciju pritiska. U jedna~ini (8) φ ozna~ava transportnu veli~inu, npr.

Kartezijsku komponentu vektora brzine iv .

Zna~enje pojedinih veli~ina kao {to su φ i φB Γ

u op{toj transportnoj jedna~ini (8) dato je u tabeli 1.

3. NUMERICAL METHOD All the conservation equations (except for the continuity equation) can be written in the form of the following generic transport equation, [2]:

∫∫

∫ ∫∫+⋅

+⋅=⋅+

VV

SS

S SV

Vq

ρVρBt

dd

dgradΓdddd

φφ

φφ φφ

sq

ssv

(8 )

while the continuity equation is combined with momentum equation to obtain an equation for

pressure or pressure correction. In equation (8) φ

stands for the transported property, e.g. Cartesian components of the velocity vector . The meaning

of the quantities and

iv

φB φΓ in generic transport

equation (8) are given in Table 1.

Tabela 1: Zna~enje pojedinih ~lanova u op{toj transportnoj jedna~ini (8) Tabl 1: The meaning of various terms in the generic transport equation (8)

e

φ φB φΓ Sφq Vqφ

iv iv effµ ( ) iT div

32grad iIvv ⋅⎥

⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +− pµµ effeff ibf ,

� � κ

t

σµµ + 0 ρεPP B −+

ε ε ε

t

σµµ + 0

vdiv

)0(max

4

3

2

21

ρεCκε,PC

κερC

κεPC B

−

+−

gdje µ , effµ i tµ ozna~avaju dinami~ku,

efektivnu i turbulentnu viskoznost, I ozna~ava

proizvoljni ntrolni volumen dobija slijede}i oblik:

jedini~ni tenzor a ii jedni~ni Kartezijski vektor. Jedna~ina (8), kada se napi{e za ko

where µ , effµ and tµ stands for dynamic,

effective and turbulent viscosity, I is unit tensor

arbitrary control volume, gets the following form:

and ii is unit Cartesian vector Equation (8), when written for an

∫∑ ∫

∑ ∫∑ ∫∫

+⋅

+⋅=⋅+

=

==

VV

n

j SS

n

j S

n

j SV

Vq

ρVρBt

f

j

f

j

f

j

dd

dgradΓdddd

1

11

φφ

φφ φφ

sq

ssv

(9)

- 170 -


gdje prvi i drugi izraz sa lijeve strane jedna~ine (9) predstavljaju vremenski i konvektivni ~lan, dok izrazi na desnoj strani predstavljaju difuzivni i izvorni ~lan, respektivno. U zadnjoj jedna~ini je

broj povr{ina koje okruùju kontrolni volumen.

fn

Za simulacije strujanja u difuzoru kori{ten je CFD (Computational Fluid Dynamic) softver COMET razvijen na ICCM Insitute Hamburg, Germany. COMET je baziran na metodi kona~nih volumena. Softver primjenjuje RANS pristup sa razli~itim modelima turbulencije. U COMET-u je pimjenjen razdvojeni algoritam rje{avanja. Integrali u jedna~ini (9) ra~unati su pomo}u pravila centralnog pravougaonika. Za aproksimaciju vremenskog ~lana kori{tena je Eulerova implicitna {ema. Zapremina proizvoljnog kontrolnog volumena ra~unata je primjenom Gaussove teoreme. Ra~unski ~vorovi gdje se ra~unaju vrijednosti zavisne varijable smje{teni su u centru kontrolnog volumena, dok su grani~ni ~vorovi neophodni za specifikaciju grani~nih uslova smje{teni u centrima grani~nih povr{ina. Jednostavan i efikasan na~in za prora~un gradijenata u centrima kontrolnih volumena primjenjen u ovom radu je ta~nosti drugog reda i baziran je na Gaussovoj teoremi divergencije i pravilu centralnog pravougonika. Pritisak je ra~unat koriste}i SIMPLE (Semi Implicit algorithm for Pressure Linked Equations) algoritam. Za rje{avanje sistema linearnih algebarskih jedna~ina primjenjen je metod konjugovanih gradijenata CG (Conjugate Gradient) sa IC (Incomplete Cholaski) prekondicioniranjem.

where first and second expressions on the left side of equation (9) represents transient rate of change and convection and expressions on the right side represents diffusion and sources, respectively. In the last equation is the

number of faces enclosing the control volume.

fn

For the simulation of the flow in diffuser CFD (Computational Fluid Dynamic) computer code COMET, developed at ICCM Hamburg, Germany is used. COMET is a finite volume flow simulation program. The software is based on the RANS equations with various models of turbulence. In COMET a segregated solution algorithm is applied. Integrals in equation (9) are calculated with help of midpoint rule. Transient term is approximated with help of the Euler implicit scheme. The volume of an arbitrary control volume is calculated using Gauss' theorem. Computational nodes at which the variable values are to be calculated are placed at the center of control volume, while boundary nodes necessary for the specifications of the boundary conditions are placed at the centers of boundary faces. A simple and efficient way of calculating gradients at control volume centers applied in this work is a second-order accuracy and it is based on the Gauss' divergence theorem and the midpoint-rule. The pressure is calculated by SIMPLE (Semi Implicit algorithm for Pressure Linked Equations) algorithm. For the solution of the linear equation systems a conjugated gradient method CG(Conjugate Gradient) with an IC (Incomplete Cholaski)-preconditioning is used.

4. OPIS PROBLEMA Izabrana geometrija je naj~e{}e kori{teni dizajn kod Kaplanovih hidroturbina u 40 i 50 godinama pro{log vijeka, tzv. ERCOFTAC (European Research Community On Flow Turbulence And Combustion) difuzor. Ima pravilan geometrijski oblik sa o{trim ivicama te je potencijal za unapre|enja u odnosu na moderne difuzore. Eksperimentalna mjerenja ra|ena su u laboratoriji Vattenfall Utveckling, Älvkarleby, Sweden, na modelskoj turbini ~iji je omjer 1:11, [1]. Eksperimentalni podaci su dati za dva razli~ita jedini~na protoka (reìma rada) kroz turbinu,

i . Eksperimenti su ra|eni pri

optere}enju od 60% {to odgovara najboljoj iskori{tenosti posmatrane Kaplanove turbine. Visina (konstruktivni pad) H u oba slu~aja je 4.5 m. Osnovni podaci o eksperimentima dati su u tabeli 2.

00.1q T = 04.1q R =

4. PROBLEM DESCRIPTION The diffuser geometry chosen was a commonly used design for Kaplan turbine plants during the 1940's and 1950's, so called ERCOFTAC (European Research Community On Flow Turbulence And Combustion) diffuser. It was a straight forward design with sharp corner and it is a potential for improvements compared to modern diffusers. Experiments were performed at Vattenfall Utveckling Laboratory, Älvkarleby, Sweden, on model turbine with 1:11 ratio, [1]. Experimental data for two different unit flows (mode of operation) through turbine are given, 00.1q T = and . Experiments were

conducted at 60% load which is the best efficiency point of studied Kaplan turbine. Height (constructive head) in both cases was 4.5 m. Basic data about experiments are given in Table 2.

04.1q R =

Tabela 2: Osnovni podaci o eksperimentima Table 2: Basic data about experiments

Naziv slu~aja (reìm rada) Case name (mode of operation)

Broj obrtaja NNumber of

revolution N

Protok QFlow Q

Jedini~na brzina obrtaja

radnog kola HDN

Unit impeller speed HDN

Jedini~ni protok HQD 2

Unit flow HQD 2

ON–DESIGN 595 o/min 0.522 m3/s 140 1.00 OFF–DESIGN 595 o/min 0.542 m3/s 140 1.04

- 171 -


Generisanje numeri~ke mreè je aktivnost koja tro{i najvi{e vremena u cjelokupnom procesu numeri~kih simulacija. Numeri~ka mreà je integralni dio cjelokupnog sistema numeri~ke simulacije i za efikasne i ta~ne numeri~ke simulacije vrlo je bitno razumjevanje procesa generisanja numeri~ke mreè. Geometrija difuzora diskretizovana je multi-blok strukturiranom mreòm koja se sastoji od pribli`no 350000 KV heksahedralnog oblika (slika 1). Za generisanje numeri~ke mreè difuzora kori{tene su metode uvoza podataka, generisanje mreè metodom nivoa }elija i multi–blok metoda. Da bi rje{ili numeri~ki problem potrebno je definisati grani~ne uslove koji vladaju na zidovima, ulazu i izlazu difuzora. U najve}em slu~aju prakti~nih primjera koristi se no–slip grani~ni uslov za zid. Ovo je idealizirana situacija i na zidovima difuzora specificiran je no–slip grani~ni uslov defini{u}i vrijednosti brzina na zidovima. Ako je strujanje turbulentno i numeri~ka mreà previ{e rijetka da rije{i velike varijacije brzine u regionima uz zid, neophodna je posebna interpolacija koja vodi do realisti~ne vrijednosti napona smicanja. Interpolacija je bazirana na tzv. funkciji zida, [3]. Ulazni grani~ni uslov je specificiran na dijelu domena rje{avanja gdje fluid ulazi u domen i gdje je poznat raspored brzina. Pritisak na ulazu je dobijen ekstrapolacijom iz unutra{njosti domena rje{avanja. Za nekompresibilne fluide maseni protok na ulazu je odre|en rasporedom brzina na granici i gusto}om fluida. U slu~aju turbulentnog strujanja κ i ε su definisani u vidu intenziteta turbulencije

i duìne turbulencije . BI BlIzlazni grani~ni uslov je specificiran na dijelu domena rje{avanja gdje fluid napu{ta domen. Aproksimirani grani~ni uslov podrazumjeva da su gradijenti svih zavisnih varijabli jednaki nuli u pravcu strujanja. Preporuka je da se ovaj grani~ni uslov stavlja na mjesto gdje je tok potpuno razvijen i {to dalje od regiona interesa.

5. UPORE\IVANJE SA EKSPERIMENTOM

Da bi se verifikovao neki numeri~ki model, potrebno je izvr{iti pore|enje sa eksperimentalnim rezultatima. Upore|ivani su dostupni eksperimentalni podaci o rasporedu pritisaka na izlazu i koeficijent pretvorbe pritiska sa rezultatima simulacija. pC

Na izlazu iz difuzora poznate su vrijednosti pritiska. Simulacije pokazuju dobro slaganje pritisaka sa eksperimentalnim rezultatima na zidovima izlaza iz difuzora (slika 2).

The numerical grid generation is the activity that consummating much time in the overall process of numerical simulation. Numerical grid is integral part of whole numerical simulation process and for efficient and exact numerical simulations it is very important to understand a process of numerical grid generation. The geometry of the diffuser is discretized with multi-block structured grid consisting of approximately 350000 control volumes with hexahedral cell shape, see Fig.1. For the numerical grid generation of the diffuser data import approach, cell-layer approach and multi-block grid generation method are used. To solve numerical problem, it is necessary to define boundary conditions on the walls, inlet and outlet of the diffuser. The no-slip wall boundary conditions are used in most practical situations. This is idealized situation, and no-slip boundary condition at walls is specified by prescribing the values of velocity on the walls. If the flow is turbulent and the numerical grid is too coarse to resolve a large velocity variation in the near-wall region, a special interpolation, which leads to realistic values of the shear stress, is necessary. The interpolation is based on the so-called wall functions, [3]. The inlet boundary condition is specified at portion of the domain where the fluid enters the solution domain and where the velocity distribution is known. The pressure at the inlet boundary is obtained by extrapolation from inside the solution domain. For incompressible flow, the mass flow rate through the inlet boundary is determined by the velocity distribution at the inlet and the density of the fluid. In the case of turbulent flow κ and ε are defined in sense of turbulence intensity and turbulent length

scale . The outlet boundary condition is specified

at portion of the domain where the fluid leaves the solution domain. Approximated boundary condition means that the gradients of all dependant variables in flow direction are equals zero. It is recommended that this boundary condition should be placed where the flow is fully developed and as far as possible from region of interest.

BI

Bl

5. COMPARASION WITH EXPERIMENTS

In order to verify some numerical model, it is necessary to make comparison with experimental data. Experimental pressure distribution on the outlet of diffuser and the pressure recovery factor

are compared with simulations. pC

At the diffuser outlet pressure distribution is known. Simulations showed good agreement with experiments on the outlet diffuser walls (Fig.2).

- 172 -


Slika 1: Numeri~ka mre`a difuzora, 359320 kontrolnih volumena, 3D pogled Figure 1: Numerical grid of the diffuser, 359320 control volumes, 3D view

Vrijednost pritisaka u ta~kama u kojima su poznate eksperimentalne vrijednosti padaju u podru~je ograni~eno sa pravim linijama pmin=pexp–pdev i pmax=pexp+pdev, gdje je pexp srednja vrijednost pritiska a pdev vrijednost maksimalnog odstupanja u toku eksperimenta. Sa psim ozna~en je pritisak kao rezultat simulacija Kao standardna mjera performansi difuzora kori{ten je koeficijent pretvorbe pritiska , [5]. Ovaj koeficijent

predstavlja balans energije izme|u pojedinih sekcija difuzora; ve}a vrijednost koeficijenta zna~i ve}e performanse i bolju iskori{tnost ~itave hidroelektrane. U ovom radu razmatran je globalni koeficijent pretvorbe pritiska koji predstavlja razliku energija pritiska na izlazu i ulazu podjeljenu sa kineti~kom energijom na ulazu:

pC

2

21

ulaz

ulazizlazp

ρv

ppC

−= (12)

gdje i predstavljaju srednje vrijednosti

pritiska na zidu na ulazu i izlazu difuzora. Ove vrijednosti su rezultat simulacija strujanja fluida u difuzoru. je srednja vrijednost brzine na ulazu

definisana sa:

ulazp izlazp

ulazv

ulaz

ulazulaz A

Qv = (13)

gdje je protok na ulazu a povr{ina

ulaza u difuzor. ulazQ ulazA

Vrijednost koeficijenta kao rezultat simulacija

je 0.9346 za slu~aj ON–DESIGN, dok za slu~aj OFF–DESIGN ta vrijednost iznosi 1.0694. Eksperimentalana vrijednost je 1.12.

pC

The value of pressure in points where they known from experiment falls in area bounded by straight lines pmin=pexp–pdev and pmax=pexp+pdev, where pexp is mean pressure and pdev maximum deviation during the experiment. psim represents pressure obtained by numerical simulation. A standard measure of the performance of diffuser the static pressure recovery factor is used, [5].

This coefficient represents the energy balance between the sections of the diffuser; a higher value means a higher performance and corresponding power plant efficiency. In this paper global pressure recovery factor is used which represents the difference of pressure energy at the inlet and the outlet divided by the inlet kinetic energy:

pC

2

21

inlet

inletoutletp

ρv

ppC −= (12)

where and stands for mean inlet

and outlet wall pressure. These values are results of the simulation of fluid flow in diffuser. is

mean inlet velocity defined as:

inletp outletp

ulazv

inlet

inletinlet A

Qv = (13)

where represent inlet mass flow and

is inlet surface. inletQ inletA

The value of coefficient obtained from

simulation for case ON-DESIGN is 0.9346 and for case OFF-DESIGN this value is 1.0694. Experimental value is 1.12.

pC

- 173 -


Razlika izme|u eksperimentalne vrijednosti i vrijednosti dobijene simulacijom koeficijenta je

pribli`no 16% za slu~aj ON–DESIGN i 4% za slu~aj OFF–DESIGN.

pCThe difference between experimental and simulated value of coefficient is approximately 16% for

case ON-DESIGN and 4% for case OFF-DESIGN.

pC

Slika 2: Vrijednosti pritisaka na donjem izlaznom zidu difuzora Figure 2: Pressure values at the bottom outlet wall of diffuser

6. REZULTATI SIMULACIJA Na slici 3 prikazana su polja brzina u presjeku

za slu~aj ON-DESIGN. Polje brzina jasno

pokazuje glavni trend strujanja u difuzora (smanjenje brzine prema izlazu iz difuzora {to je i jedna od glavnih funkcija difuzora).

0=y

Kao efikasan na~in analize strujanja u difuzoru isti~e se prikazivanje strujnih linija ili putanje ta~aka od ulaza prema izlazu difuzora, [4]. Tangenta na strunju liniju na nekoj ta~ki du` strujne linije predstavlja pravac vektora brzine u toj ta~ki. Na slici 4 prikazane su strujne linije od ulaza prema izlaza za slu~aj ON-DESIGN gdje se jasno vidi pojava vrtlo`nog strujanja u koljenu difuzora. Cilj modifikacije geometrije difuzora je pove}anje koeficijenta . Konus difuzora nije bio predmet

redizajna zbog ~injenice {to je on dimenzionisan prema postoje}oj turbini. Koljeno difuzora tako|e je zadr`alo isti dizajn. Modifikacije geometrije ra|ene su na izlaznom divergentnom dijelu difuzora. Originalni difuzor je odsje~en u zoni prelaza koljena u divergentni dio (

pC

726.0=x m).

6. RESULTS OF SIMULATIONS On Fig. 3 velocity filed for case ON-DESIGN in

0=y cross section is presented. Velocity field

clearly indicated main flow trend in diffuser (reducing velocity to the diffuser outlet which is one of the main role of the diffuser). As an efficient way of flow analysis in the diffuser the streamlines or paths of particles from the inlet to the outlet is emphasized, [4]. Tangent on the streamline in some point along the streamline represent velocity vector direction in this point. Fig. 4 shows the streamlines from the inlet to the outlet for case ON-DESIGN where the swirling structure of the flow in diffuser bend can be clearly recognized. The aim of geometrical modifications is increasing of coefficient . The diffuser cone was not subject of

redesign because of fact that it was designed according to the existing turbine. The diffuser elbow also kept same design. Geometry modifications are performed at the outlet diverging part of diffuser. Original diffuser is cut in zone between elbow and the diverging part (

pC

726.0=x m).

-0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -350

-300

-250

-200

-150

-100

-50

0

50

100

150

Visina izlaza difuzora (m)

Priti

sak

(Pa)

pexp pmin=pexp-pdev pmax=pexp+pdev psim

- 174 -


U orginalnoj izvedbi (ORIG) {irina izlaznog dijela difuzora je bila 1.0 m a visina 1.063 m. Povr{ina popre~nog presjeka izlaza je 1.063 m2. U prvom slu~aju modifikacije {irina izlaza iz difuzora je pove}ana sa 1.0 m na 1.2 m. Povr{ina izlaza je 1.2756 m2 (MOD–1). U drugom slu~aju {irina difuzora je 1.6 m a povr{ina izlaza iz difuzora je 1.7008 m2 (MOD–2). U tre}em slu~aju zadr`ana je ista {irina, dok je visina difuzora smanjena sa 1.063 m na 0.8 m. Povr{ina izlaza je 0.8 m2 (MOD–3). Za svaki slu~aj modifikacije generisana je nova numeri~ka mre`a. Grani~ni uslovi su isti kao i kod originalne geometrije. Uticaj modifikacije geometrije na koeficijent pretvorbe pritiska sumiran je u tabeli 3.

In the original configuration (ORIG) width of outlet part is 1.0 m and height 1.063 m. The outlet surface cross-section is 1.063 m2. In the first case of modification diffuser outlet width is increased from 1.0 m to 1.2 m. The outlet surface cross-section is 1.2756 m2 (MOD–1). In the second case diffuser width is 1.6 m and the outlet surface cross-section is 1.7008 m2 (MOD–2). In the third case the same width is kept but the outlet height is reduced from 1.063 m to 0.8 m. The outlet surface cross-section is 0.8 m2 (MOD–3). For each case of modification new numerical grid is generated. Boundary conditions were the same like in original configuration. The influence of geometrical modifications on the pressure recovery factor is summarized in Table 3.

Slika 3: Polje brzina, slu~aj ON–DESIGN, presjek 0=y

Figure 3: Velocity field, case ON–DESIGN, section 0=y

Slika 4: Strujne linije od ulaza prema izlazu, 64 ta~ke selektovane na ulazu difuzoru Figure 4: Streamlines from the inlet to the outlet, 64 points at the inlet selected

- 175 -


Tabela 3: Vrijednost koeficijenta za razli~ite tipove geometrije za slu~ajeve ON–DESIGN i OFF–DESIGN pC

Table 3: Values of coefficient for different types of geometry for cases ON–DESIGN i OFF–DESIGN pC

Tip geometrije Geometry type

Povr{ina izlaza iz difuzora Aizlaz Diffuser outlet surface Aizlaz

pC

ON–DESIGN

pC

OFF–DESIGN ORIG 1.0630 m2 0.9368 1.0694 MOD–1 1.2756 m2 0.9083 1.0420 MOD–2 1.7008 m2 0.8773 1.0277 MOD–3 0.8000 m2 0.9515 1.0895

7. ZAKLJUÂK U ovom radu je prezentovana stacionarna trodimenzionalna numeri~ka simulacija strujanja u difuzoru sa koljenom i uticaj geometrijskih modifikacija na koeficijent pretvorbe pritiska. Istraìvanje je vr{eno pomo}u programa COMET baziranog na metodi kona~nih volumena. Rezultati simulacija su upore|ivani sa dostupnim eksperimentalnim rezultatima o pritisku na izlaznom zidu difuzora i koeficijentu pretvorbe pritiska. Sa inènjerske ta~ke gledi{ta postignuto je dobro slaganje rezultata. Razlika izme|u numeri~ke i eksperimentalne vrijednosti koeficijenta pretvorbe pritiska je 16% za slu~aj kada turbina radi sa najve}im koeficijentom iskori{tenja, dok je slu~aj kada turbina radi sa smanjenim koeficijentom iskori{tenja ta razlika 4%. Rezultati numeri~kih simulacija prikazani su u dijagramskoj formi, formi polja raspodjela i u formi strujnih linija. Standardni–ε model turbulencije pokazao je dobro osobine u modeliranju turbulentnog toka u difuzoru. Da bi ispitivali uticaj geometrijskih modifikacija na koeficijent pretvorbe pritiska, ura|ene su tri modifikacije originalne geometrije difuzora. U prvom i drugom slu~aju modifikacije pove}avana je povr{ina popre~nog presjeka na izlazu iz difuzora, dok je u tre}em slu~aju ta povr{ina smanjenja. Najve}a vrijednost koeficijenta pretvorbe pritiska dobivena je u tre}em slu~aju modifikacije difuzora i ona je ve}a za pribli no 2% u odnosu na originalnu izvedbu difuzora.

7. CONCLUSION In this paper steady three-dimensional numerical flow simulation of an elbow diffuser and influence of geometrical modifications on the pressure recovery factor is performed. Investigation is carried out with finite volume software COMET. Results of simulation were compared with available experimental results for diffuser outlet walls pressure and pressure recovery factor. From engineering point of view, a good agreement is obtained. The difference between experimental and numerical value of the pressure recovery factor is 16% for the case when turbine operate with the highest efficiency and 4% for the case when turbine operate with reduced efficiency. Numerical results were presented in diagram form, distribution field form and in streamlines form. Standard �-ε model of turbulence showed good performance in modeling of turbulent flow. In order to investigate influence of geometry modification on the pressure recovery factor, three modifications of original diffuser geometry are performed. In the first and second cases of modifications diffuser outlet cross section is increased, while in the third case this cross section is reduced. The highest value of the pressure recovery factor is obtained for the third modification of the diffuser and it is approximately for 2% highest when compared it to the original diffuser.

8. LITERATURA - REFERENCES [1] U. ANDERSSON, Turbine 99-Task and

Boundary Conditions, Vattenfall Utveckling, Älvarleby, Sweden, 1999.

[2] DEMIRD@I], S. MUZAFERIJA, Numerical method

for coupled fluid flow, heat transfer and stress analysis using unstructured moving mesh with cells of arbitrary topology. Comput. Methods Appl. Mech. Engrg., 125:235-255, 1994.

[3] S. MAURI, J.L. KUENY, F. AVELAN, Numerical

Prediction of the Flow in a Turbine Draft Tube-Influence of the Boundary Conditions,

Proceedings of FEDSM'00 ASME 2000 Fluids Engineering Division, Summer Metting, June 11-15, Boston, Massachusetts, USA, 2000.

[4] M. ROTH, R. PIEKERT, Flow Visualization for

Turbomachinery Design, Swiss Center for Scientific Computing ETH Zürich, 1999.

[5] EISINGER, A. RUPRECHT, Automatic Shape

Optimization of a Hydro Turbine Components Based on CFD, Seminar “CFD for turbomachinery application”, Gdansk, 2001.

- 176 -

Ma{instvo 3(8), 177 – 188, (2004) C.Pinca-Bretotean,...: ISTRA@IVANJE IZDR@LJIVOSTI ZAGRIJANIH...

ISTRA@IVANJE IZDR@LJIVOSTI ZAGRIJANIH VALJAONI^KIH VALJAKA U EKSPLOATACIJI

Conf. Eng. Camelia PINCA-BRETOTEAN, Dr. Es Sc.; Lect. Eng. Imre KISS, Drd. Es Sc.; Prof. Eng. Teodor HEPUT, Dr. Es Sc.; Assist. Lect. Eng. Ovidiu TIRIAN, Drd. Es Sc., University “POLITEHNICA” Timisoara, Faculty Engineering Hunedoara, Mechanical Department, Revolutiei 5, Hunedoara, 331128, Romania

REZIME

Istraìvanja o izdr`ljivosti vru}ih valjaoni~kih valjaka u toku eksploatacije predstavljaju va`no nau~no i ekonomsko pitanje. Studija prikazuje detaljni pristup utjecaja razli~itih tehnolo{kih aktora na izdr`ljivost valjaoni~kih valjaka u toku eksploatacije napravljenih od razli~itih vrsta ~elika i sirovog gvo`|a, te predlaè rje{enja s ciljem pove}anja izdr`ljivosti valjaka u toku eksploatacije. Do danas nije napisana referentna lite atura koja bi se detaljno bavila teorijskim i eksperimentalnim aspektima teme ovog istraìvanja.

f

r

r t

il

f rr

r rlit

f t

Prijedlog iznesen u radu da se analiza valjaka za valjanje vr{i kori{tenjem tehnike elektronskog izra~unavanja za izu~avanje toplotnog reìma zagrijanih valjaka za valjanje je novina s nau~nog, ali i eksperimentalnog stanovi{ta. Cilj rada jeste predstaviti nekoliko smjernica koje se odnose na pobolj{anje kvaliteta valjaka, a u nastojanju da se pove}a izdr`ljivost i sigurnost u toku rada.

Klju~ne rije~i: zagrijani valjaoni~ki valjci, izdr`ljivost u toku eksploatacije, toplotni zamor, eksperimentalna instalacija, ~elik, gvo`|e, ciklusi naprezanja

RESEARCHES UPON THE DURABILITY IN EXPLOITION OF THE HOT ROLLING MILL CYLINDERS

Conf. Eng. Camelia PINCA-BRETOTEAN, Dr. Es Sc.; Lect. Eng. Imre KISS, Drd. Es Sc.; Prof. Eng. Teodor HEPUT, Dr. Es Sc.; Assist. Lect. Eng. Ovidiu TIRIAN, Drd. Es Sc., University “POLITEHNICA” Timisoara, Faculty Engineering Hunedoara, Mechanical Department, Revolutiei 5, Hunedoara, 331128, Romania

ABSTRACT

The researches on the durability in exploitation of hot rolling mill cylinde s represent an importan scientific and economical issue. The study represents a detailed approach of the influence of various technological factors on the durability in exploitation of rolling m l cylinders made of different steel and pig iron grades and suggests solutions meant to increase the durability of the rolls in exploitation. Up to this moment, there is no reference publication to minutely deal with the theoretical and experimental aspects o this theme of resea ch. The paper propose rolling cylinde analysis by utilization of electronic calculation technique to study the hot rolling cylinde s the mal regime is a novelty from scientific and experimental viewpoint. The purpose of this work is to present few directions concerning the qua y improvement of the rolls, aiming the increasing o durability and safety in opera ion.

Key words: hot rolling mill cylinders (rolls), durability in exploitation, thermal fatigue, experimental installation, steel, iron, stress cycles

- 177 -


1. UVOD

Valjaoni~ki valjci su dijelovi u valja~kim prugama koji su najpodlo`niji habanju i imaju potro{nju od 0,8 kg/tona valjanog ~elika. U zemlji (Rumuniji) se izvalja 4,5 miliona tona ~elika godi{nje. To predstavlja potro{nju od 3600 tona valjaka, u vrijednosti od 40...50 milijardi leja, {to istraìvanjima daje velik ekonomski i nau~ni zna~aj. O izdr`ljivosti u eksploataciji valjaoni~kih valjaka se malo govori u relevantnoj literaturi, kako u Rumuniji, tako i u svijetu. Do danas nije napisana referentna literatura koja bi se detaljno bavila teorijskim i eksperimentalnim aspektima teme ovog istraìvanja. Na{a istraìvanja nastoje dati odgovore na ve}inu stvarnih problema u vezi s pove}anjem tvrdo}e valjaoni~kih valjaka. Njih karakteri{e sloèn sistem pucanja povr{inskog kalibarskog sloja ili jednostavno pucaju usljed toplotnog {oka kojeg izaziva kontakt vru}eg metala i valjaka hla|enih vodom. Studija predstavlja detaljan pristup utjecaja razli~itih tehnolo{kih faktora na izdr`ljivost kod eksploatacije valjaoni~kih cilindara napravljenih od razli~itih vrsta ~elika i sirovog gvo`|a, te predlaè rje{enja ~iji je cilj pove}anje tvrdo}e valjaoni~kih cilindara u toku eksploatacije.

1. INTRODUCTION The rolling mill cylinders are the parts most subjected to wear in the rolling trains and they represent a consumption of 0.8 kg/tone of rolled steel. Nationwide, 4.5 million ton steel is being rolled every year. This represents a consumption of 3,600 tone rolls, worth of 40… 50 billion lei, which imposes researches with an important economic and scientific impact. The durability in exploitation of the rolling mill cylinders is little approached in the reference literature, both in Romania and worldwide. Up to this moment, there is no reference publication to minutely deal with the theoretical and experimental aspects of this theme of research. Our researches are trying to give answers to most actual problems related to the increase of hardness of rolling mill cylinders. They are characterized by a complex system of cracking of the superficial caliber layer or they simply break because of the thermal shocks caused by the contact of the hot metal with the water-cooled cylinders. The study represents a detailed approach of the influence of various technological factors on the durability in exploitation of rolling mill cylinders made of different steel and pig iron grades and suggests solutions meant to increase the hardness of rolling mill cylinders in exploitation.

2. PREDSTAVLJANJE EKSPERIMENTALNE OPREME

Istraìva~i koriste podatke prikupljene u industrijskoj upotrebi u Fabrici za integralnu proizvodnju ~elika u HUNEDOARI (Rumunija), kao i laboratorijske eksperimente provedene na jedinstvenoj, kompleksnoj i originalnoj instalaciji. Slika 1. predstavlja konstrukcijski plan instalacije za odre|ivanje izdr`ljivosti zagrijanih valjaoni~kih valjaka. Instalacija daje mogu}nost dodatnog izu~avanja kao i odre|ivanja izdr`ljivosti u eksploataciji svih vrsta valjaka koji se trenutno koriste u industrijskim fabrikama. Eksperimenti su ura|eni sa grupama od {est prstenova vanjskog pre~nika od 250 mm, koji su preuzeti iz izu~avanih vrsta industrijskih valjaka (Slika 2.). Imaju}i na umu istraìvanje, napravljene su tri armature uzoraka, svaka sa {est prstenova koji su se sastojali od sljede}ih materijala:

65 VMoCr15 – ~elik koji se koristi za proizvodnju valjaka prethodno valjanje;

55 VMoCr12 – ~elik koji se koristi za proizvodnju valjaka za sekcije te{kog valjanja (te{ke pruge);

90 VMoCr15 – ~elik koji se koristi za proizvodnju valjaka za sekcije te{kog valjanja;

2. PRESENTATION OF THE EXPERIMENTAL EQUIPMENTS

The researches use data collected from the industrial use at the IRON AND STEEL INTEGRATED PLANT of HUNEDOARA (ROMANIA), as well as laboratory experiments carried out on a unique, complex and original installation. Figure 1 presents the construction plan of the installation for determining the durability of the hot rolling mill cylinders. This installation provides the possibility of further studiers and also to establish the durability in exploitation for all types of rolls used presently in industrial mills. The experiments are made on groups of six rings, with a 250 mm exterior diameter, carried out from the studied types of industrial rolls (Figure 2). Having in view the research, three armatures of specimens were made, each with six rings and every ring made of the following materials:

65 VMoCr15 – steel used to manufacture rolls from semi-finished mills;

55 VMoCr12 – steel used to manufacture rolls from heavy section mills;

90 VMoCr15 – steel used to manufacture rolls from heavy section mills;

- 178 -


OTA3 – ~elik koji se koristi za proizvodnju valjaka na sekcijama za te{ko, srednje i lahko valjanje;

FNS2 – gvo`|e koje se koristi za pravljenje valjaka na sekcijama za te{ko valjanje;

FD2 – gvo`|e koje se koristi za pravljenje valjaka na sekcijama za te{ko i lahko valjanje i ì~anim valjaonicama.

OTA3 – steel used to manufacture rolls from heavy, medium and light section mills;

FNS2 – iron used in the making of rolls in heavy section mills;

FD2 – iron used in the making of cylinders in heavy and light section and wire mills.

Slika 1. Konstrukcijski plan instalacije za odre|ivanje izdr`ljivosti zagrijanih valjaoni~kih valjaka 1. glavna osovina 2. eksperimentalni uzo ci; 3,4. leàjev ; 5. asinhroni elektri~ni motor; 6. otpornik

elektri~ne pe} ; 7. kolektor toplo nog napona; 8. osovinica 9,10. spojnice; 11. metalni skelet ; r ii t ;

ill t r i; . , t

FIGURE 1. The construction plan of the installation for determining the durability of the hot rolling m cylinders: 1. main axis; 2. experimen al tryouts (samples); 3,4. bea ings; 5. asyncron electric eng ne; 6.

electric rezistance furnace 7 thermotension collector; 8. pin; 9 10. couplings; 11. metallic skele on

Slika 2. Montaà glavne osnovine i uzoraka u obliku prestena u toku ispitivanja izd `ljivosti rFigure 2. Assembly of main axis and ring shaped samples, Under durability tests

Ovi prstenovi su izloèni razli~itim cikli~nim termi~kim poticajima, pri ~emu su se, s jedne strane, prilikom rotacije glavne osovine zagrijavali u elektri~noj pe}i na razli~itim temperaturama, a s druge strane rashla|ivali u razli~itim okruènjima, odnosno zraku, vodi i uglji~nim snje`nim mlazovima.

These rings were subject to different cyclical thermal solicitations, which, during the period of a rotation of the main axis, on one hand warm up in an electric furnace at different temperatures, and on the other hand cool in different environments, respectively in air, water and carbonic snow jets.

- 179 -


U toku eksperimenata, nakon odre|enog broja ciklusa naprezanja, povr{ina o{trih stranica prstenova pokazuje znakove pucanja usljed termi~kog zamora. Do pucanja dolazi u razli~itim intervalima u toku naprezanja, prema kojima }e se odrediti broj ciklusa. Ovi ciklusi se razlikuju, u zavisnosti od vrste materijala koji se ispituje. U toku eksperimenata biljeì se varijacija temperature na uzorcima u obliku prstena, kao i temperatura u elektri~noj pe}i s automatskim pode{avanjem i odràvanjem na prethodno utvr|enim vrijednostima. Da bi se izmjerila varijacija temperature u ekperimentalnim prstenovima, implantiran je jedan od njih s kupastom osovinicom prvobitno opremljen Pt-Pt/Rh termoelementima. Pre~nik ìce je 0,06 mm, a moment inercije je ispod desetine sekunde. Ovi termoelementi mjere varijaciju temperature na povr{ini uzorka i na dubinama od ∆r = 0; 1,5 i 3 mm. Predstavljeni su zajedno sa unutra{njom montaòm na Slici 3. Osovinica s montiranim termoelementima koja se uklapa u prsten je predstavljena na Slici 4. Nakon utvr|ivanja broja ciklusa naprezanja dok se ne pojavi prva pukotina koju izaziva termi~ki zamor prave se histogrami izdr`ljivosti za svaku vrstu materijala koji se koristi za proizvodnju valjaoni~kih cilindara i za svaku vrstu naprezanja. Rezultati }e se uporediti s onim dobivenim pri industrijskoj upotrebi u vlajaonicama preduze}a «ISPAT SIDERURGICA» u HUNEDOARI.

During the experiments, after a certain number of stress cycles, the surface of the sharp sides of the rings present signs of cracks because of the thermal fatigue. They appear at different intervals during the stress, intervals according to which the number of cycles are to be established. These cycles differ, depending on the type of materials studied. During the experiments the temperature variation is recorded in the ring shaped specimens (samples), as wells as the temperature of the electric furnace with authomatic adjustment and maintenance at previously established values. To perform the measurements of temperature variation in the experimental rings, one of them is implanted with a conical pin with initiaally equipped Pt-Pt/Rh thermocuples. The wire diameter is 0,06 mm and the inertion response under a tenth of a second. These thermocuples measure temperature variation on the surface of the sample and the ∆r = 0; 1,5 and 3,0 mm depths. Tthey are presented together with the interior assemblage, in Figure 3. The pin with the assembled thermocouples to be fit in the ring, is presented in Figure 4. After establishing the number of stress cycles, untill the first thermal fatigue caused cracks appear, durability histograms are done to each type of material, used to manufacture rolling mill cylinders and to each type of stress. The results are to be compared with those in the industrial exploitation of the “ISPAT SIDERURGICA” COMPANY of HUNEDOARA, in the ROLLING MILLS sectors.

-

Slika 3. Montaà kupaste osovinice ugra|ene u ptpt/rh termospojnice

Figure 3. Assembly of conical pin fitted Pt-Pt/rh thermocouplings

Slika 4. Osovina termospojnice montirana i pripremljena za instaliranje u uzorak prstena

Figure 4. Thermocoupled pin assembled and prepared for installation in the ring sample

- 180 -


3. RE@IM RADA Rezultati ispitivanja termi~kog reìma zagrijanih valjaoni~kih valjaka, gdje je zabiljeèno pet izohronih dijagrama koji predstavljaju varijacije temperature valjaka za valjanje u toku jednog obrtaja, {to odgovara vrijednosti ugla 2π pokazuju da se maksimalne asimetri~ne temperature javljaju pri smanjenoj brzini valjanja, odnosno smanjenom broju obrtaja valjaka. Kod pove}anja broja obrtaja valjaka na visokim temperaturama valjanja smanjuje se asimetri~na temperatura polja, pa se smanjuje i termi~ki zamor optere}enja kalibara. Na osnovu toga je izabrana minimalna vrijednost broja obrtaja uzoraka koja je ograni~ena na test izdr`ljivosti, a to je 30,6 o/min, {to izaziva najve}i termi~ki zamor zato {to su termi~ka naprezanja koja se javljaju kao posljedica varijacija temperature maksimalna, a nakon relativno malog broja obrtaja javljaju se prve pukotine usljed termi~kog zamora. Kad je u pitanju temperatura srednje elektri~ne pe}i namijenjena zagrijavanju eksperimentalnih prstenova, ona bi trebala biti {to je mogu}e vi{a kako bi uzorci dostigli stabilan reìm do maksimalne mogu}e temperature. U na{em slu~aju, izra~unata temperatura dvaju otpornika srednje elektri~ne pe}i, od kojih svaki ima ~etiri umotane spirale, je 1000°oC, mi smo dobili 960°±100°oC, a eksperimenti su ostvareni na 910°±100°oC. Kako bi se pove}ao broj ciklusa optere}enja do pojavljivanja prvih pukotina usljed termi~kog zamora, poku{ali smo odràti {to je mogu}e vi{u temperaturu kod testiranja i ubrzati i naglasiti hla|enje. Svaki od tri seta uzoraka koji se sastoje od {est prstenova su bili ograni~eni na reìm rada, teè}i izra~unatom trenutku pojavljivanja prvih pukotina usljed termi~kog zamora i biljeè}i broj ciklusa optere}enja. Na osnovu prethodno predstavljenih podataka odabrali smo tri eksperimentalna termi~ka reìma, ~iji su glavni elementi predstavljeni u tabeli 1. Red eksperimenta je bio: reìm A, B i C. U toku eksperimenata neprestano je biljeèna temperatura srednje elektri~ne pe}i u stacionarnom reìmu (910°oC,) i varijacije temperature u toku jednog obrtaja prstenova, kako na vanjskoj povr{ini tako i u povr{inskom sloju na dubini od ∆r = 1,5 i 3 mm. U toku eksperimentalnog procesa ispitivanja termi~kog zamora primijenjena je tehnika eletronskog ra~unanja koriste}i program – adamth.cpp, na jednom IBM PC kompjuteru sa ADAM-4018 modulom na ulazu i ADAM.4520 konvertorom na izlazu. Na ovaj na~in su zabiljeène cikli~ne varijacije temperature u ta~kama, na povr{ini i u povr{inskom sloju. Dobiveni rezultati su prikazani u tabelama 2, 3, 4. i dijagramima na slikama 5, 6, 7.

3. WORKING REGIMES

From the study of the thermal regime of the hot rolling cylinders, where have been registered five isochronal diagrams representing the rolling cylinders temperature variations during one rotation, corresponding to a 2π circular measure angle, resulted that the maximal asymmetrical temperatures appears at diminishing rolling speed, respectively diminishing rotation numbers of the rolls. Once with the increment of the rotation numbers of the rolls, at high rolling speeds decrease the asymmetric temperature fields so, decrease also the calibers thermal fatigue loading. On these bases, we chose the minimal value for the rotation number of the tryouts constrained to durability test being as 30.6 rot/min, producing the highest thermal fatigue because the thermal tensions appearing as effect of temperature variations are maximal and, after a relative small number of rotations, appear the first thermal fatigue cracks. Regarding the temperature of the electric furnace medium intended for experimental rings warming, this has to be as high as possible in order that the tryouts reach a stabilized regime to a maximal possible temperature. In our case, the temperature of the two resistors electric furnace medium, having four curled spirals each, was calculated to 1000°oC and we obtained 960°±100°oC, but the experiments were effectuated at 910°±100°OC. In order to increase the number of the loading cycles, until the first thermal fatigue cracks appear, we have tried to maintain as high as possible temperature for tryouts, and the cooling fast and accentuated. Each of the three sets of tryouts consisting in six rings were constrained to a working regime, pursuing the calculated moment of the appearance of the thermal fatigue first cracks, registering the number of loading cycles. Based on the previous data presented, we chose three experimental thermal regimes, having the main elements presented in table 1. The order of the experiments was regime A, B and C. During the experiments, was registered permanently the temperature of the electric furnace medium in stationary regime (910°C) and the temperature variations to one revolution of the rings, on the exterior surface as well in the superficial layer at ∆r = 1,5 and 3 mm depth. During the experimental process of durability at thermal fatigue was utilized the electronic calculus technique using a program – adamth.cpp – working on one IBM PC computer, for ADAM–4018 modules at the entrance and ADAM-4520 converter to the exit. In this way has been registered the cyclic temperature variations in points, at the surface and in the superficial layer, the obtained results from the file being showed in tables 2, 3, 4 and the diagrams from figures 5, 6, 7.

- 181 -


Tabela 1. Eksperimentalni re`imi Table 1. The experimental regimes

EKSPERIMENTALNI REŽIMI EXPERIMENTAL REGIMES

NAZIV KARAKTERISTIČNIH ELEMENATA IZ EKSPERIMENTALNOG REŽIMA

THE NAME OF THE CHARACTERISTIC ELEMENTS FROM THE EXPERIMENTAL REGIME

M.U. A B C

Broj obrtaja uzoraka postavljanih na glavnoj osovini

Rotation number of the tryouts mounded on the main axle

[rot / min] 30.6 30.6 30.6

Temperatura srednje lektrične peći The temperature of the electric furnace

medium [°C] 910±100°C 910±100°C 910±100°C

Vrijeme zagrijavanja uzoraka The tryouts warming time

[s] 0.98 0.98 0.98

Vrijeme hlađenja uzoraka The tryouts cooling time

[s] 0.98 0.98 0.98

Ugao uvođenja toplote The heat introduction angle

[rad] π π π

Odstranjenje hla|enja The cooling evacuation

[rad] π π π

Medij za hlađenje The cooling medium

- zrak air

voda koja kruži circulated water

ugljenični snijeg carbonic snow

Tabela 2. Cikli~ne varijacije temperature na povr{ini i u povr{inskom sloju uzoraka u toku re`ima eksploatacije A, sa n=30,6 o/min, pri temperaturi pe}i (960°±10°oC) Table 2. Cyclical temperature variation on the surface and in the superficial layer of samples, exploited in regime A, with n = 30.6 o/min, at a furnace temperature (910± 100C)

VARIJACIJA TEMPERATURE, [°C] THE TEMPERATURES VARIATIONS, [°C]

VARIJACIJA TEMPERATURE, [°C] THE TEMPERATURES VARIATIONS, [°C] No.

∆r = 0mm ∆r = 1.5mm ∆r = 3mm No.

∆r = 0mm ∆r = 1.5mm ∆r = 3mm 0 265.0 246.6 231.3 21 768.4 631.3 502.6 1 338.3 271.1 249.6 22 748.1 620.8 502.3 2 391.2 325.0 264.0 23 708.6 607.0 500.0 3 424.0 355.2 285.3 24 678.3 580.8 490.0 4 482.3 378.0 306.8 25 649.6 555.4 478.2 5 515.2 397.7 329.7 26 605.3 529.8 456.5 6 536.5 416.4 348.3 27 542.5 478.6 420.2 7 586.4 449.0 363.1 28 516.4 458.0 403.0 8 610.2 466.7 373.1 29 479.5 432.7 384.0 9 638.2 480.1 388.0 30 449.7 398.2 358.6 10 655.0 501.4 403.0 31 421.5 365.6 338.2 11 679.8 515.3 419.0 32 392.6 337.5 312.0 12 699.3 537.6 430.0 33 362.4 325.4 300.0 13 713.6 550.3 443.6 34 339.8 302.7 288.1 14 738.5 568.6 452.1 35 309.4 285.0 271.0 15 750.8 582.0 459.6 36 288.1 269.3 258.6 16 765.6 595.3 468.2 37 280.0 257.3 245.3 17 780.2 609.8 477.6 38 271.3 247.0 233.0 18 790.3 618.3 484.4 39 252.6 241.4 228.0 19 795.3 620.5 490.1 40 250.8 242.2 229.2 20 785.3 629.6 497.2 41 265.0 246.6 231.3

- 182 -


Tabela 3. Cikli~ne varijacije temperature na povr{ini i u povr{inskom sloju uzoraka u toku re`ima eksploatacije B, sa n = 30,6 omin, pri temperaturi pe}i (960°±10°oC) Table 3. Cyclical temperature variation on the surface and in the superficial layer of samples, exploited in regime B, with n = 30.6 o/min, at a furnace temperature (910± 100C)

VARIJACIJA TEMPERATURE, [°C] THE TEMPERATURES VARIATIONS [°C]

VARIJACIJA TEMPERATURE, [°C] THE TEMPERATURES VARIATIONS [°C] No.

∆r = 0 mm ∆r = 1.5 mm ∆r = 3 mm No.

∆r = 0 mm ∆r = 1.5 mm ∆r = 3 mm 0 229.7 202.0 162.2 21 764.0 530.4 332.0 1 303.6 251.0 180.4 22 685.3 515.1 330.1 2 345.5 278.0 195.3 23 585.2 498.3 329.0 3 385.2 302.0 210.8 24 520.1 476.2 323.2 4 429.0 333.0 229.3 25 486.7 450.2 318.1 5 460.7 355.6 240.0 26 452.5 427.5 309.9 6 500.6 379.4 253.4 27 408.5 402.3 295.4 7 546.2 403.8 266.5 28 380.9 380.4 278.6 8 573.0 425.9 277.6 29 316.0 346.0 262.8 9 590.0 444.0 284.4 30 293.0 323.0 251.6 10 627.4 460.5 293.3 31 274.0 303.8 235.2 11 645.3 475.0 300.2 32 255.3 295.2 223.4 12 662.0 488.2 309.5 33 244.0 280.4 207.8 13 683.0 499.6 313.6 34 229.6 262.8 195.4 14 709.8 509.7 320.0 35 216.0 248.9 190.0 15 728.6 518.8 323.3 36 208.6 236.0 182.4 16 748.2 524.1 327.6 37 197.0 218.6 175.3 17 760.1 529.5 329.3 38 195.2 209.7 165.2 18 773.0 532.0 331.1 39 204.1 197.0 160.0 19 788.5 537.2 333.4 40 207.3 189.0 165.4 20 788.0 539.1 335.6 41 229.7 202.0 162.2

Tabela 4. Cikli~ne varijacije temperature na povr{ini i u povr{inskom sloju uzoraka u toku re`ima eksploatacije C, sa n = 30,6 o/min, pri temperaturi pe}i (960°±10°oC) Table 4. Cyclical temperature variation on the surface and in the superficial layer of samples, exploited in regime C, with n = 30.6 o/min, at a furnace temperature (910± 100C).

VARIJACIJA TEMPERATURE, [°C] THE TEMPERATURES VARIATIONS [°C]

VARIJACIJA TEMPERATURE, [°C] THE TEMPERATURES VARIATIONS [°C] No.

∆r = 0 mm ∆r = 1.5 mm ∆r = 3 mm No.

∆r = 0 mm ∆r = 1.5 mm ∆r = 3 mm 0 223.1 180.3 140.6 21 737.0 492.4 293.0 1 280.6 200.1 152.3 22 654.4 474.6 286.0 2 357.0 232.4 173.5 23 542.7 450.0 275.2 3 410.3 259.7 190.2 24 472.9 429.0 270.5 4 470.7 287.6 207.5 25 428.7 405.6 265.4 5 506.3 314.0 220.2 26 382.8 375.1 256.3 6 529.0 332.3 229.4 27 321.0 350.2 243.1 7 564.2 358.3 241.6 28 284.0 310.9 234.0 8 586.8 375.1 250.2 29 250.6 291.0 220.5 9 612.5 393.4 260.1 30 233.4 270.0 205.6 10 635.8 413.1 269.3 31 223.2 255.0 196.5 11 653.2 430.0 275.9 32 212.9 248.4 186.3 12 672.4 445.3 280.1 33 205.7 233.0 180.1 13 688.3 455.2 285.6 34 202.1 222.0 175.6 14 702.5 469.7 290.0 35 198.5 212.0 168.0 15 715.6 480.2 293.1 36 193.4 202.0 163.0 16 729.7 485.4 294.2 37 195.7 195.0 152.6 17 739.8 494.5 298.1 38 199.8 183.0 150.1 18 751.6 500.0 299.0 39 204.0 176.0 142.0 19 762.6 503.3 299.6 40 208.0 174.0 140.0 20 760.0 503.0 298.3 41 223.1 180.3 140.6

- 183 -


Slika 5. Cikli~ne varijacije temperature u ta~kama, na povr{ini i u povr{inskom sloju (za reìm A, B i C)

Figures 5. The cyclic temperature variations in points, at the surface and in the superficial layer (for the regime A, B and C)

Slika 6. Cikli~ne varijacije tempe ature u ta~kama, na povr{ini i u povr~inskom sloju (za reìm A, B i C) r

.Figures 6 The cyclic temperature variations in points, at the surface and in the superficial layer (for the regime A, B and C)

Slika 7 Cikli~ne varijacije tempe ature u ta~kama, na povr{ini i u povr{inskom sloju (za reìm A, B i C) . r

.Figures 7 The cyclic temperature variations in points, at the surface and in the superficial layer (for the regime A, B and C)

- 184 -


4. REZULTATI I ANALIZE Analiziraju}i dijagrame varijacija temperature, koje se smatraju izohronim stanjima, u toku termi~kog zamora eksperimentalnih stanja testiranja kod reìma A, B i C, moèmo primijetiti da je najvi{a temperatura zabiljeèna na vanjskoj povr{ini prstenova i iznosi 795,3°oC, pri reìmu A kad je hla|enje vr{eno na otvorenom zraku. Pri reìmu B, sa sistemom hla|enja kruènjem vode, krivulja varijacija temperature ima manje nagla{en pad u oblasti ugla hla|enja, dostiù}i maksimalnu temperaturu na povr{ini prstena od 788,5°oC i minimalnu temperaturu izme|u 160°oC...192,2°oC. Pri reìmu optere}enja C kori{ten je led od karbon dioksida koji uduvava distributivni kolektor, krivulja varijacija temperature postaje, u podru~jima hla|enja, jo{ vi{e nagla{ena, s maksimalnom temperaturom na povr{ini prstenova od 762,6°oC i minimalnom temperaturom u povr{inskom sloju od 140°oC. Sinteze karakteristi~nih podataka za biljeènje varijacije temperatura u eksperimentalnim reìmima optere}enja A, B i C predstavljene su u tabeli 5.

4. RESULTS AND ANALYSES

Analyzing the temperature variations diagrams, considered as isochronal estates, during the thermal fatigue experimental estates of the tryouts in A, B and C regime, we can observe that the highest registered temperature on the exterior surface of the rings was 795,3°C, in the A regime when the cooling has been effected in open air. In the B regime, having a recycling water bath cooling system, the temperature variations curves have a less accentuated downgrade in the area of the cooling angle, reaching the maximal temperature on the rings surface 788,5°C, and the minimal temperature between 160°...195,2°C. In the C loading regime was used carbon-dioxide ice blasted in by a distributive collector, the temperature variations curves becoming, in cooling area, even more accentuated, the maximal temperature on the rings surface being 762.6°C, and the minimal temperature in the superficial layer 140°C. The synthesis of the characteristic data for the registered temperature variations in the experimental loading regime A, B and C are presented in the table 5.

Tabela 5. Sinteza karakteristi~nih podataka cikli~ne varijacije temperature sa povr{inskog sloja vrste uzorka prstena eksperimentalno eksploatisanih u reìmu A, B, C Table 5. Sinthesis of the characteristic data for cyclical variation of temperature, from the sauperficial layer of ring typed tryout experimentally exploited in A, B, C regime

GRANIčNA VARIJACIJA TEMPERATURE KOJA JE REZULTAT EKSPERIMENATA [0C] LIMIT TEMPERATURE VARIATION

RESULTING FROM EXPERIMENTS [0C]

DIJAGRAM CIKLIČNIH VARIJACIJA TEMPERATURE PREMA

EKSPERIMENTALNOM REŽIMU EKSPLOATACIJE

DIAGRAM OF THE CYCLICAL VARIATION OF THE TEMPERATURES, ACCORDING TO EXPERIMENTAL EXPLOITATION REGIME

DUBINA POVRŠINSKOG SLOJA ∆r [mm] DEPTH OF THE

SUPERFICIAL LAYER ∆r [mm] Maximal Minimal

0 793.3 250.8 1.5 631.3 241.4

EKSPERIMENTALNO NAPREZANJE U REŽIMU "A" (vidi SLIKU 5.)

EXPERIMENTAL STRESS „A” REGIME (see FIGURE 5) 3.0 502.6 228.0

0 788.5 195.2 1.5 539.1 187.0

EKSPERIMENTALNO NAPREZANJE U REŽIMU "B" (vidi SLIKU 6.)

EXPERIMENTAL STRESS „B” REGIME (see FIGURE 6) 3.0 335.6 160.0

0 762.6 204.0 1.5 503.3 174.0

EKSPERIMENTALNO NAPREZANJE U REŽIMU "C" (vidi SLIKU 7.)

EXPERIMENTAL STRESS „C” REGIME (see FIGURE 7) 3.0 299.6 140.0

Op}a opservacija jeste da za sva tri registrirana dijagrama, krivulje varijacije temperature imaju izvjesno pomjeranje na apscisi, {to pokazuje vrijeme prijenosa toplote u masi prstena, odnosno povr{inskom sloju. Sli~na je situacija u procesu koji je suprotan hla|enju, gdje je nagla{enost u reìmima B i C, kad se povr{ina prstena brè hladi, a povr{inski sloj na dubini od ∆∆r = 1,5 mm ostaje zagrijan vi{im temperaturama od onih na povr{ini.

As a general observation, for all the three registered diagrams, the temperature variations curves peaks have a certain displacement on the abscissa, fact that indicates that the heat transmitting time in the rings mass, respectively in the superficial layer. The situation is similar in a reverse way to the cooling process too, being more accentuated in the B and C regimes, when the rings surface cools faster and the superficial layer at the ∆r = 1,5 mm depth remains warm up by higher temperatures that the surface ones.

- 185 -


U toku eksperimenata sa izdr`ljivo{}u, reìmi A, B i C su odvojeno primijenjeni kod svakog skupa uzoraka formiranih od {est prstenova, a koji predstavljaju 6 ispitivanih vrsta ~elika i livenog gvo`|a s ciljem vizualizacije pojavljivanja prvih pukotina usljed termi~kog zamora. Prve pukotine usljed termi~kog zamora javljaju se na o{trim bo~nim vanjskim krajevima s maksimalnim pre~nikom φ250 mm na svakom prstenu montiranom na zaptivku, nakon odre|enog utvr|enog broja ciklusa termi~kog optere}enja. Vizualizacije koje su napravljene s ciljem posmatranja pukotina termi~kog zamora ra|ene su dva puta dnevno, ra~unaju}i broj ciklusa koji pro|e nakon svake vizualizacije. Nakon eksperimenata za testiranje izdr`ljivosti kod eksploatacije, koji su ocijenjeni u ciklusima termi~kog optere}enja, napravljeni su histogrami izdr`ljivosti za svaki reìm optere}enja i za svaku vrstu ispitivanog materijala, a rezultati su predstavljeni na slikama 8, 9 i 10.

During the durability experiments, after the A, B and C regime, applied separate for each set of tryouts formed of six rings, representing the 6 studied steel and cast iron marks, aiming by visualization the appearance of the first thermal fatigue cracks. These first thermal fatigue cracks appear on the sharp lateral exterior edges at a Φ250 mm maximal diameter, on each ring assembled in the packing, after a certain determined number of thermal loading cycles. The visualizations made in order to observe the thermal fatigue cracks were made twice per day, calculating the number of cycles passed after each visualization. After the experimental exploiting durability tests, evaluated in thermal loading cycles, were made durability histograms, for each loading regime and for each mark of studied material, the results being presented in figures 8, 9 and 10.

5. ZAKLJU^CI

• Kod reìma optere}enja A ispitivani materijali su imali najve}i otpor pri ciklusima naprezanja do pojavljivanja prvih pukotina usljed termi~kog zamora (reìm optere}enja);

• Kod reìma optere}enja B, prve pukotine usljed termi~kog optere}enja su se javile kod manjeg broja ciklusa naprezanja (srednji reìm);

• Kod reìma optere}enja C, pukotine usljed termi~kog zamora su se javile kod najmanjeg broja ciklusa naprezanja (te{ki reìm);

• Vrsta naprezanja koja je dala najbolje rezultate u vezi sa stabilno{}u termi~kog zamora – ispitivana u reìmu naprezanja C – jeste vrsta ~elika OTA3;

• Vrste 65 VMoCr15, 55 VMoCr12 i 90 VMoCr15 relativno su dobro izdràle naprezanje termi~kog zamora kod reìma naprezanja A i prihvatljivo dobro kod reìma naprezanja B i C;

• U slu~aju dvaju vrsta gvo`|a kori{tenih u ekperimentalnom istraìvanju, uo~eno je bolje pona{anje kod FNS2, koji je pro{ao 173.000 ciklusa pod reìmom naprezanja C do pojavljivanja prvih pukotina usljed termi~kog zamora;

• Vrsta gvo`|a FD2 se pona{a prihvatljivo i koristi se za proizvodnju tvrdih valjaka na stalcima za zavr{nu obradu.

5. CONCLUSIONS

In stress regime A, the materials under study resisted longest at stress cycles, untill the first thermal fatigue cracks appeared (loading regime);

In stress regime B, the first thermal fatigue cracks appeared in a smaller number of stress cycles (medium regime);

In regime C, the thermal fatigue cracks appeared at the lowest number of stress cycles (heavy regime).

The type of stress which gave the best results regarding stability to thermal fatigue – studied in stress regime C – is the OTA3 steel type;

Types 65 VMoCr15, 55 VMoCr12 and 90 VMoCr15 underewent relatively well to the stress of thermal fatigue in stress regime A and acceptably well in stress regimes B and C;

In the case of the two types of iron used in experimental research, a better behaviour was noticed at FNS2, which underwent to 173,000 cycles in stress regime C, until the first cracks of thermal fatigue;

Iron type FD2 behaves acceptably and is used to produce hard rolls from finishing stands.

- 186 -


Slika 8. Histogrami izdr`ljivosti (za reìm A, B, i C) Figures 8 Durability histograms (for the regime A, B and C) .

Slika 9. Histogrami izdr`ljivosti (za reìm A, B, i C) Figures 9 Durability histograms (for the regime A, B and C) .

Slika 10 Histogrami izdr`ljivosti (za reìm A, B, i C) .Figures 10. Durability histograms (for the regime A, B and C)

- 187 -


6. LITERATURA - REFERENCES

[1] Camelia PINCA-BRETOTEAN, Ştefan TOADER, Dorin PLESA: Considerations Concerning the Impact of Thermal Fatigue Upon the Hot Rolling Cylinders, VIIth International Symposium Interdisciplinary Regional Research – ISIRR2003, Hunedoara, pg. 494…499;

[2] Camelia PINCA-BRETOTEAN, Ştefan

TOADER, Dorin PLESA: Considerations Concerning the Impact of Thermal Fatigue Upon the Hot Rolling Cylinders, Annals of the Faculty of Engineering Hunedoara, 2004, Tom II, Fascicola 1, pg. 63...69;

[3] Camelia PINCA-BRETOTEAN: Cercetări de

durabilitate în exploatare a cilindrilor de laminor la cald, GRAN CNCSIS, TEMA 6/24;

[4] Ştefan TOADER, Camelia PINCA-

BRETOTEAN: Cercetări asupra rezistenţei la

oboseală termică a cilindrilor de laminor la cald, GRAN CNCSU, 5004/1996


BRETOTEAN: Consideraţii privind determinarea analitică a tensiunilor termice în cilindri de laminor la cald, Buletinul Stiintific al UPT, 1998, Fasc.1, pg. 15…22;

[6] Camelia PINCA-BRETOTEAN: Consideraţii

privind determinarea tensiunilor termice ce acţionează în cilindri de laminor din caja unui laminor blooming Ø1300 mm, Buletinul Stiintific al UPT, 1999, Fasc.2, pg. 33…41;


BRETOTEAN: Cercetări asupra variaţiilor de temperatura în cilindri cajei pregatitoare dintr-un laminor de profile grele Ø650 mm, Annals of the Faculty of Engineering Hunedoara, 1997, Tom I, pg. 282...289;

- 188 -

Ma{instvo 3(8), 189 – 194, (2004) V.Barto{ova,...: O EFIKASNOSTI PROIZVODNOG PROGRAMA...

O EFIKASNOSTI PROIZVODNOG PROGRAMA MA[INSKIH URE\AJA KOMPANIJE

Ing. Viera Bartošová, PhD., Žilinská univerzita v Žiline, Fakulta PEDaS, Katedra ekonomiky, Univerzitná, 8215/1, 010 26 Žilina, Slovenská republika, tel.: +421 41 5133 224, e-mail: [email protected] doc. Ing. Jozef Majerčák, PhD., Žilinská univerzita v Žiline, Fakulta PEDaS, Katedra ekonomiky, Univerzitná, 8215/1, 010 26 Žilina, Slovenská republika, tel.: +421 41 5133 410, e-mail: [email protected]

REZIME

Svaka kompanija organizira svoj process proizvodnje tako da osigura trajni prosperitet u datim vanjskim uslovima i prema svojim unutra{njim mogu}nostima. U na{em doprinosu obavje{tavamo vas o iskustvima preduze}a za proizvodnju ma{inskih ure|aja srednje veli~ine.

Klju~ne rije~i: proizvodni program, konkurentna sposobnost, pore|enje proizvoda, tehni~ko-ekonomski nivo proizvoda.

TO THE ISSUE OF EFFICIENCY OF MACHINERY COMPANY’S PRODUCTION PROGRAM

Ing. Viera Bartošová, PhD., Žilinská univerzita v Žiline, Fakulta PEDaS, Katedra ekonomiky, Univerzitná 8215/1, 010 26 Žilina, Slovenská republika, tel.: +421 41 5133 224, e-mail: [email protected] doc. Ing. Jozef Majerčák, PhD., Žilinská univerzita v Žiline, Fakulta PEDaS, Katedra ekonomiky, Univerzitná 8215/1, 010 26 Žilina, Slovenská republika, tel.: +421 41 5133 410, e-mail: [email protected]

SUMMARY

Each company organizes its production process in such a way, that in the given external conditions and in its real internal possibilities, the permanent prosperity is secured. In our contribution we inform about the experiences of a middle size machinery company .

Key words: production program, competitive ability, product comparison, technical-economic level of the product.

1. UVOD Postoji nekoliko ideja o poslovnim strategijama iz kojih se razvijaju poslovne aktivnosti preduze}a. Jedna od najva`nijih, ako ne i primarnih i osnovnih pretpostavki o uspje{noj strategiji svake kompanije jeste imati dobar, komercijalno uspje{an proizvodni program. Koji je uslov ili nu`na pretpostavka dugoro~nog trajnog postojanja kompanije? Postoji niz internih, ali i vanjskih situacija i uslova za koje je potrebno sistemati~no provo|enje mota “biti trajno uspje{an”.

1. INTRODUCTION

There are more business conception strategies from which rise business activities of the company. One of the most important, if not the primary and fundamental assumption of the successful strategy of every company, is to have good, commercially successful production program. What is a requirement or necessary assumption of long-term permanent existence of the company? There are number of inter nal but also external situations and conditions which need motto „to be permanently successful” systematically implement.

- 189 -


Ne postoji, i mo`da nikad ne}e ni postojati, univerzalni i potpuno konkretan vodi~. Svaka kompanija mora izabrati i provesti svoj vlastiti na~in prema prosperitetu u budu}nosti u odnosu na broj specifikacija, te unutra{njih i vanjskih uslova i situacija. Ovo je realnost ve} mnogo godina, {to je i generalizirano u teoriji organizacijskog i strate{kog upravljanja, kao i upravljanja ~injenicama. Strategija prosperiteta se s pravom smatra najprogresivnijim na~inom ka budu}nosti kompanije. Iskustvo mnogih poslovnih subjekata dokazuje da ova strategija ima i imat }e ~itav spektar varijanti s razli~itom dubinom i karakterom svog ostvarivanja, po~ev{i od malih sistemati~nih koraka ka kontinuiranom razvoju trì{ta i produktivnog standarda kompanije.

There is no, and maybe hardly ever will be, available universal and entirely concrete guide. Each company must with regard to number of specifications, internal and external conditions and situations choose and implement its own way toward the future and prosperity. This is many years’ history reality and it is generalized in theory of organizational and strategic management as well as in managing changes. Prosperity strategy is correctly considered as the most progressive way toward the future of the company. An experience of many business subjects prove, that this strategy has and will have whole spectrum of variants with different depth and character of its accomplishment, from the small systematic steps toward the continuing development of market and productive standard of the company.

2. METODE IZRAÛNAVANJA EFIKASNOSTI I STOPE POTENCIJALNOG POSLOVNOG USPJEHA PROIZVODA

Efikasnost ili stopa potencijalnog poslovnog uspjeha proizvoda je unutar tipa fukncionalnog tro{ka koji se smatra efektivno{}u i koristi mjerenog proizvoda ili usluge. U pragmati~nom smislu potrebna efikasnost zna~i “RADITI STVARI I USLUGE DOBRO”, a potrebno kori{tenje zna~i “RADITI DOBRE STVARI I USLUGE”. Povezivanjem ova dva faktora efektivnosti proizilazi princip “RADITI DOBRE STVARI DOBRO”. Prije nego {to kompanije i organizacije lansiraju novi proizvod ili uslugu na trì{te, one ispituju po kojoj }e se cijeni proizvod vjerovatno prodavati, te da li njihova cijena «probija» cijenu kojom vlada trì{te. Konkurentno pore|enje proizvoda ima zna~ajan utjecaj na funkcije odredi{ta inovativnih namjera. To je danas uslovni korak ne samo za objektiviziranje poloàja proizvoda na trì{tu (konkurentna sposobnost) nego i za specificiranje inovativnih namjera, tehnolo{ku pripremu proizvoda i pronalaènje finansijskih resursa za restrukturiranje proizvodnih programa s kojima je proces revitalizacije obi~no povezan. Pore|enje industrijskih proizvoda predstavlja jedan od nu`nih koraka objektivizacije u~enja o konkurentnoj sposobnosti proizvodnje na relevantnom trì{tu. Bez te analize ne moèmo znati ni slabosti ni ja~e strane proizvoda i njegov tehni~ko-komercijalni potencijal na relevantnom trì{tu.

2. METHODS OF EFFICIENCY CALCULATION AND POTENTIAL BUSINESS SUCCESS RATE OF THE PRODUCT

Efficiency or potential business success rate of the product is within a functionally expenditure type considered as an effectiveness and utility of measured product, service. In pragmatic conception a required effectiveness means „TO DO THE THINGS AND SERVICES WELL“ and required utility means „TO DO GOOD THINGS AND SERVICES“. From connection of two effectiveness factors the principle „TO DO GOOD THINGS WELL“ rises. Before companies and organizations launch a new product and service to the market they examine at what price will be a product probably sold and whether their costs „do not bust“ the price ruled by market. A competitive comparing of the products considerably influences destination functions of innovative intentions. It is conditional step not just for the objectification of the product position at the market nowadays (competitive ability), but also for the specification of the innovative intentions, technological preparation of the production and finding the financial resources at the restructuring of the production programs, which the revitalization process is often connected with. The comparing of industry production products ranks among necessary steps for the objectification of learning the competitive ability of production at the relevant market. Without this analysis we are not able to know objectively both the weakness and strengths of the production and its technically-commercial potential at the relevant market.

- 190 -


Za one koji vr{e pore|enje proizvoda neophodno je u metodologiji naglasiti rije~i na relevantnom trì{tu jer nije mogu}e porediti parametre proizvoda na razli~itim trì{tima gdje vrijede druk~ije navike korisnika u dràvama i proizvodi se koriste u razli~itim tehnolo{kim i prirodnim uslovima. Primjena matematike u pore|enju proizvoda generira nekoliko statisti~kih metoda koje se mogu koristiti u analizi poloàja ispitivanog proizvoda u odnosu na proizvode na konkurentnom poloàju. Sljede}e multikriterijske metode su me|u najzna~ajnijim metodama statisti~kog pore|enja proizvoda:

Korter, ra~unanje tehni~ko-ekonomskog nivoa

proizvoda kompanije je za potrebe rangiranja jedna od najboljih metoda s obzirom na to da funkcionira i s tehni~kim i s trì{nim parametrima:

• po~inje traènjem proizvoda s kojim se moè vr{iti pore|enje (proizvod iz iste klase, ista komercijalna upotreba od strane vi{e vode}ih svjetskih proizvo|a~a),

• izbor parametara koji najbolje odslikavaju njegov tehni~ki, cjenovni i nivo kori{tenja. To predstavlja jedan od najva`nijih koraka metodologije i iziskuje poznavanje nauke o robi,

• dati va`nost sastavnim parametrima u kompleksu parametara,

• obrazloìti trì{nu cijenu proizvoda spektru tehni~kih parametara i izra~unati tehni~ko-ekonomski nivo.

Procenti, ra~unanje procentualnog izraàvanja

kvaliteta parametara u odnosu na najbolji kvalitet ili vrijednost. Ra~unanje uklju~uje parametre svih uporedivih proizvoda; dobivene procentualne vrijednosti kvaliteta proizvoda koji se odnose na najve}i zbir bit }e izbrojani za pojedine proizvode. T-ta~ke, princip ovog metoda se zasniva na

pore|enju istih kvaliteta kori{tenja proizvoda metodom: “standardne skale Z”, rad sa aritmeti~kom sredinom i srednjim kvadratnim odstupanjem; metod omogu}ava rangiranje:

• proizvoda uzimaju}i u obzir izabrani parametar,

• datih parametara proizvoda (supernormalnost ili mediokritet),

• cijelog tehni~kog nivoa. Mjerni indikatori, ove statisti~ke metode su

adekvatno opisane u literaturi koja se bavi statisti~kim problemima. Njihovi algoritmi su dio softverskih proizvoda (aplikacijskih programa), ali i ra~unara namijenjenih za nau~no i statisti~ko ra~unanje.

For the solvers of comparing the products it is necessary to emphasis in the methodology the words at the relevant market, because it is not possible to compare the parameters of the product at different markets, where are valid different national user habits and these products are used under the different technological and natural conditions. Mathematics appliance generates for the comparing of products a few statistic methods applicable for analysis of the examined product position with products in competitive position. Among the most significant methods of the statistic comparing the products are multi-criterial methods:

Korter, the calculation of technically-economic

level of the company’s product is for the needs of the ranking one of the best methods, as it works with both technical and market parameters:

• it stars with looking up the comparable products (same class product, same commercial usage by more leading world producers),

• choosing of the parameters the best portraying its technical, price and utility level. It ranks among the most important steps of the methodology and requires the science of commodities knowledge,

• give weight to constituent parameters in the complex of parameters,

• account for the market price of the products to the spectrum of technical parameters and the calculation of technically-economic level.

Percentiles, the calculation of percentile quality

expression of the parameters in relation to the best quality or value. The calculation includes parameters of all comparable products; the acquired percent values of products’ qualities relating to the highest sum will be counted up for individual products. T-points, the principle of this method is based on

the comparing of the same utility qualities of products through the method: “of standard scale Z”, working with the arithmetic mean and mean square deviation; the method enables to do ranking:

• of products considering to choosen parameter, • given product parameters (supernormality or

mediocrity), • of complete technical level.

Mensural indicators, these statistics methods are

adequately described in literature with statistic issue. Their algorithms are part of software products (application programs), but also calculators intended for scientific and statistics calculations.

- 191 -


3. OSNOVNE KATEGORIJE FUNKCIONALNOG PRISTUPA TRO[KOVIMA

Funkcionalnost proizvoda, a time i zadovoljstvo kupca datim proizvodom ili uslugom, je jedan od osnovnih kriterija kojima se mjeri potencijalni komercijalni uspjeh proizvoda ili usluge. Prava funkcionalnost proizvoda se shva}a kao stopa njegove cjelisho-dnosti i kvaliteta. Ona nastaje iz primarne kategorije funkcionalnog pristupa tro{kovima, odnosno funkcije. Funkcija proizvoda ili usluge predstavlja odnos izme|u potrebe kupca i kvaliteta proizvoda ili usluge. Ona, dakle, pokazuje koje su potrebe kupca i, s druge strane, istovremeno odraàva ono u ~emu je proizvod zapravo dobar i {ta on radi. Stopu kori{tenja ili obim kori{tenja proizvoda ili usluge moèmo otkriti samo iz onih kvaliteta kori{tenja koji su kupcu va`ni, a ~iji se efekat izraàva kao funkcija. Ti relevantni kvaliteti kori{tenja proizvoda su kvantitativne i kvalitativne prirode i u literaturi se nazivaju karakteristikama.

Parametar je aritmeti~ki iznos odre|enog kvaliteta opisan odgovaraju}om koli~inom mjernih jedinica ili posebno kvalitativnim opisima (u odnosu na izabranu skalu bodovanja).

Funkcionalnost i njena stopa omogu}avaju ocjenjivanje proizvoda i usluge kao jednog kompleksa i to na~in na koji ih kupac vidi. Kad se rangira funkcionalnost dobro je slijediti skup funkcija, njihove karakteristike i parametre koji opisuju i mjere proizvod ili uslugu kao cjelinu. Op}enito se odnosi na ove funkcije i karakteristike:

• interne funkcije – opisuju nivo proizvoda i usluge,

• potro{a~ke funkcije – opisuju glavnu svrhu kori{tenja,

• funkcije usluge – karakteriziraju pouzdanost, mogu}nost upravljanja, vijek trajanja, sigurnost pri radu i sli~no,

• komercijalne funkcije – karakteriziraju nivo priloga i aparata, dostupnost rezervnih dijelova, brzinu isporuke, blagovremenu isporuka, sloène usluge itd.,

• socijalno-ekonomske funkcije koje izraàvaju odnose proizvoda ili usluge i ìvotnih stavova, te odnos njih i ekologije i sl.

Svaki proizvod i svaka usluga (i njihova funkcionalnost) karakteriziraju razli~ite osobine izraène razli~itim koli~inama i njihovim mjernim jedinicama. Sa stanovi{ta metodologije nije mogu}e izra~unati tako razli~it skup kvantitativnih podataka i izraziti ukupnu stopu funkcionalnosti proizvoda ili usluge.

3. BASIC CATEGORIES OF THE FUNCTIONALLY EXPENDITURE APPROACH

Product functionality, thus the rate of consumer’s satisfaction with given product or service is one of the two basic criteria, which measure the potential commercial success of the product or service. The proper functionality of the product is understood as its expediency and quality rate. It arises from primary category of a functionally expenditure approach, that is function. The function of the product or service represents relation between a customer need and product or service qualities. It shows then, what are the needs of the customer and simultaneously on the other hand it reflects at what is product actually good and what it is doing. To find out the utility rate or utility extent of the product or service is possible only from those, for customer important utility qualities, whose effect is expressed as the functions. These relevant utility qualities of the product are by quantitative and qualitative nature and in literature they are called characteristics.

Parameter is an arithmetic amount of a certain quality described with appropriate quantity in measure units or specifically in qualitative descriptions (according to choosen point scale). Functionality and its rate enable to apprise the product and service as a complex, in the customer’s way of looking at it. When ranking the functionality, it is appropriate to follow the set of functions, their characteristics and parameters which describe and measure the product or service as a whole. Generally it concerns these functions and characteristics: • internal functions – describing the level of

products and service, • consumable functions – describing the main

purpose of using, • service functions – characterizing the

reliability, manageability, lifetime and operation safety and the like,

• commercial functions – characterizing the level of attachment and accessories, spare parts accessibility, delivery promptness, on time delivery, complex services and so on,

• socially-economic functions expressing relations of the product or services towards life attitudes, towards ecology and so on.

Each product and each service (and their functionality) characterize various qualities expressed through the different quantities and their measure units. From the methodology point of view it is not possible to count such a diverse set of quantitative data, and articulate a total rate of the product or service functionality.

- 192 -


O~igledno je da se razmatra zajedno s pitanjem multikriterijskog rangiranja, {to mora biti pra}eno metodolo{kom primjenom za odre|ivanje vrijednosti funkcionalnog zna~enja (kriterija) zajedno sa metodolo{kom primjenom koja kvantificira vrijednosti kori{tenja (korisnost ili funkcionalnost). Na{ doprinos se zasniva na poznavanju ~injenice da je prvenstvena, ali ne i jedina pretpostavka uspje{ne strategije ma{inskih ure|aja kompanije imati dobar, komercijalno uspje{an proizvodni program. Da bi zaista postao polazi{te prosperiteta jedne kompanije, proizvodni program bi trebalo da se postepeno transformira u proizvodni plan, u nivo konkretne narud`be, i u konkretne proizvode, koji se proizvode u uslovima konkretne proizvodno-tehni~ke baze. Ovi proizvodi moraju prona}i na~in prema kupcu na osnovu obostrano najbolje cijene. Proizvodnja shva}ena kao veza proizvodnih faktora (posebno rada i kapitala) u odlu~ivanju o stopi ima utjecaja na efikasnost kompanije i njenu konkurentnu sposobnost. Vrh rukovodstva kompanije mora neprestano stvarati harmoniju horizontalnih odnosa, tj. izme|u proizvodnje, prodaje, ulaganja, finansiranja, kupovine i produktivne saradnje. To sve iziskuje stvaranje i trajno aùriranje informacijske baze, uklju~uju}i i informacije o: • ekonomskoj i finansijskoj situaciji u kompaniji

(dijagnostika i finansijsko zdravlje), • relevantnom trì{tu (veli~ina, razvoj, potraìvanja), • konkurentnosti (glavni konkurenti, mogu}e

zamjene, prepreke ulasku u tu granu), • prodaji i marketingu odlu~uju}ih proizvoda i

njihovom pore|enju sa konkurentima podijeljeni prema sljede}em: - cijene i prora~uni cijena, - reklame i tro{kovi u vezi s njima, - istraìvanje, razvoj i unapre|enje pripreme za tehni~ku proizvodnju (sposobnost trgovca – dizajnera da ubijedi kupca u visoki nivo tehni~kih rje{enja i tehni~ke pouzdanosti tipi~ne za produktivno-tehni~ku osnovu, - proizvodni nivo, kontrole, proizvodni

kapaciteti i stabilnost tro{enja, dobavlja~i i logisti~ki nivo,

- radnici – kvalifikacije i dobna struktura, zadovoljstvo, raspoloènje, radna klima, stabilnost i obrt.

Iz osnovnog iskustva se mogu izvu}i op}i zaklju~ci, a to je da bi strate{ka analiza proizvodnog programa svake kompanije trebalo da teì prvenstveno sljede}em:

• analiza i prognoza relevantnog trì{ta, • analiza konkurenata i unutra{njih resursa i

potencijala kompanije, • prognoza prihoda kompanije.

Evidently it is concerned in issue of the multi-criterial ranking, that must be accompanied with methodical appliance for determination the values of functionally meaning (criteria), added with methodical appliance quantifying the utility values (usefulness or functionality). Our contribution is based on knowledge, that primary, but not the only assumption of the successful machinery company’s strategy is to have good, commercially successful production program. To become really the starting point of the company’s prosperity, a production program needs to be gradually transformed into the production plan, into the level of concrete order, into the concrete products, producing under the conditions of concrete productively-technical base and these products must find the way forward to a customer on the base of a mutually best price. Production understood as connection of factors of production (especially work and capital) in deciding rate influences the efficiency of the company and its competition ability. Top management of the company must continually creates harmony of horizontal relations, i.e. among production, sales, investment, financing, purchase and productive cooperation. That all requires to creates and permanently actualizes an informative base including information about:

• economic and financial situation of the company (diagnostics of the financial health),

• relevant market (bigness, development, demands), • competition (main competitors, possible

substitutions, barriers to enter the branch), • sales and marketing of decisive products and

their comparing with competitors ones divided for: - prices and price calculations, - advertisement and costs related with it, - research, development and advance of technical production preparation (ability of businessman – designer to persuade the customer about high level of technical solutions and technological reliability typical of productively-technical basis, - production-level of its controlling,

production capacities and steady of their depleting, suppliers and logistics level,

- workers – qualifying and age structure, satisfaction, moods, working climate, stability and stuff turnover.

In the basic of experiences can be drawn the generalized conclusions, that strategic analysis of production program of each company needs to pursue primarily:

• analysis and prognosis of relevant market, • analysis of competitors and internal resources

and company’s potential, • prognosis of company’s revenues.

- 193 -


4. ZAKLJUÂK Proizvodni program i strategija razvoja kompanije traè stalnu pa`nju. Pogodno je, ukoliko vlasnici i faktori kompanije li~no upravljaju i trajno nadgledaju da se ne mije{a u imovinske osobine kompanije i da strategija predstavlja njene namjere, funkcije i ciljeve, tj. da kompanija radi u datom segmentu trì{ta kao tehni~ki najbolja i istovremeno svojim rezultatima obavje{tava o dobrom finansijskom zdravlju i prosperitetu.

Kao primjer ja~ih strana kompanije moè se spomenuti sljede}e:

● visok nivo konstruktivne i tehnolo{ke proizvodne pripreme koja omogu}ava dostizanje visokog kvaliteta prozvoda, {to kupac traì,

● visoka radna kultura i stvaranje klime za kreativni timski rad u kompaniji,

● prilago|avanje kvalifikacijske structure osoblja,

● efektivno kori{tenje proizvodnih kapaciteta,

● efektivan sistem motivacije i ekonomskog interesa,

● visok nivo organiziranja logisti~kih procesa i partnerstva s dobavlja~ima,

● finansijska stabilnost kompanije i konstantna briga o finansijskom zdravlju kompanije, dobrom partnerstvu s bankama {to omogu}ava ponovno finansiranje svake narud`be.

4. IN CONCLUSION Production program and strategy of company’s development need continual pay attention. It is appropriate, if proprietors and factors of the company personally manage and permanently supervise for not to be interfered into property virtue of the company and for its strategy representing its intentions, functions and goals, i.e. so as the company performs in given market segment as a technically best and at the same time with its results informs about a good financial health and prosperity.

As an example of company’s strengths is possible to mention following:

● high level of constructive and technological production preparation enabling to achieve high quality of products which is required by a customer,

● high working culture and making of climate for creative teamwork in the company,

● accommodating staff qualifying structure,

● effective using of production capacity,

● effective system of motivation and economic interest,

● high level of organizing the logistics processes and partnership with suppliers,

● financial stability of the company and constant solicitude about company’s financial health, good partnership with the banks allowing to re-finance every order.

5. LITERATURA - REFERENCES [1] Szilágyi, M.. Re{trukturalizácia slovenských strojárskych podnikov. In: Zborník z medzinárodnej

konferencie Management a ekonomika firmy, Praha 1999

- 194 -

KLASIFIKACIONA [EMA METODA ZA RJE[AVANJE PROBLEMA...

Documents

Transcript of KLASIFIKACIONA [EMA METODA ZA RJE[AVANJE PROBLEMA...