MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei...
Transcript of MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei...
![Page 1: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/1.jpg)
Željko Andreić – Mehanika fluida: P8 1
MEHANIKA FLUIDA
dio 8
prof. Željko Andreić
Rudarsko-geološko-naftni fakultet
Sveučilište u Zagrebu
http://rgn.hr/~zandreic/
![Page 2: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/2.jpg)
Željko Andreić – Mehanika fluida: P8 2
sadržaj 1-2-3!
Istjecanje
1. istjecanje kroz različite otvore
2. nestacionarno istjecanje
3. mlazovi
![Page 3: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/3.jpg)
Željko Andreić – Mehanika fluida: P8 3
Istjecanje kroz male otvore
h držimo konstatnim. BJ za točke A i B daje:
pa
pa
B
A
0 0
h
![Page 4: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/4.jpg)
Željko Andreić – Mehanika fluida: P8 4
Istjecanje kroz male otvore 2
ako je otvor malen, , pa se ovo svodi na:
ovo je Torricelli-jeva jednadžba za brzinu istjecanja.
vB=brzina pada sa visine h!!!
![Page 5: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/5.jpg)
Željko Andreić – Mehanika fluida: P8 5
Istjecanje kroz male otvore 3
stvarna brzina istjecanja je manja zbog gubitaka na izlaznom
otvoru:
pa BJ postaje:
sa rješenjem:
![Page 6: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/6.jpg)
Željko Andreić – Mehanika fluida: P8 6
Istjecanje kroz male otvore 4
ako se izlazni ureñaj sastoji samo od otvora na posudi, svi gubici
dolaze od njega i opisuju se koeficijentom smanjenja brzine:
pa izraz za brzinu postaje:
ϕ se odreñuje eksperimentalno, a pravilnim zaobljenjem
rubova dostiže se ϕ=0,98.
![Page 7: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/7.jpg)
Željko Andreić – Mehanika fluida: P8 7
Istjecanje kroz male otvore 5
pa je protok kroz otvor
presjek kontrahiranog
mlaza je As
presjek izlaznog
otvora je A
![Page 8: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/8.jpg)
Željko Andreić – Mehanika fluida: P8 8
Istjecanje kroz male otvore 6
pa izraz za protok postaje:
uvodimo koeficijent istjecanja
Prednost: koeficijent istjecanja možemo direktno
eksperimentalno mjeriti preko protoka!
primjer: za okrugli otvor oštrih rubova je αααα=0,61.
![Page 9: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/9.jpg)
Željko Andreić – Mehanika fluida: P8 9
Istjecanje ispod razine tekućine
B.J. sad glasi
pa
pa
B
A
0 0
hA
hB
![Page 10: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/10.jpg)
Željko Andreić – Mehanika fluida: P8 10
Istjecanje ispod razine tekućine 2
no,
a, kao i prije,
pa za ovaj slučaj Torricelli-jeva brzina iznosi:
![Page 11: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/11.jpg)
Željko Andreić – Mehanika fluida: P8 11
Istjecanje ispod razine tekućine 3
stvarnu brzinu nalazimo kao i kod jednostavnog istjecanja uz
pomoć koeficijenta protoka izlaznog otvora i koeficijenta istjecanja:
i
![Page 12: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/12.jpg)
Željko Andreić – Mehanika fluida: P8 12
Istjecanje kroz veliki otvor
B.J. od A do B (uz i ) bi dao
pa
pa
B
A
hB
pa
h2
h1
![Page 13: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/13.jpg)
Željko Andreić – Mehanika fluida: P8 13
Istjecanje kroz veliki otvor 2
Ovo nije sasvim točno jer je strujnica zakrivljena pa postoji
doprinos centrifugalne komponente tlaka.
Istovremeno, za točke 1 i 2 na rubovima otvora bi našli da je:
i
naravno, kako je je i
![Page 14: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/14.jpg)
Željko Andreić – Mehanika fluida: P8 14
Istjecanje kroz veliki otvor 3
i
pa protok moramo naći integracijom preko površine otvora.
Radi jednostavnosti uzmimo da je otvor pravokutnog
presjeka, širine b:
![Page 15: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/15.jpg)
Željko Andreić – Mehanika fluida: P8 15
Istjecanje kroz veliki otvor 4
integracija kao konačni rezultat daje ovaj izraz:
![Page 16: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/16.jpg)
Željko Andreić – Mehanika fluida: P8 16
Istjecanje kroz otvor pred kojim tekućina ne miruje
brzinu va sad ne možemo zanemariti, pa BJ postaje:
pa
B
A
0 0
hpa
vA
vB
![Page 17: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/17.jpg)
Željko Andreić – Mehanika fluida: P8 17
Istjecanje kroz otvor pred kojim tekućina ne miruje 2
sa rješenjem:
kod velikog otvora protok opet nalazimo integriranjem.
![Page 18: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/18.jpg)
Željko Andreić – Mehanika fluida: P8 18
Istjecanje kroz otvor pred kojim tekućina ne miruje 3
npr., za pravokutni otvor nalazimo:
odn. za prelijev (tzv. Eytelwein-ova formula):
![Page 19: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/19.jpg)
Željko Andreić – Mehanika fluida: P8 19
Istjecanje iz posuda pod tlakom
patm
pp
B
A
0 0
h
![Page 20: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/20.jpg)
Željko Andreić – Mehanika fluida: P8 20
Istjecanje iz posuda pod tlakom 2
BJ za ovaj slučaj je:
sa rješenjem:
![Page 21: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/21.jpg)
Željko Andreić – Mehanika fluida: P8 21
Istjecanje u nestacionarnim uvjetima
npr. bazen koji se neravnomjerno puni i prazni (rezervoar za
vodu i sl.)
h
Qd
A
S
Qvan
![Page 22: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/22.jpg)
Željko Andreić – Mehanika fluida: P8 22
Istjecanje u nestacionarnim uvjetima 2
protok kroz rupu na dnu ovisi o dubini tekućine iznad otvora:
ako je dubina veća od ho, ona se smanjuje prema ho, a ako je
manja, onda raste prema ho!
![Page 23: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/23.jpg)
Željko Andreić – Mehanika fluida: P8 23
Istjecanje u nestacionarnim uvjetima 3
da nañemo vrijeme potrebno da se dubina promijeni sa h1 na h2
postavimo jednadžbu sačuvanja volumena:
a otuda integracijom nalazimo traženo vrijeme:
![Page 24: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/24.jpg)
Željko Andreić – Mehanika fluida: P8 24
Istjecanje u nestacionarnim uvjetima 4
ako je posuda prizmatična, S je konstantan i rješenje lako
nañemo:
prekinemo li dotok (ho=0!) rezervoar se prazni:
![Page 25: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/25.jpg)
Željko Andreić – Mehanika fluida: P8 25
Istjecanje u nestacionarnim uvjetima 6
a potpuno će se isprazniti (h2=0!) za:
Ovo vrijeme je dva puta duže od vremena potrebnog da isti
volumen tekućine istekne uz konstantnu dubinu h1!
![Page 26: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/26.jpg)
Željko Andreić – Mehanika fluida: P8 26
Mlazovi
hmax
lmax
vo α
![Page 27: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/27.jpg)
Željko Andreić – Mehanika fluida: P8 27
Mlazovi 2
- mlaz je struja fluida potpuno omeñena slobodnom površinom
- fluid se slobodno giba prostorom i podvrgnut je samo sili teži i
silama trenja s okolinom (često su zanemarive)
- mlaz se giba po zakonima mehanike materijalne točke (kosi
hitac)
-ako je početna brzina mlaza:
onda je (uz zanemarivanje otpora zraka):
![Page 28: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/28.jpg)
Željko Andreić – Mehanika fluida: P8 28
Mlazovi 3
sa rješenjem:
vrijeme postizanja max. visine je:
a vrijeme leta (horizontalna podloga):
![Page 29: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/29.jpg)
Željko Andreić – Mehanika fluida: P8 29
Mlazovi 4
domet mlaza:
max. domet je za α=45o !
ovo postaje:
uz
![Page 30: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/30.jpg)
Željko Andreić – Mehanika fluida: P8 30
Mlazovi 5
i na kraju, omjer dometa i max. visine:
u stvarnosti, mlazovi dostižu do 3/4 teoretske visine/dometa.
![Page 31: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/31.jpg)
Željko Andreić – Mehanika fluida: P8 31
Horizontalni mlaz
ht
hm
![Page 32: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/32.jpg)
Željko Andreić – Mehanika fluida: P8 32
Vertikalni mlaz
h
vy raste pa se presjek mlaza
smanjuje, a može doći i do
njegovog raspada.
![Page 33: MEHANIKA FLUIDA dio 8 - rudar.rgn.hrrudar.rgn.hr/~zandreic/studenti/fluidi/mf_p8.pdfŽeljko Andrei ć – Mehanika fluida: P8 1 MEHANIKA FLUIDA dio 8 prof. Željko Andrei ć Rudarsko-geološko-naftni](https://reader031.fdocument.pub/reader031/viewer/2022021509/5a84a0e47f8b9a001c8bb4a4/html5/thumbnails/33.jpg)
Željko Andreić – Mehanika fluida: P8 33
Vertikalni mlaz 2