УДК 532.54 Сосуд с постоянным притоком жидкости и

advertisement
F&$%#"/ ?NB?C. OH')/)$@":&$/+- %&A#"/+. 2012. P 32
CDE 532.54
'.). =9.T1,1, $.*. (.470.
L,#*('.3 0$8.&0$-D063 .((-,+&/$",-D('.3 )&-.",70.4,('.3 =0./,#(.","
= ">%$#$2$()K >$%)"#' '6;"="2$<'()@ M%"6(?
6 !"!M#$ >%) )!;$A$()) &)#="!;) A$%$Q !)J"(
>$((*$"#./$,"(1 E$+$4$ 0$7&2+,0.1 ),#.&+$ #,-$'($8.&0067 $/"&'&-,%$0.3 / 5.+#$/-.4,('&3 (.(",*,, .E/,("0&3 )&+ 0$E/$0.,* «"$0"$-&/ (&(=+» . (&("&1C,3 .E (&(=+$ ( )&("&1006* )#."&'&* 2.+'&(". . &)&#&20,0.,* 4,#,E (.;&0. L#. :"&* =4."6/$,"(1 /#,*1 &)&#&20,0.1 (&(=+$ 4,#,E (.;&0. L&-=4,0$ %&-,, "&40$1 ;&#*=-$ +-1 ),#.&+$ $/"&'&-,%$0.3. L#./&+1"(1 '#./6,, )&E/&-1BC., &8,0."D &A.%'= / (-=4$, )#,0,%#,2,0.1 /#,*,0,* &)&#&20,0.1
(&(=+$.
=F35:9G: HF79.: (&(=+ ( )&("&1006* )#."&'&* . .(",4,0.,* 4,#,E (.;&0, $/"&'&-,%$",-D0$1 (.(",*$ 0$'&).",-D0&5& ".)$, '&:;;.8.,0" #$(7&+$ (.;&0$, /#,*1 E$)&-0,0.1 /&(7&+1C,3 /,"/. (.;&0$, /#,*1 &)&#&20,0.1 (&(=+$ 4,#,E (.;&0, ),#.&+ $/"&'&-,%$0.3 =#&/01
2.+'&(". / (&(=+,.
A.I. Kvashnin, E.M. Naboka
Perm National Research Politechnic University
TO THE IDENTIFICATION OF THE SELF-OSCILLATING PERIOD
OF THE LIQUID LEVEL AT THE SIPHON DISCHARGING
It is considered period of relaxation self-oscillations in hydraulic system known as “tantalum container”
which consist of vessel with a constant inflow and outflow through a siphon. The emptying time of the vessel
through the siphon is taken into account. More accurate formula for the period of self-oscillations is derived.
Curves for error estimation at neglect of the emptying time of the vessel is pointed.
Keywords: vessel with a constant inflow and outflow through a siphon, self-oscillating system of
accumulating type, siphon discharging coefficient, time of filling of the siphon ascending branch, the
emptying time of the vessel through the siphon, self-oscillating period of liquid level in a vessel.
T)$.3 $ =)$%)-##;@ ='"%)/)@ 9"3/)$%" " "$%&:&#"&@ :&'&, $">)# ('"$. 1) ='&3$%+(5-&% $)*)1 +(%)/)5&*+%&56#.8 $"$%&@. #+/)="%&56#)7) %"=+ ]1^.
C')(&#6 z 9"3/)$%" ( $)$.3& =)(;<+&%$- 3) ,+=)5#&#"- ()$A)3-I&1 (&%(" $">)#+ – z $ z2 , + ,+%&@ =')"$A)3"% )=)')9#&#"&
" $#"9&#"& .')(#- 3) (&5":"#; z $ z1 , ='" /)%')1 (),#"/+&% '+,';(
=)%)/+ ( $">)#&. D+5&& =')4&$$ #+=)5#&#"- " )=)')93&#"- $)$.3+ =&'")3":&$/" =)(%)'-&%$-. Y+'+/%&' ",@&#&#"- .')(#- 9"3/)$%" ( $)$.149
O.B. E(+<#"#, X.!. N+*)/+
3& () ('&@&#" ='&3$%+(5&# #+ '"$. 2. ?&'")3 +(%)/)5&*+#"1 T '+(&#
$.@@& ('&@&#" ,+=)5#&#"- $)$.3+ (()$A)3-I&1 (&%(" $">)#+) t1
" ('&@&#" )=)')9#&#"- t2 .
0"$. 1. T)$.3 $ "$%&:&#"&@ 9"3/)$%" :&'&, $">)#:
1 – =)3()3-I+- %'.*+; 2 – $">)#; 3 – $)$.3
0"$. 2. 0&5+/$+4")##;& +(%)/)5&*+#".')(#- 9"3/)$%" ( $)$.3&
F '+*)%+A ]2, 3, 4^ ='"#"@+&%$- ='&3=)5)9&#"& ) %)@, :%) '+$A)3
9"3/)$%" :&'&, $">)# @#)7) *)56<& '+$A)3+ :&'&, =)3()3-I.8 %'.*.
" =&'")3 +(%)/)5&*+#"1 )='&3&5-&%$- %)56/) ('&@&#&@ ,+=)5#&#"- $)$.3+ T 6 t1 . G+/)& 3)=.I&#"& #& ($&73+ $='+(&35"(). F #+$%)-I&1 $%+%6& '+$$@+%'"(+&%$- #+A)93&#"& =&'")3+ +(%)/)5&*+#"1 $ .:&%)@ ('&@&#" )=)')9#&#"- $)$.3+ :&'&, $">)#.
F'&@- ,+=)5#&#"- $)$.3+ @)9#) )='&3&5"%6 $5&3.8I"@ )*'+,)@:
150
E )='&3&5&#"8 =&'")3+ +(%)/)5&*+#"1 .')(#- ( $)$.3& ='" "$%&:&#"" 9"3/)$%"
S* " z2 + z1 #
,
Q0
t1 $
73& S* – =5)I+36 =)=&'&:#)7) $&:&#"- $)$.3+; z1 – (&'%"/+56#+- /))'3"#+%+ (A)3#)7) $&:&#"- $">)#+ )%#)$"%&56#) ='"#-%)1 =5)$/)$%"
$'+(#&#"- " z $ 0 # ; z2 – (&'%"/+56#+- /))'3"#+%+ (&'A#&1 %):/" ()$A)3-I&1 (&%(" $">)#+.
?'" #+A)93&#"" ('&@&#" )=)')9#&#"- $)$.3+ :&'&, $">)# *.3&@ =)5+7+%6 3("9&#"& 9"3/)$%" /(+,"$%+4")#+'#;@ ]5^. G)73+ '+$A)3
:&'&, $">)# *.3&% ,+("$&%6 )% .')(#- z 9"3/)$%" ( $)$.3&:
Q " z # $ IS$ 2 gz , z % ] z1 , z2 ^ ,
73& I – /)H>>"4"&#% '+$A)3+ $">)#+; S$ – =5)I+36 9"()7) $&:&#"$">)#+; g – .$/)'&#"& $()*)3#)7) =+3&#"-.
B, 3">>&'&#4"+56#)7) .'+(#&#"- '+$A)3)(
S*
dz
$ + LNQ " z # + Q0 MO , Q " z # 8 Q0 , `z % ] z1 , z2 ^ ,
dt
=)5.:"@
dt $ +
S * dz
IS$ 2 gz + Q0
,
73& dz – ",@&#&#"& .')(#- 9"3/)$%" ( $)$.3& ($@. '"$. 1) ,+ *&$/)#&:#) @+5;1 =')@&9.%)/ ('&@&#" dt.
M%$83+ =.%&@ "#%&7'"')(+#"- #+A)3"@ ('&@- "$%&:&#"- 9"3/)$%" :&'&, $">)#
t2
z1
t2 $ 5 dt $ + 5
z2
0
73& a $
Q0
IS$ 2 g
,
S* dz
IS$ 2 gz + Q0
$
'
z2 + a (
) z2 + z1 0 a ln
*,
2 g ),
z1 + a *-
2 S*
IS$
z2 8 z1 8 a.
D5- =&'")3+ +(%)/)5&*+#"1 .')(#- 9"3/)$%" ( $)$.3& $ .:&%)@
('&@&#" ,+=)5#&#"- ()$A)3-I&1 (&%(" $">)#+ #+A)3"@
T $ t1 0 t2 $
S* " z2 + z1 # L
z2 + a ( M
2a '
Z1 0
) z2 + z1 0 a ln
*[ .
Q0
z1 + a *- [O
ZN z2 + z1 ),
151
O.B. E(+<#"#, X.!. N+*)/+
?)5.:&##.8 >)'@.5. @)9#) ='"(&$%" / ("3.
S* " z2 + z1 # L
Q " z2 # + Q0 ( M
Q02 ' Q " z2 # + Q " z1 #
2 z2
ln
T$
0
Z1 0
)
*[ ,
2
)
Q0
Q0
Q " z1 # + Q0 *- O[
NZ z2 + z1 Q " z2 # ,
73& Q " z1 # " Q " z2 # – '+$A)3 :&'&, $">)# ='" $))%(&%$%(.8I"A ,#+:&#"-A z1 " z2 , Q " z2 # 8 Q " z1 # 8 Q0 .
N+ '"$. 3 ='&3$%+(5&#; /'"(;&, =),()5-8I"& )4&#"%6 )<"*/.
( )='&3&5&#"" =&'")3+ +(%)/)5&*+#"1 .')(#- 9"3/)$%" ( $5.:+& ='&#&*'&9&#"- ('&@&#&@ "$%&:&#"- :&'&, $">)# ( ,+("$"@)$%" )% '+$A)3+
:&'&, =)3()3-I.8 %'.*. ='" $5&3.8I"A =+'+@&%'+A 7"3'+(5":&$/)1
$"$%&@;: z1 $ 0, 05 @ ; z2 $ 0, 2 @ ; I $ 0,35; Sc $ 1>10+4 @ 2 ; S* $ 1 @ 2 .
0"$. 3. J'+>"/ ,+("$"@)$%" =&'")3+ +(%)/)5&*+#"1 )% '+$A)3+ 9"3/)$%"
:&'&, =)3()3-I.8 %'.*.: 1 – $ .:&%)@ ('&@&#" )=)')9#&#"- :&'&, $">)#;
2 – *&, .:&%+ ('&@&#" )=)')9#&#"- :&'&, $">)#
G+/"@ )*'+,)@, ='&35+7+&@+- >)'@.5+ 35- )='&3&5&#"- =&'")3+
+(%)/)5&*+#"1 .')(#- ( $)$.3& $ =)$%)-##;@ ='"%)/)@ 9"3/)$%"
" "$%&:&#"&@ :&'&, $">)# .:"%;(+&% ('&@- )=)')9#&#"- $)$.3+ :&'&,
$">)# " 3+&% *)5&& %):#;1 '&,.56%+%.
E'B+'(,6$F'2".%': .A'.(%
1. !+7#.$ E. E)5&*+#"-: F(&3&#"& ( "$$5&3)(+#"& /)5&*+%&56#;A
$"$%&@. – !.: !"', 1982. – 304 $.
2. W')13 B.B. N&%'+3"4")##;& 7"3'+(5":&$/"& ='"/5+3#;&
,+3+:" " %&A#)5)7"". – !.: B,3-() OTF, 2008. – 245 $.
152
E )='&3&5&#"8 =&'")3+ +(%)/)5&*+#"1 .')(#- ( $)$.3& ='" "$%&:&#"" 9"3/)$%"
3. !"',+39+#+3,& O.Y., W+1/)( F.O. ?+'+3)/$; #&>%-#)1 >","/" /
B#-% /)@=68%. "$$5&3. – !.; B9&($/. – 223 $.
4. !"',+39+#,+3& O.Y., E&'"@)( K.J., E)=&1/"$ !.J. G&)'"- /)5&*+#"1 ( #&>%&=')@;$5)()@ 3&5& / B#-% /)@=68%. "$$5&3. – !.;
B9&($/, 2005. – 363 $.
5. J"3'+(5"/+, 7"3')@+<"#; " 7"3')='"()3;: .:&*#"/ 35- (%.,)( / G.!. W+<%+ [" 3'.]. – !.: O56-#$, 2010. – 423 $.
References
1. Magnus K. Kolebaniya: Vvedenie v issledovanie kolebatelnykh sistem [The vibrations: introducrion to investigation of oscillating systems].
!oscow: Mir, 1982, 304 p.
2. Broyd I.I. Netraditsionnye gidravlicheskie prikladnye zadachi i
tekhnologi [Non-traditional hydraulic applications and technology].
!oscow: OTF, 2008, 245 p.
3. Mirzadzhanadze A.Kh., Baykov V.A. Paradoksy neftyanoy fiziki
[Paradoxes of oil physics]. Moscow, Institut kompyuternykh issledovaniy,
223 p.
4. Mirzadzhanzade A.Kh., Kerimov Z.G., Kopeykis M.G. Teoriya
kolebaniy v neftepromyslovom dele [Theory of oscillations in the oilfield].
Moscow, Institut kompyuternykh issledovaniy, 2005, 363 p.
5. T.M. Bashta [et al.] Gidravlika, gidromashiny i gidroprivody:
uchebnik dlya vtuzov [Hydraulics, hydraulic machines and hydraulic actuators: the textbook for technical universities]. !oscow: Alyans, 2010,
423 p.
GB $0/(6$H
>0$7)') V+"%.$)#6 N0$)(0'2 (?&'@6, 0)$$"-) – /+#3"3+%
%&A#":&$/"A #+./, 3)4&#% /+>&3'; «J"3'+(5"/+ " 7"3'+(5":&$/"& @+<"#;» 2JWMC F?M ?NB?C (614990, 7. ?&'@6, E)@$)@)56$/"1 ='.,
29, e-mail: ggm@pstu.ru).
I$B(%$ S0,")': C'H$:+(0'2 (?&'@6, 0)$$"-) – 3)/%)' %&A#":&$/"A #+./, =')>&$$)', ,+(&3.8I"1 /+>&3')1 «J"3'+(5"/+ " 7"3'+(5":&$/"& @+<"#;» 2JWMC F?M ?NB?C (614990, 7. ?&'@6, E)@$)@)56$/"1 ='., 29, e-mail: ggm@pstu.ru).
153
O.B. E(+<#"#, X.!. N+*)/+
About the authors
Kvashnin Aleksandr Ivanovich (Perm, Russian Federation) – Ph. D. in
Technical Sciences, Ass. Professor, Department of Hydraulics and Hydraulic
Machines, Perm National Research Polytechnic University (29, Komsomolsky
av., Perm, 614990, Russian Federation, e-mail: ggm@pstu.ru).
Naboka Evgeniy Mikhaylovich (Perm, Russian Federation) – Doctor of
Technical Sciences, Professor, Head Department of Hydraulics and Hydraulic
Machines, Perm National Research Polytechnic University (29, Komsomolsky
av., Perm, 614990, Russian Federation, e-mail: ggm@pstu.ru).
?)5.:&#) 4.04.2012
154
Download