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����� ���������� �"! � #%$'&)(+*,(+-.$0/2143'(+5 � 6'187:9)$'�';<(>=��?9)1�6'#4@A1:;B1�9)$C@A$D5)(E7:9)$'�';F/2#+GH1I/2#>! #C;B1�9)� J+�K5)()L• M%N?OQP2R,S:TQUQVXWYRZV+T\[2R^] ;B149)$C@A$_5)(F=�*%(X6 �`7:9)$'�';B5 �a;B149)�)b`$'5 $'Jc(X�a! #%=I/d(+J+�B(+5 1feI(+!)� 1I/2� 5)10gX1IeD5 1I6,#%h;B� � 1I6 J+5 #%hF7�� e��)& 6'#Xhi6'#X5 Jj(X6 Jj(lkm=4�?9<(_e�1I/Qn�� 5 1oJ+14GH1Xpq� 5 1I� r �j@s(+9)5)(a=4� 9B( pq149B1�6 J+�)#%/d(+hI5 1�JX5)(a=4�?9<(%tvuw *X1�=4�?9B1x=418#Xg%� & 5 #H� eI�)(jG>(+*%(4@A$E! #:@A1v;B�)5 � r �F/d(%=41Ey
N/kg z u• OQ{B|<P\}_TQUQVXWYR�VXT~[dR:] ;B149)$C@A$85)(�#C;BhI#X*%(X�)(v@A$'b 1�! #X*%� ���x�H#%=�J+*%(X� $C@A$8=41�;B� � 1I6 J+5 �)/^6'#X5 Jj(X6 J+#%/�k�;B1s@s=4J+*%#5 1�6'1:& *X�B=4J+1�! #%*X� ���05)(H7:9)$'�';\p?�?9B�D;B1s@s=4J+*%#E7:9)$'�';<(�5)(�g%�?9)#H6'#4@A$EeI(%/2� �xn�1�5)$D! #X*%� �Ht4u���(�#%*X1�=4�?9B1=41E/2#X�B(f;B1I7�� 5)�)=�(XJ+�^�_#C;BhI#X*,(+�)(4@A$,bB(D! #%*X� ��5)(K6'#4@A$i;B149)$C@A$�u0��� ��! � #%$'&)(+*,(+-.$F! #X*%� ���)5)=46 � 3^=4�?9<(;B1I7��)5 � ��1>=41H! #�@s(X/ |�R,W\� {BPQN �?9)� � R,U\M,{)P2N 5B(+! #%5)( ] ! #%*X� ��� 5)=46?(f=4�?9B(f$i5 1I6,#�@�Jj(+& 6)��=�1H=�*X#C;B�F5B(#C;Bh�#%*%(X�)(v@A$'b $>� 5)7��)5 � J+1Ie��)/d(+9)5 $0!)#X*X�)��6'#�@s(�@A18#C;<� 1��'1�5B(�=�*X#4@A�B/�*%1�6 J+#%� #%/�5 #X�B/d(j9)1
~nu
~pn =d~F
dA− *X1�6)J+#X�"5)(+!)#X5)( kA�%u��ct
�8(oJj(4@H5)(X& � 5�=41%p�$`5)1�6'#%/�#X! ��J+1I/�=v9)$'&)(4@A$_;B#%gX�j@s(_7�� eI� & 6?(�*X149B� & � 5)(�eC(o6'#4@A$F@A1i! #XJ+�)1�gX5)#o;<1�*X1IJ=46?(j9<(+� 5 � 3D*X149B� & � 5)(�;<(Hg%�fg%�?9B(H$D! #%J+! $'5 #%=�J+�K#C;B� 1I�,1I5)(�$D5 1�6'#�@.Jj(+&)6 �f$f7:9)$'�';B$�u���#%/2#%b $EJ+1�5)e�#X�B(5)(+!)#X5)(�=�1�5)(�*%�?9)#�1I7�1I6 Jj(+585)(X& � 5x/2#XG�1�#%! �)=�(XJ+�8! � 1I5 #X��1�-:12;B1s@s=4J+*,(�! #X*%� ���)5)=46 � 3:=4� 9B(�6 � #%e�7:9)$'�';\u
�l#X6 *%� � $x=�Jc(XJ+� 6'1q7:9)$'�';<(�! � #X$'&)(X*%(�=41q(+!)=�#+9)$'J+5 #����)149B(XJ+� *X5)#:/2� �)#X*%(X-�1q7:9)$'�';<()u\�8(X6,#�b 1q��5)(4@s/d(+-:�Jj(+5)h�1�5)r �j@s(j9B5 ��5)(+! #%5�;B#X*%1I=4JX��;B#Z6)� 1�Jj(+-�(Z7:9)$'�';<()p�! � ��/2� � #%*%(+-.$�7:9B$,�?;<(lJc(X5 h�1I5 r �j@�9<(+5 ��5)(X! #X5)��$7:9)$'�';<$�/2#X�B(v@A$�g%� J+��@A14;B5)(+6)��5 $C9)�Ypx��5)(X! #X5B=46'#Z=4Jj(+-:1a$�-:1I/2$�=41`=�*X#C;B��=�(%/2#Z5)(�5 #%�)/d(j9)5)1a5)(+! #%5 1Xu�8(X6,#H7:9)$'�';f5 1�/2#+GH1�;�(��+JX� ! �)�X5 #X�)/d(+9)5 1:5)(+!)#X5 1��)=4J+1IeI(X-:("k�g%� b 1:5)(+� $%��1I5)(8! � 1IJ+! #,=4Jj(+*%6'(�#�7:9)$'�';<$E6'(X#6'#X5)J+� 5 $'$?/2$ tvpQ! � �?9)�)6,#,/�/2�)� #X*,(+-�("7:9B$,�?;<("! #%=�Jc(4@s(+b 1E=I(X/2#K5 #X�B/d(j9)5 �_5)(X! #X5)��!)� � J+�)=�6?( ] Jv@4uD5)(X! #X5B=46'#=�Jc(X-�18$f7:9)$'�';B$f=�18/2#+G�1�#%! �)=�(XJ+�H@A1v;B5)#%=4Jj(X*X5 #,/`� 1�9B(+r �+@A#%/qL
~pn = −p~n, Jv@4uP = −p
1 0 0
0 1 0
0 0 1
≡ −pE ks�Xu��Xt
h�;B1q@A1�=�(~pn
#XeI5)(+&)1�5K*X1I6 J+#X�K5)(X! #X5B( p =�(PJ+1I5 e�#%�i5)(+!)#X5)()u2 �1�Jj(jn�5)(E(X5)(j9)�)eI(H5)(+!)#X5)=�6,#%h8=4Jj(+-�(>$f7:9)$'�';<$<p
6?(+#K�^;B1C7�� 5 � r �+@A1D*X1I6 J+#X�B(f�_J+1I5 e�#%�)(i=4$F;<(+JX��5B(K! � 1v;�(+*%(X-��B/d( uH �(+6 9)1Xp\$o=4Jj(+JX� r �`7:9)$'�';<(K=��?9)$o6'#�@A#%/7:9)$'�';K;B1�9)$C@A185)(H5 1�6)$D! #X*%� �0p?�B/d(v@A$'b �K$0*%�';B$oks�Xu¡�Ct.�_kA�Xu��Xt4p #C;B� 1I�'$I@A1C/2#>! #%/2#%b $D� eI�)(+eC( L
~F = −∫
A
p~ndAkA�Xu ¢%t
£ �?9B$ ~Fb)1I/2#E& 1C=4J+#E#%e�5)(X&)(+*,(+J+�f�"=I( ~P
�K5)(+eI� *%(XJ+� V+T\[2{<S¤OQP\TQ�?TQVXWYN u
¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬xF¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼���¾½ ¿�À�Á�ÂiÃÄÂiÀ8ÅiÃiÂiÃÄÆFÇKÈ.�^ÉqÊ�À8�8Ë�À8ÌÍÀÎÆFÇKÃKÊ8ÃiÉ�ÏDÀÐÒÑ�ÓFÐ0Ãf�oÐ
�8(X#���J+#:@A18� 1�& 1I5 #E$D$'*X#C;<$<p'! �)�K/2�)� #X*,(+-.$f7:9)$'�';<(H5)(X! #X5B=46'#H=�Jj(+-:1:$"7:9)$'�';B$0=�18=4*X#C;<�f5)(H! �)� J+�)=I(+6Yuw =�5 #X*%5)(.@A1v;B�)5 � r)(�$
SI=4�)=�J+1I/2$>eI(x! � � J+�B=�(+6x@A1 O2N V+WYN'[_] [Pa]
u ¥ 9)JX1�� 5)(XJ+� *%5)(�@A14;B� 5 � rB(xeI(x! � � J+�B=�(+6~p,6'#�@s(=�1E& 1I=�J+#K6,#%� �)=�J+�_$^/2143'(+5 �)r ��7:9B$,�?;<(�@A1
bar ] 1bar = 100000 Pa ≡ 105 PauH �1I7��)5 � ���B/2#K=I(c;<("! #�@s/2#X*%1
*%1�eI(X5 1�eI(H! � �)J+�)=�(X6Yp)(H6'#4@A18b 1C/2#>& 1I=�J+#D=4� 1IJj(+J+�Yu�Ô~#D=4$�L• « � S:{<VXÕiR,P2V+WQT�OQP\TQ�?T2VXN?W_] ! � 14;<=4Jj(X*+n8(8! � � JX�)=�(X6f#%6,#X9)5 #%h�*%(Xev;B$,3,()pX�D#%5x@A1�7�$'5 6 r �j@s(�*X� 1C/21�5)=�6 �?3$%=49)#X*,(���5B(c;</2#%�)=46'1.*X�)=�� 5 1Xu w e�5)(X&)(+*,(�=�1.=I(
pa�?9)�
pbu~��5)(X���)/o! � #%�)(+& $'5 �B/d(�5)(4@A& 1���b 1q=41.$,eI�)/d(+JX�
;<(>@A1pa = const
p2��;<(oJc(F*X� 14;B5 #%=�Ji� e�5)#%=4�a!)� � g+9B�?GE5 #pa ≈ 1 bar
p2Jj(+& 5)�j@A1pa = 101325 Pa
u w *%(*X� 14;B5 #,=4JH! � 1v;�=4Jj(+*Xn�(HJ+eI*)u�=4Jj(X5';<(+�%;B5)$0(+JX/2#,=�7�1I� $�u
• « O2VX{)[Q¸��?UQT_O\PQTQ�?TQV%N?WF] (X!)=4#X9)$'J+5 �H! � � J+�B=�(+6E=�1q/21I� �H$x#C;B5 #,=4$85)(�(+!B=4#+9B$,JX5 $x5 $C9)$x!)� � J+�)=�6?(Ek�*X�?;B�=v9)�)6 $o�%u �%tvp?�"#X5K$07:9)$'�';B$f/2#+G�18g%� J+�i/d(+-:�Ypj@A14;B5)(X6"� 9)�K*X1�b)�f#C;F(+JX/2#,=�7�1I�)=46'#Xh%u w e�5)(X&)(+*,(�=418=I(
pu
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pmu
• ¯ {)�?OQPQT\�?T2VXN?W�] (+6'#i@A1�$�5 1�6'#�@DJc(X& 6 ��$�7:9B$,�?;B$�! � � J+�B=�(+6 S�N?Ø0T #C;Ù(XJX/2#%=I7�1��)=�6,#%hXp�#%5';<(�=�1o$J+#4@"Jj(+& 6 �Ù/2#+GH1o;<1I7�� 5 �B=�(+JX��! #%J+! � � JX�)=�(X6�6'#4@A�Ù! � 14;<=4Jj(X*+n8(��)(Xe49)�)6 $�(+JX/2#,=�7�1I�)=46'#Xh^! � � JX�)=46?(l�(+!)=�#+9)$'J+5)#Xh:! � � J+�B=46?(>$0J+#�@.Jj(X& 6 �Yu w eI5)(+&)(X*%(�=41x=�(
pvu
p
pA
(pm)A
(pv)B
pB
pa
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pA = pa + (pm)A − (+!B=4#+9B$,JX5 �K! � � J+�B=�(+6K$fJc(X& 6 � ¥pB = pa − (pv)B − (+!)=�#+9)$'J+5 �K! �)� J+�)=I(+6K$0Jj(+&)6 �"ð
(pm)A − 5)(XJ+! � � JX�)=�(X6"$0Jc(X& 6 � ¥(pv)B − ! #%J+! � �)J+�)=�(X6K$0Jj(+& 6)�Kð
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ρkê=49)� 6?(Ö�Xu��Xtvu���#C;�6'#4@A�)/�$,=v9)#%*X#%/Ùb 1"#X*,(KeI(+!)� 1I/2� 5B(o7:9)$'�';<(ogX� JX�`$`=4Jj(+-.$
/2� �)#X*%(X-:(,ù w ;BhI#X*%#X��@A1:;B#Xg%� #>! #XeI5)(+J>�>@A1v;<5 #%=�Jc(X*%(X5 ] *%1�6 JX#X�)=�6 �Ke�gX�)�K=�*X�?3"=4�?9B(>6'#�@A185)(>-�$0;B1�9)$C@A$f/2#X�B(g%� J+�H@A1v;B5B(+6K5 $C9)�Yu
A
V
dA
dV
~n
−p~ndA
~f
£ 9)�)6'("�%u � L2���)#X� eI*X#+n�5)(�eI(X! � 1I/2�)5)(>7:9)$'�';<(H$f=�Jc(X-�$f/2� � #X*,(+-�(
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��6 $'! 5)(>eC(+! �)1I/2� 5)=�6?(H=��?9B(=
∫
V
ρ ~f dV
;B#%6H@A18$,6)$,!)5)(H! #X*%� ��� 5B=46?(�=��?9B(Kk�;<149)$C@A1x5)(H! #%*X� �At.#C;B� 1��'1I5)(��)e��)(Xe�#%/qL
��6 $'! 5)(>! #%*X� ��� 5)=�6'(H=4� 9B(= −
∮
A
p~n dA
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∫
V
ρ ~f dV +
−∮
A
p~n dA
= 0kA�Xu û,t
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A6'#4@s(H#XhI�)(+5 �)&)(+*,(�eC(+! � 1C/2� 5 $
V$,JX� & $05B(�!)� #%/21�5)$0J+1x7�� eI� & 6'18*X1�9)� & �)5 1�$,5)$,Jj(X�
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A
p~n dA =
∫
V
∇ p dV, ∇ =∂
∂x~i+
∂
∂y~j +
∂
∂z~k − ý (X/2� 9)J+#X5)#X*>#X! 1I�)(+JX#X�
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Jj(+6'#�;<(>=41q@A14;B5)(X& � 5)("�Xu ûE/2#+GH1:5B(+! �)=I(+J+�K$0=49)1v;<1�b 1C/�#%g+9)� 6)$<L∫
V
(
ρ ~f −∇ · p)
dV = 0kA�Xu�ÿXt
�8(X6,#H@A1DeC(+! �)1I/2� 5)(V
! � #X�)e�*X#Xn:5)#i� eI(XgX�)(X5)( p~9)1�*,("=�J+�)(X5)(>@A1v;<5)(+& �)5 1okA�%u û't:b 1Dg%� J+�K@A1v;<5)(+6?(K5 $C9)�`$^5)(v@ ]#%! ��J+�+@A1I/�=v9)$'&)(4@A$D6?(c;<(�@A18! #C;<� 5 J+1Ih��)(+9)5)(>7�$'5 6 r �j@s(�@A14;B5)(+6?(H5 $C9)�Yu
ρ ~f −∇ · p = 0 =⇒ ρ ~f = grad pks�Xu �%t
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ρ(
fx~i+ fy~j + fz ~k)
=∂p
∂x~i+
∂p
∂y~j +
∂p
∂z~k
h�;B1�=4$fxpfy
�fz
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ρ2
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pA + ρgx = pB + ρg(H + x− hm) + ρmghm
⇒ ∆p = (ρm − ρ)ghm + ρgH
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��#XeI5)(+J+�^=4$F=v9B1v;B1Ib �^!)#C;<(+r �~Lρ1 = 1000 kg/m3
pρ2 = 1200 kg/m3
pρ3 = 800 kg/m3
pρ4 = 900 kg/m3
pρm = 13600 kg/m3
ph1 = 1 m
ph2 = 4 m
ph3 = 2 m
ph4 = 2 m
ppm = 1 kPa
�pv = 2 kPa
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hipi = 1, . . . 5
pρm
�ρu
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h1 = 0.9 mph2 = 2 m
ph3 = 0.7 m
ph4 = 1.8 m
pH = 2.5 m
pρ = 1000 kg/m3
pρm = 13600 kg/m3
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ρ1 = 1000 kg/m3p
ρ2 = 800 kg/m3p<(>$D;<1I=45)#4@�=41x5B(j9B(Xe��KJ+1�&)5 #%=�JEh�$%=�J+� 5 1
ρ3 = 870 kg/m3u�ð.�)=4�)5 1x5 � *%#%(HJ+1I& 5 #%=�J+�o=�$<L
H1 = 0.8 mpH2 = 0.6 m
�H3 = 1.1 m
pY("� eI5)(c;a5 � *%#%(fJ+1�&)5 #%=�J+�^*X9B(c;�(v@A$i! � �)J+�)=4r)�pv = 0.15 bar
�pm = 0.25 bar
u w ;B�)1v;B� JX�o! #%6'(Xe�� *,(+-:1x/d(+5 #,/21�J+�B(DeI(EG>�)*X#%/�kρm = 13600 kg/m3
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H = 0.5 mpα = 45◦
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D
C
P
v
ξ
η
u
zzC
ρα
∆vC
vD
vC
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d~P
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P = (pc − pa)AkA�Xu¡�I¢,t
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• ð.1�� J+�)6'(+9)5)(�6'#X#X�%;B�)5)(+Jj(�J+14GE� ��Jc(H6'#4@s(>=�1�/21I� �"#C;i=49)#Xg%#C;B5 1�!)#X*X�)���K=41�#C;B� 1I�,$C@A1:5)(H#%=�5 #X*%$E� e��)(XeI()L
zc =pc − paρg
kA�Xu¡��û't
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� e��B(+e�#,/qL∆vc =
IcξvcA
kA�Xu¡�Cÿ%t
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ξ
vc =zc
sinα
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;<(E=4J+#�@A�cosα
tvu¥ 6'#Z! #%*X� �Î5 �+@A1�=4�)/21IJ+� � &)5)( p�#X5';<(�!)#%=4JX#4@A����! #,/21��)(X-�1a5)(X!)(c;<5 1`Jj(+& 6'1a! #Z6'#%#X�%;B� 5)(XJ+�
uuÔY#
�)(X=�J+#4@s(X-�1�@A1�#C;B� 1I�,1I5 #8� eI�)(+eI#%/∆uc =
IξηucA
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pmpm pvpv
h1
h1h1h1
h2h2
h2
h2
h3
ρ1 ρ1
ρ1ρ1
ρ2
ρ2ρ2
ρ2 ρ3P P
PP1
P2
∆vc ∆vc∆vc� � ¢ û
C C C2
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pc = pa − pv + ρ1gh1 + ρ2gh2
£ �?9B(H! � �)J+�)=46?()LP = (pc − pa)A = (−pv + ρ1gh1 + ρ2gh2)A
��#X#%�X;<� 5)(+Jj(�J+14GE� ��Jc()Lzc =
pc − paρ2g
= − pvρ2g
+ρ1
ρ2
h1 + h2
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α = 90◦t
∆vc ≡ ∆zc =IcξzcA
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¥ 6'#q@A1�*%� 1v;B5)#%=4J8!)#XJ+! �)� J+�)=�6'(pv
*X1�bB(�#C;EeIgX� �B(ρ1gh1 +ρ2gh2
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pc = pa + pm + ρ1gh1 + ρ2gh2
£ �?9B(H! � �)J+�)=46?()LP = (pc − pa)A = (pm + ρ1gh1 + ρ2gh2)A
��#X#%�X;<� 5)(+Jj(�J+14GE� ��Jc()Lzc =
pc − paρ2g
=pmρ2g
+ρ1
ρ2
h1 + h2
�8(+!)(j;B5)(�Jj(+& 6?(>=4�?9B1P ] r 1I5 Jj(+�K! � � JX�)=46?(ik
α = 90◦t
∆vc ≡ ∆zc =IcξzcA
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pc = pa − pv + ρ1gh1 + ρ2gh2 + ρ3gh3
£ �?9B(H! � �)J+�)=46?()LP = (pc − pa)A = (−pv + ρ1gh1 + ρ2gh2 + ρ3gh3)A2
��#X#%�X;<� 5)(+Jj(�J+14GE� ��Jc()Lzc =
pc − paρ2g
= − pvρ3g
+ρ1
ρ3
h1 +ρ2
ρ3
h2 + h3
�8(+!)(j;B5)(�Jj(+& 6?(>=4�?9B1P ] r 1I5 Jj(+�K! � � JX�)=46?(ik
α = 90◦t
∆vc ≡ ∆zc =IcξzcAw *j;B1�*,(jGE�f�B=4Jj(H! � � &)(>6?(X#�! #C;a�Xu
û)u ¥ !)=�#+9)$'J+5)�"!)� � J+�)=I(+6K$0JX14GE� ��JX$<L
pc = pa + pm + ρ1gh1 + ρ2gh2
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£ �?9B(H! � �)J+�)=46?()LP = (pc − pa)A = (pm + ρ1gh1 + ρ2gh2)A2
��#X#%�X;<� 5)(+Jj(�J+14GE� ��Jc()Lzc =
pc − paρ2g
=pmρ2g
+ρ1
ρ2
h1 + h2
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P1 = pmAJv@4u
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kê(,tD� eI#XJ+1I�)/d=46?(k�g'tD! #X9)� J+�)#X!)=�6'(��#XeI5)(+J+�K=4$f=v9)14;B1�b �K! #C;�(+r �YL
GpFpDpHpnppa
�ρu
�?u�ï21�eI1�� *%#%(+�:=�(q6'#X5B=4Jj(+5 JX5 �)/f5 � *X#%#%/f$Cn�()pc=�! #4@A1I5.@A1�=�(�3?�';B�)(c$C9)�)& 6'# ] ! 5 1�$?/d(XJX=46 �B/f!)#4@s(+&B(+*%(X& 1I/0! � � JX�)=46?( u��e�5)(j;�$Cn8(�*+9<(c;<(.5)(XJ+! � � JX�)=�(X6
pm0 = 0.2 baru w ;<� 1v;B�)J+�8eI(q6'#+9B� 6'#q=�1d6?9)� !8!)#C;B� h�5)1�(+6'#�5)(+JX! � � J+�B=�(+6
$Cn�(o$��)1�e�1I� *X#,(+� $�5)(X�)(X=�J+1D5)(pm1 = 0.4 bar
u^ðq(Xev;B$,3a/21�-�(^=�Jc(X-�1f� e�#%J+1��B/d=46 �Yuo �(+JX�`! #C;<(Xr �`=4$�LD = 100 mm
pd = 40 mm
pH = 1 m
ph = 120 mm
pG = 50 N
ph0 = 100 mm
ppa = 1020 mbar
�ρ = 900 kg/m3
u¢ u�ï21�eI1�� *%#%(+�8!)� � 6?(+eC(+5H5)(�=v9)� r)�H=I(X=4JX#4@A�>=41q� e�;B*X1.6'#%/2#%� 1XuQ�l9)1�*%#4@\6'#,/2#X� ��5)(j9B(Xe�1q=41qJ+1I& 5 #%=�J+�>h�$,=4J+� 5)1ρ1 = 1000 kg/m3
�ρ3 = 3000 kg/m3
pq;B#X6`@A1^*%� 1v;B5)#%=4Ja5B(+J+! �)� J+�)=�6'(Ö� eI5)(c;�JX1�& 5 #,=4J+��h�$,=4J+� 5)1ρ1
pm = 0.05 baru.�´;B1I=�5 #4@�6'#%/2#%� �H@A1�J+1I& 5 #,=4JDhI$%=4JX� 5 1
ρ2 = 800 kg/m3� eI5)(c;o6'#�@A1x*+9B(j;<(>! #%J+! � � JX�)=�(X6
pv = 0.1 baru��ú! � 1Ih��)(j;B5 #,/�e��?;B$F� eC/21��'$F6'#%/2#%�)(0=�1H5B(j9B(Xe��o!)�)(+*%#X$'h4(+#%5 �F#XJX*X#X�F;B�)/21I5 e��j@s(
2a × bp
6'#4@A�i@A1KeI(XJ+*X#%� 1�5aeC(+J+*,(+�)(X& 1I/�#%gX� JX5 �)/�#%6,#^#%=�106'#4@s(o! � #+9<(+e��Ö6 � #%e0Jj(+& 6)$ w ��$'! �)(X*X5)(D@A1f5)(o�)(+*,(r � J+1�GH()u w ;B� 14;B� J+��/2�)5 �)/d(j9B5 $l=��?9)$
F!)� �Z6'#4@A#4@>b 1ieC(+J+*,(+�)(X&leC(+J+*,(+�)(XJ+�l#XJ+*%#X�Yu��#Xe�5B(+J+�Z=4$�
=v9)14;B1�b)�"!)#C;<(+r �~La = 200 mm
pb = 1 m
ph1 = 1.5 m
ph2 = 1 m
�α = 45◦
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$'! �)(+*%5)(�@A1H5)(D�B(+*%(X5Fr � JX14GH()u w ;<� 1v;B�)J+�^!)#+9)#XGH(v@�Jj(+&)6,1 w k��)e��)(X& $'5)(+J+�^�)(X=�J+#4@s(X-�1ht�! � �^6'#%/21�b)1
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�H2 = 3 m
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Pz1 = ρgVz1 = ρgVz1 = ρ g
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= ρgR2π
[H
2+R
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[H
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[H
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P =√
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z =
√
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= ρgR2π
√
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(
R2πH +1
2· 4
3R3π
)
= ρgR2π
(
H +2R
3
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J+1�&)5 #%=�J+�`kρ1 = 800 kg/m3
�ρ2 = 1000 kg/m3
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R = 0.2 mpL = 2 m
ph = 2 m
ppm = 0.1 bar
pρ = 1000 kg/m3
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A
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A
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D
Vρ
Pv
P ∗N
PP
A1A1
A2
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5)("J+$6 �)� *X$f! #X*%� �0u2��J+#,/�r � n.$"/2#+G�1C/2#>! � �)/21I5 � J+�F�K=v9B1v;B1Ib �K! #%=�J+$'!)(+6Yu
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A1
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L
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~Pv = −∮
A
p~ndA = −∫
A1
p~n dA1
︸ ︷︷ ︸
~P
+
(
−∫
A2
p~n dA2
︸ ︷︷ ︸
~P∗
N
)
= ~P + ~P ∗N
��#%��J+#�G�1�9)�)/2#�;<(�#C;B� 1v;<�)/2#E=4�?9)$Pp?�f�)/d(4@A$'b �K�f*%�';B$>;<(�@A1 ~P ∗
N = −~PNp,;B#Xg%�j@s(�=�1�6,#%5)(+&B(+50�)e��)(Xe�eC(
#C;B�)1��'� *%(X-�18=��?9)1�! � � JX�)=46?(>5)(H6 � �)*X$0!)#X*X�,30/21IJ+#C;B#%/Ö! #%J+�)=46?()L~P = ~Pv + ~PN = ~PN − ρ~gV
ks�Xu��jû,th�;B1q@A1
VeC(+! � 1C/2� 5)(H6'#4@s(>=41:;<#XgX�+@s(�eC(+J+*,(+�)(X-�1C/`6 �)� *X1�! #%*X� ���"=I(�#C;<h�#X*,(+�)(4@A$'b �)/ag%� #4@A1C/��)(X*X5 � 3D! #X*%� ���Yu
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ρ
PvPv
Pv
PNPN
PN
P
P
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¥ 6,#�G�1�9)�)/2#�;<(`5)(X! � ��1I/2#F@A14;<(+5�#X! ��J+���)e��)(XeXp�6?(c;�(�=41�eI(X! � 1C/2� 5)(V
;B#%gX�j@s(a$'*%#j�,1I-�1C/n
�)(X*X5 �?3! #%*X� ���Yp?� eI�)(+e�eI(E=��?9)$D!)� � J+�)=�6?(>5)(H6 � � *%$0! #%*X� �Ù@A1XL
~P =
n∑
i=1
~PNi + ~Pv =
n∑
i=1
~PNi − ρ~gVks�Xu��XÿXt
��#X9)� 6'#��)(+*%5 �?3�!)#X*X�)���H� eI(XgX�)(XJ+�>eI(�� 1I��(+*,(+-:1q! � #XgX9)1I/d(.@A1�=�J+*%(X�>9)� & 5)#Xh�� eIgX#X�B( u\ü'1I5 1��B(j9)5 #BpX! #,=4J+#�@A�g%1I=46'#%5)(+&)(X50g%� #[email protected])(+& �)5)(Kk�� eIgX#X�B(��B(+*X5)�?30! #%*X� ���Bt�5)(H6,#�@A�i/2#+G�1IJ+18� 1I��� J+�KeC(c;<(XJ+�f!)� #XgX9)1I/qu
�%1v;B5B(+& � 5B(EkA�Xu��Xÿ%t)@A1�*X1I6 J+#X�B=46?(.@A1v;B5B(+& � 5B(��E! � #4@A1I6 J+#%*%(+-:1I/^5)(�;B*,(�� eI(XgX�)(X5)(:$'! �)(+*%5)(:! �)(X*Xr)(�;B#%gX�j@s(%/2##C;BhI#X*,(+�)(4@A$,b)1�! � #�@A1�6 r)�j@A1Xu
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¯ PQTQS�R,P��� 5 u w ;<� 1v;B�)J+���)5 J+1�5)e�� J+1IJ8=4�?9)1.! � �)J+�)=46?(:5)(�6 � � *%$8! #X*%� ��=I(�=49)� 6'1Xpj#%g+9)�)6'(�& 1�JX*X� J+�)5 1q#%/2#XJj(X&)(r � 9)� 5';B�B(>! #+9)$'! �)1�& � 6?(
R�0;B$,GE� 5 1
Lu�ð.1�9)� & � 5)1
ρ�H
Jj(+6'#j�'18=�/d(+JX�)(+J+�K! #%e�5)(XJ+�)/qu
x
y
R
H
ρ
Pv1 Pv2
P =?
V1
V2
PN1
PN2
PN3
kê(,t(b)
£ 9)�)6'("�%u��"� L w ;B� 1��'� *,(+-:18=4�?9)18!)� � J+�)=�6?(>5)(H6 � � *%$D! #X*%� �¤/21�J+#C;<#%/Ö! #XJ+�B=46?(
��� #%g+9)1C/`b 18g%� J+�i� 1I��1I5"5B(�;B*%(H5)(X& � 5)()p #XeI5)(+& 1I5)(>=�(Kkê(,tq�ak�g't.5)(E=v9)�)r �a�Xu¡���)ud�´=v9)$'&)(4@A$0! #C;o=4� 9)1Dkê(,tPN1
�Pv1
=4$f#C;B� 1I�,1I5 1:� e��)(Xe��)/d()L
PN1 = (pc1 − pa)A1 = ρg
(
H − R
2
)
R√
2L
Pv1 = ρgV1 = ρg
(R2π
4− 1
2R2
)
L = ρgR2
(π
4− 1
2
)
L
��� #�@A1�6 r)�j@A1x=4�?9B1P
5)(>#,=41:6,#%#X�%;B� 5B(+J+5 #%h8=4�)=�J+1I/d(x− y
=4$0#C;B� 1��'1I5 1�� e��)(Xe��)/d()L
Px = PN1 cos 45◦ = PN1
√2
2= ρg
(
H − R
2
)
RL
Py = PN1 sin 45◦ − Pv1 = ρg
(
H − R
2
)
RL− ρgR2
(π
4− 1
2
)
L = ρg
(
H − R
4π
)
RL
��5 JX1�5 eI� J+1�JD=��?9)1P@A1XL
P =√
P 2x + P 2
y = ρgRL
√(
H − R
2
)2
+
(
H − R
4π
)2
�´=49)$'&)(v@A$D!)#C;(b)
� 5 JX1�5 eI� J+1�JX�i#C;Bh�#%*%(+�B(v@A$'b �?3D=4� 9B(E=4$D#C;<� 1��'1�5 �"� eI�)(+eI#%/qL
PN2 = ρg
(
H − R
2
)
RL , PN3 = ρgHRL
Pv2 = ρgV2 = ρg · 1
4R2πL
��� #�@A1�6 r)�j@A1x=4�?9B1P
=4$�LPx = PN2 = ρg
(
H − R
2
)
RL
Py = PN3 − Pv2 = ρgHRL− ρg · 1
4R2πL = ρg
(
H − R
4π
)
RL
�8(X�)(+*%5 #)pX;B#Xg%�j@A1�5)�K=�$D�)=�J+�i� 1�eI$C9)Jj(+J+�Yu�
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G�H��DH(' KMLON1L�PRQ�Xu� �*X1�6 $'h¾9)1:@A1v;B5)(X6 �?3o!)� 1�& 5)� 6?( p
d = 1.2 mp��^J+14GE� 5)(
G1 = 12 kN�G2 = 4 kN
p~=4!)#4@A1�5)1H=�$F$,G�1�JX#%/qu¥ 6'#�=�1�#%*X1�6 $'h¾9)1:=�Jj(+*X1�$E*X#C;B$Kk
ρ = 1000 kg/m3t4p,#C;<� 1v;B�)J+�"=��?9)$EeC(+J+1IeI(+-�(x$E$,GH1�J+$
Tp,6?(+#��06'#+9)�)6 �
=41�;B1I#HeC(+! �)1I/2� 5 1�9<(+6'��1�6 $'h¾9)1�5)(+9B(+eI�f� eI5)(c;F5 �)*X#%(H=v9B#XgX#C;<5 1�!)#X*X�)���Yu�?u���$'h¾9)� r)(�k
ρm = 7800 kg/m3t0! #+9B$,!)� 1�& 5)� 6?(
r = 6 mm5)(+9)14GH1^5)(�#XJ+*%#X��! � 1�&)5 � 6?(
d = 10 mmu
w ;B� 14;B� J+�_=4�?9B$F6'#4@A#%/�6 $'h¾9)� r)(D;B1�9)$C@A1H5B("=�1v;B�)��J+1>*X1I5 J+�?9<( p~(X6,#K=4$o! #XeI5)(+J+��=v9)14;B1�b)�^! #C;�(+r �YLF =
100 Npρ = 1000 kg/m3
pD = 30 mm
pH = 500 mm
u
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% 149B�)/2#x;�(�#C;B�)1v;B�)/2#H=4�?9B$E! � � J+�B=46?(�5B(�6 �)� *X$E! #%*X� ��=�(�=v9)� 6'1H�Xu¡���)uQ��#C;<=�1�J+�B/2#�=41�;<(H=�/2#H! � �?9B� 6'#%/� eI*X#j�'1�-�( w @�9)1I� #X*%1d@A1v;<5)(+& �)5 1�� eC(+g%�)(j9)�E!)#XJ+! $'5 #�! � #%� e�*%#+n�5 $�eC(+! �)1I/2� 5 $E7:9)$'�';�(x�>;<(�=�/2#x6'#%5)=4Jj(+JX#X*%(+9)�;<(�@A1�! #XJX� 1�g%5 #:;<(x=�$'/d(�=4*X� 3H=4� 9B(86'#4@A1.;B1�9)$C@A$�5B(8J+$HeI(+!)� 1I/2� 5)$�g%$�;<1d@A1v;B5B(+6?(�5 $C9)�>;<(xgX�>#X5)(�g%�?9B(8$>=4Jj(X-�$/2� �)#X*%(X-:()u
ρ PG
P
PN
R
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��eC(+gX1I� �)/2#>=�(j;<(�JX$0eI(X! � 1C/2� 5 $D6?(+#H5)(>=v9B� r �_�Xu¡���)p,Jj(X6,#�;<(�@A1v;�(+5f;B1�#H! #%*X� ���f6'#4@A��@A18#XhI�)(+5)� &)(+*,(xg%$�;<16 �)� *%(i!)#X*X�)�·5)(i6'#4@A$^J+�)(jGE�)/2#^=4�?9)$�! � �)J+�)=46?()p2(F#%=4Jj(+9)�^;B1�#E@A14;B5)(i� 9)�a*X� ��1D�)(X*X5 �?3�! #X*%� ���~u0 �(X6?9)1%pJ+1I& 5 #,=4J>6'#4@s(>=4185)(+9B(+eI�f$'5 $'Jj(+�"J+18eC(+! � 1C/2� 5 1q@A18$0=�Jj(+-.$f/2� � #%*%(X-:()p?Jv@4u
~PN + ~R︸ ︷︷ ︸/ Ðvë çlâ Ì çzè�ê ä{ñ ælâzå ÐvÌ åãælß Ì + ~PG
︸︷︷︸ËÍÌ ç ä àµá6æ{â{å ÐvÌ åãælß Ì = 0ks�Xu��r�,t
h�;B18=�$<L ~PN ] =4�?9<(>! � � J+�B=46?(>5)(H�)(+*%5 $D! #%*X� �q�~PG = ρ~g V ] J+14GE� 5B(>J+1�& 5)#%=4JX�"6'#�@s(H=�1�5)(j9B(Xe��"$'5 $'Jc(X�KeI(+!)� 1I/2� 5)1v�~R ] =��?9B(H6'#4@A#,/`6 � �)*%(>! #%*X� ��;B149B$I@A185B(>J+1�& 5)#%=4J>$'5 $'Jj(+�"eI(X! � 1I/2�)5 1Xu
¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬xF¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼)ñ
£ �?9<(PG
;<149)$C@A18$"J+14GE� ��J+$KeI(+!)� 1I/2� 5)1Vp?;<#X6F=4� 9B(
PN;B149B$I@A1��)=4! #C;FJ+1�GE� ��Jj(E�)(X*X5 18!)#X*X�)���ak�! #%h¾9)1v;�(v@
! �)� & $0*%1�eC(+5 $DeC(>=4�?9)1�! �)� J+�)=�6'(>5B(>�)(+*%5 1�!)#X*X�)���Bt4u�8(X6,#�@A18! #>J+� 1Ib 1I/10�$'J+5 #%*X#,/`eI(X6'#X5 $�p ~P = −~R p';B#+9<(+e��B/2#x;B#E�)e��)(XeI(�eI(>=4�?9B$0! � �)J+�)=46?()L
~P = ~PN + ~PG = ~PN + ρ~g Vks�Xu��l�+t
�ú#X!)��J+1C/�=v9B$,&B(v@A$�pY6?(c;<("=41HeI(X! � 1C/2� 5)(K=�(%=4J+#�@A�o#C;n�)(+*%5 �?3F! #%*X� ���o��6 � � *%1>! #X*%� ���^5)(06'#�@A$FJ+�)(+G>�B/2#
=��?9)$0!)� � J+�)=�6?( p � eI�)(+e8=I(H=��?9)$f! � � J+�B=46?(:@A1XL
~P =
n∑
i=1
~PN + ~PG =
n∑
i=1
~PN + ρ~g Vks�Xu��r�%t
¯ PQTQS�R,P��ÝF6� w ;B� 14;B� J+�K� 5)J+1�5 eI� J+1IJ0=4�?9B18! � � J+�B=46?(>5)(H6 � � *%$D! #%*X� �¤=�(>=49)� 6'1E�Xu��r�)p,#%g+9)�)6'(>¢32cûH#%/2#XJj(X&)(r � 9)� 5';B�B(>! #+9)$'! �)1�& 5 �)6'(
R�f��� � � 5 1
Lu £ /d(+J+�)(XJ+�K! #XeI5)(+JX�)/�*%149)�)& � 5 1
HpρpR
�Lu
x
y
R
H
ρ
PG1 PG2
P =?
V1
V2
PN1
PN2
PN3
èmä%ì(b)
£ 9)�)6'("�%u �v� L w ;B� 1��'� *,(+-:18=4�?9)18!)� � J+�)=�6?(>5)(H6 � � *%$D! #X*%� �¤/21�J+#C;<#%/Ö! #XJ+�B=46?(
�8(X#���!)� 1�Jj3,#C;B5 ��! � �)/21I�Yp+�H#X*%(4@Yb 1.gX� J+��$'�)(+�'1�5�5)(�;B*%(�5)(+& �)5)( pj$'*X#j�'1I-�1C/K�)(+e�9)� & � JX�?3��)(X*X5 �?3x!)#X*X�)���$fr �?nq$E;B#%gX�j@s(X-:(HeI(X! � 1C/2� 5 1
Vu�Ô\(+6'#:@A18eI(>=v9B$,&B(v@q! #C;akm(%t4L
PN1 = ρg
(
H − R
2
)
R√
2L
PG1 = ρgV1 = ρg
(3
4R2π +
1
2R2
)
L = ρg
(3
4π +
1
2
)
R2L
��� #�@A1�6 r)�j@A1x=4�?9B18! � � J+�B=46?(>5)(H#%=�1:6'#%#X�%;B� 5)(XJ+5 #%h:=��)=4J+1C/d(x− y
L
Px = PN1 cos 45◦ = ρg
(
H − R
2
)
RL
Py = PG1 + PN1 sin 45◦ = ρg
(3
4π +
1
2
)
R2L+ ρg
(
H − R
2
)
RL = ρg
(3R
4π +H
)
RL
��5 JX1�5 eIJ+� J+1IJD� 1�eI$C9)J+$C@A$'b 1x=4�?9B18! � � J+�B=46?(:@A1XL
P =√
P 2x + P 2
y = ρgRL
√(
H − R
2
)2
+
(3R
4π +H
)2
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��(>=49)$'&)(v@.! #C;akêg,t.� 5)J+1�5 eI� J+1IJ+�o=4� 9B(PN2
pPN3
�PG2
=4$D#C;<� 1��'1�5)�f� eI�)(+eI�)/d( L
PN2 = ρg
(
H − R
2
)
RL; PN3 = ρgHRL
PG2 = ρgV2 =3
4ρgR2πL
w ;BhI#X*,(+�)(4@A$,b)1�! �)#4@A1�6)r �j@A1x=4� 9)1PL
Px = PN2 = ρg
(
H − R
2
)
RL
Py = PN3 + PG2 = ρg
(3R
4π +H
)
RL
�
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-�1Ih�#%*%(�=�J+�)(X5)(�#%6 *%(X��1I5)(q@A1:J+1�& 5)#X��b)$Dh�$,=4J+� 5)1ρp?(8;B�)$,h4(�� e49B#+G�1I5)(x;B1A@s=�J+*X$E6'#X5B=4Jj(+5 J+5)#Xh:! � � J+�B=46?( u
¥ 6'#�@A18! #%6'(Xe�� *,(+-:1:/d(X5 #%/21IJ+�)(h = 134 mm
#C;B� 14;B� J+�i=��?9)18� eIJ+1�eC(+-�(>�K=I/2� r)(X-:(>*%1�e�1 ¥ ] ¥ u' �(+JX�K=�$! #C;<(Xr �YL
h1 = 1.2 mph2 = 2.2 m
pR = 0.4 m
ppv = 8 kPa
pα = 30◦
pρ1 = 900 kg/m3
pρ2 = 1000 kg/m3
�ρm = 13600 kg/m3
u�?u�ï21�eI1�� *%#%(+�a#%g+9)�)6'(o! #+9B$I9B#X! J+1_k
R = 1 mt�! � � & *%� ��b)1�5^@A1f*%1�e�#,/ ¥ ] ¥ eI(^e��';Z6'#4@A�F@A1"5)(Xh�5 $'J�! � 1I/d(
3%#%� � e�#%5 Jj(j9)��eI(�$,h4(X#α = 45◦
uQ��� 1�eI1�� *%#%(X� $8=41q5)(j9<(+e���*X#C;<()pj(�5B(v@A*X�)��#4@\Jc(X& 6 �H� 1�eI1�� *%#%(X�)(�/2#X5)J+� �)(X5@A1�� 5B=4J+� $?/21I5 JfeC(0/21I� 1�-:1�!)� � J+�)=�6?( u w ;B� 1v;<� J+�^=��?9)1H6'#4@A1�#%! J+1I� 1�b $C@A$K*X1Ie�$ ¥ ] ¥ eI(H;B*,(D! #%6?(+e��)*%(+-�(� 5)=�J+� $?/21�5 Jj()L�kê(%t
pm = 10 kPa�ak�g,t
pv = 20 kPau
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ρ = constt
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V̇ = constkm� u��ct
h�;B1�@A1f=I(V̇
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V̇ =
∫
A
(~v, ~n) dAk �?u��Xt
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zzz
v = v(r, z)v = v(r, z) v = v(z)
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5)()LV̇ =
∫
A
v dAkm�?u ¢%t
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vs A =
∫
A
v dA =⇒ vs =1
A
∫
A
v dAkm�?u û,t
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v1 v2
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A2
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V̇ = const =⇒ v1A1 = v2A2
h�;B18=�$v1
�v2 V+P2RX¹2Ø"RF»)PQMXTQU2R $0! �)1I=41Ir �)/d(H! #%*X� ��� 5)(
A1
�A2
u'DH�'DH(' �����������¤Q��t���ÔL1�t��N��RL�YøQ��RL
��#,=�/2J+�)(4@s/2#0@A14;B5 $`r)1�*o6)� #XeK6'#4@A$a=4JX� $C@A�lJX1�& 5 #,=4J���$%#X& �)/2#o;B*,(F! � 1C=41I6'(Ö� ] �0��� ] �?p�Jj(X6,#F;<(^$a-��)/d(=�J+� $C@s(+-:1��)/d(8@A1v;B5 #C;<�)/21�5 eI�j@s=46)�F6?(+�)(X6 J+1��akê#X*X#�@A1�*X�?9)#0*%(jGE5 # ] eI(E! �)1I=41I6'18eI(E6'#4@A1x=�1x! � ��1�P.1�� 5)$I9B�j@A1�*,(@A14;B5)(+&)� 5)(�J+#�/2#%�)(qgX�)J+�8eI(j;B#X*%#+n�1�5 # � t4u���e�$'*X14;B1�5)1Q!)� 1�J+!)#%=4Jj(X*X6'1XpCeI(q! �)1I=41I6'1��d�8�.*,(jGE�8=v9)14;B1�bB(Q@A1v;B5)(X& � 5)()L
Y1 = Y2 + Yg1−2
km� u ÿ%th�;B1.@A1%L
Yi =piρ
+ gzi + αiv2i
2
km� u �,tYi ] $'6 $'! 5)(H1�5)1�� hI�j@s(H! #�@A1v;B�)5 � r �i/d(X=�186'#4@A$07:9)$'�';i! #,=41v;<$I@A18$
i ] J+#,/`! � 1C=41I6 $0r 1I*X�Ypi = 1, 2
upiρ
] ! � �)J+�)=45B(>1�5 1I� h��j@s(>!)#�@A14;B� 5 �)r �F/d(X=�1 ] 1I5 Jj(j9)!)�j@s(gzi ] ! #+9B#+GH(4@A5)(H1�5 1I� h��+@s(�!)#�@A14;B� 5 �)r �F/d(X=�1αiv2i
2] 6 � 5)1�J+� r 3,(>1�5)1�� hI�j@s(>! #:@A1v;B�)5 � r �F/d(%=41
ð.149B� & � 5)(αi
=�1�5B(+e��)*%(0® {BPQTQ{)[QT2VX{B|lWY{BR'ÕKT � Tn�CR%UQ� �?9B� WY{BPQR,W � T2{)UQT�WY{BR'ÕKT � Tn�CR,U\�`WQTQUQR,�?T�}\WYRR,U2R%P�¡+Tn�CR u w *%�)/�6'#X1I7��)r �j@A1�5)J+#%/�=�106'#X� �)h�$C@A1fhI� 1���6?(i6'#4@s(o=�10& �)5 �Ö(X6,#�=41f6 � 5 1IJ+� & 6?(^1�5 1I� h��j@s(^�)(X& $'5)(! �)1�6'#�=4�)1v;B-:1FgX�)e�� 5 1o6'(X#
v2i /2
pQ@A1I��7�� e��)& 6 �ZgX�)e�� 5)(a/21I-:(`! #`! #%! � 1�&)5 #%/�! � 1C=41I6 $Ùkm=4� 14;B-�(agX� eI� 5)(K@A1
¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬xF¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼¼X
eC(X/2� ��n:1I5)(>gX� eI� 5)(,tvudÔ~(X6,#Eb)1:#%*%(4@�6'#X1C7�� r �j@A1I5 J>eI(+*%�)=4�)J+�f#C;i� 1�G>�B/d(E=4J+� $C@s(X-:()L
α =
{
1.058,JX$,�)gX$C9)1�5)J+5 #D=4JX� $C@s(+-:1
2,9B(X/2�)5)(+� 5)#>=�J+� $C@s(+-:1 km� u��Xt
ð.�';B�B/2#0;�(x@A1>6'#C;^J+$'� g%$I9B1�5 J+5)#XhE=4J+�)$I@s(X-:(α ≈ 1
k�$'*%1�6ob)1I/2#K$%=�*%(v@s(XJ+�α = 1
eI(fJ+$'� g%$I9B1�5 J+5)#K=�J+� $C@s(+-:1Ct4p;B#%6�@A1`6'#C;´9B(X/2�)5)(+� 5)#Xh`=4JX� $C@s(+-�(
α = 2u �(�@A1
αlam > αtur/2#+G�1C/2#l;<(ZeC(+6?nq$'& �)/2#���5B(�#%=45)#X*X$
� eIh¾9)1v;�(�!)� #%7��?9<(�g%� e��)5 1�eI(:#X*,(�;B*%(:� 1�G>�B/d( ] 5)(+�)/21%pX6'#C;EJ+$'� gX$C9)1I5 J+5 #%h�=4J+� $C@s(X-:(�$�*X1�b)1I/F;B149)$�! #X*%� ���)5 1! #%! � 1I& 5 #Xh.! � 1C=41I6'(�r 1I*X�x�)/d(%/2#8=v9)$'& $C@�;<(�@A1qgX� eI� 5)(�! � � gX9)�?GE5 #�6'#X5)=�Jc(X5 J+5)(�k�J+$'� g%$C9)1�5 JX5 #.@A1�e�hI� #,t4p+Jj(+6'#�;�(@A1�! � #%7�� 9Kg%� e��)5 1x6'#C;FJ+$'� gX$C9)1I5 J+5 #%h�=4JX� $C@s(+-�(D=v9)�)& 5 �j@A�F!)� #%7��?9B$f=4�)1v;B-:1�g%� e�� 5)1Xp)5)1�h�#0��J+#8@A1�J+#0=v9B$,&B(v@eC(K9B(X/2�)5)(+� 5)�`! � #,7��?9~u ¥ 6'#^$^eC(c;<(XJ+6 $�5 �j@A1"5)(+h¾9B(X��1I5��)14GE�)/�=4JX� $C@s(+-�( pd! #C;B�B(+e�$?/21I*%(i=41E;<(F=41f�)(c;<�a#J+$'� g%$C9)1�5 JX5 #D=4J+�)$I@s(X-�$ �
ä àV��à ä àãâ Ï â{æ�â{æ{ê"è
���b��� ��� �V������ �
v1
v2
z1
z2
�
�
�
�
£ 9)�)6'(E� u ¢)L~ �1I#�r)1�*X#%*X#C;<(�$06'#%/218=�J+� $C@A�F7:9)$'�';
��#,=v9)14;B-��8& 9B(+585)(q;B1I=�5 #4@Y=4JX�)(+5 ��@A1v;B5)(X& � 5 1�km�?u�ÿXt�! � 1v;�=4Jj(+*Xn�( ¡X¸Y»)TQ�?WYR�VX�?P\¸v�IU2R�R,UQR,P�¡+Tn�IRxO\PQT~[2TQWY{BSVX�?P\¸v�CN?Ø"No� R{}QUQ{<V+�?TZ{%¹�O\P2R'VXR,WYN�"WYNFOQP2R,V%R,W\¸l¼ p'�f#%5x@A1�@A1v;B5B(+6"e�gX�)� $>9)#X6?(+9)5 �?3Eh�$'g%� Jj(+6?(x1I5 1�� hI�j@A1��hI$,g%� J+6 $f1�5 1I� h��+@A18$%=v9B1v;iJ+� 1�-�(>! � � 9)� 6'#%/l=4J+�)$I@s(X-:(>JX1�& 5 #,=4J+�K#C;i! � 1C=41�6?("��6?(�!)� 1I=�1�6 $K�?u
Yg1−2=∑
Yglok1−2+∑
Ygtr1−2
km� u �,t {BWYN'[2U\T ¡X¸Y»)T � T�R,U2R,P�¡+Tn�CR
�8#%6'(+9)5 �EhI$,g%� r �D1I5 1��)h��j@A1�=41�;B1��H(+*,(v@A$H5)(x5 1I6'#%/a#C;B�)1��'1�5 #,/_/21I=4JX$D$>r 1�*%#X#C;B$Hh�;B1�=�1�#C;<� 1��'1�5B(81�5 1I� h��+@s(7:9)$'�';�(i�+#C;B$'eI/21C�+#C;`7:9)$'�';<(K�a#%5)(i=41E6'#X�)�)=4J+�aeC(i=4J+*,(+�)(X-�1E5)1�6'#XhD=�1�6 $'5';<(X� 5 #%hD=4J+� $C@s(X-:( ] J+#F=4$_/21I=�Jj(h�;B10r 1�*F/21I-:(i!)�)(+*,(+ra=4*%#XhE! � $,G>(+-�(`k�6'#+9B1�5 #Bp~6 � �)*X� 5)(,tvpd/21I=�Jj(ih�;B1H;B#X9B(+eI��;B#F=4$,G�1I-:(F�?9)�a! � #%��� �)1�-�(! #%! � 1I& 5 #Xh0! � 1I=�1�6?( pd/21C=4Jj(i$_r 1�*%#X*%#C;B$F5)(i6'#�@A�)/d(o=�1D5)(j9<(+e�10*X1�5)J+�?9)�Ypd�)J�;Qu £ *X�a#%*X�ah�$'g%� r �a1�5)1�� hI�j@A1f=�1#C;B�)1��'$C@A$D5)(H#%=�5 #X*%$Dð.(4@s=4g,(j3%#%*X#%h�#Xg%�)(X=�r)( L
Yglok= ζ
v2
2
km� u �,t
h�;B1�@A1�#C;BhI#X*%(X�)(v@A$'b �ζ6,#%1I7�� r)�j@A1�5 JD9)#X6?(j9B5 #Xh8hI$,g%� J+6?(E1I5 1�� hI�j@A1�� 9)�F6'#X1C7�� r �j@A1I5 JD#XJX! #X�)()u�ð.� 1v;<5 #%=�J+�
#%*X�?3`6'#X1C7�� r �j@A1I5 Jj(ob)1"$Ö=4*X�B/´eI(j;<(+r)�)/d(og%� J+��$'5)(+!)� 1v;�! #XeI5)(+J+1KeC(o6'#X5)6 � 1�Jj(X5Ör 1I*X#%*X#C;\u�Ô\(+6'#j�'1Xp�=�J+*%(X�
¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬xF¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ¼¼�
;B#%h�#%*X#X�B(H@A1D;<(D@A1 »BPQMXTQUQN v »)PQMXTQUQN�T\M,N�[d{BWYN'[QU2{�¡Ö{B�?OQ{BPQN u_Ô\(+6'#ob)1f� 1Ir �)/2#^eI(o=�$,G�1�-:10!)#X! � 1I& 5 #%h! �)1I=41I6?("km=v9)�)6'(D� u �%t�h�$'gX� Jj(X601�5)1�� hI�j@A18gX� JX�"#C;<� 1��'1�5"� eI�)(+eI#%/qL
Yg = ζsuv22
2
�(X6?9)1Xp�1�5 1I� h��j@s(^$a� eI5 #%=�$ζsu
v22
2
@A1K�+#C;<$,eI1�Jj(X�X#C;Ö7:9)$'�';B5 1K=4JX� $C@A1"eC(o=�J+*%(X�)(+-:1f*%� JC9)#Xh4(o5)(�/21I=4JX$`h�;B1�@A1;B#%�x9)#�;B#>! �)#%/21�5)18! � 1I=�1�6?( u
�%1v;B�)5 ��9B#X6?(j9)5)��#XJX! #X�l6'#C;l6'#Xh4(^b 1K5 1K6'#X� �)=�J+��ð.(4@s=4g,(j3%#%*F#Xg%�)(+eC(+ri@A1"9)#%6?(j9)5 �l#XJX! #X�Ö$%=v9B1v;Z5)(+h¾9)#%h! �)#X���)� 1�-�(>r 1�*%#X*%#C;<( p'=v9)�)6'(E� u ûBu
v1 v2
A1
A2
£ 9)� 6?(E�?u û)Ld�8(+h¾9B#�! �)#X���)� 1�-:18! #X!)� 1�& 5)#Xh�! � 1C=41�6?(
ü~$,g%� Jj(+6"1�5 1I� h��+@A18$%=v9B1v;i5)(+h¾9)#%h�!)� #X��� � 1�-�(E=41�#C;B�)1��'$C@A1:6'#%� � ��b)1�-:1I/�P�#%�%;<( ] �8(+� 5)#X#X*%1�7�#X�B/2$I9B1XL
Ygnp=
(v1 − v2)2
2
km�?u¡���,t
� ¸Y»)T � T�R%U2R,P�¡XTz�CRK¸BV+[2RX¹��?P2R,ØfNw *%��h�$'g%� r �H1�5 1I� h��j@A1�=�$x! #,=v9)14;B� r)(:J+� 1I-:(��)eI/21��'$xJX1�& 5 #,=4J+�>�>e��?;<(:r 1�*%�YuQ���)�?9)� 6'#%/^=4J+�)$I@s(X-:(:g%� e�� 5)#%/
v6)� #Xe�r 1I*H;B$,G>�)5 1
L�K! #%*X� ��� 5 1�! #%! � 1�&)5 #Xh�! � 1C=41I6'(
A#%5"=�1:#C;<� 1��'$C@A1�5)(>#%=�5 #X*%$H �(+�)=��j@A1�*%#Xh:#XgX�B(X=4rB( L
Ygtr= λ
L
4Rh
v2
2
km�?u¡�X�ct
ð.149B� & � 5)(Rh
=4185B(+e��)*%(�� TY¹dPQN%¸'[2T�}QW\T�PQN,¹dTn��¸<V �K#X5K=41:;<1I7�� 5 �)��1�6?(+#>#C;B5 #,=�! �)#XJ+#%& 5 18! #%*X� ��� 5 1��#%6 *%(X��1�5)#Xh:#XgX�B/d( p?Jv@4u
Rh = A/Ou�Ô\(+6'#:@A18eI(H6 � $,GE5 $0r)1�*>! � 1I& 5 � 6?(
D3'�?;B�)(c$C9)� & 6)�"�B(c;B�j@A$,=.@A1v;B5B(+6YL
Rh =A
O=D2π
4Dπ=D
4
!)(f=41� �(+�)=��j@A1�*,(D7�#X�B/2$I9<(DeI(f=v9)$'&)(4@�=4J+�)$I@s(X-:(f6 � #%e�r 1I*X��k�J+#"b 1�gX� J+��=v9)$'&)(4@�$F=4*%�)/�eI(j;<(+r �B/d( � tF=4*%#C;B�5)()L
Ygtr= λ
L
D
v2
2
km�?u¡�C�%tð.149B� & � 5)(
λ=�1�5)(Xe�� *,( WY{<R,ÕKT � Tz�CR,U\�"�?P2R%Ø"N �0#%5D=41�$H#%! ��JX1I/`=49)$'&)(v@A$HeI(X*X�)=��D#C;fï21A@A5 #X9?;<=�#X*X#%h�gX� #�@s(
�a� 1�9B(+JX� *X5 1D3'�B(+!)(X*X#%=�J+��r 1�*%�kuf� =49)$'&)(v@A$^9B(%/2� 5)(+�)5 #XhD=�J+� $C@s(+-�(ir 1�*%�a=41E! #%5)(+�H(v@A$o6?(X#K3'�?;B�)(c$C9)� & 6)�
h¾9B(XJ+6'1Xp�6'#%1I7�� r �+@A1�5 J`JX� 1�-�(`eC(+*X�B=4��=�(%/2#`#C;Re
��Jj(aeI(X*X�)=�5 #%=�J`=41i;<#XgX�+@s(a� eoJj(X& 5 #Xho� 1���1�-�(��8(+*%�j@A1 ]
¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬xF¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ��
£ J+#%6?=4#X*%�?3x@A14;B5)(+&)� 5)(HeI(>=v9B$,&B(v@�=4J+� $C@s(X-:(>$Dr)1�*X�B/d( p?Jv@4L
λ =64
Re− 9B(X/2� 5B(+� 5 #D=�J+� $C@s(+-:1 km�?u¡�I¢,t
�`=49)$'&)(v@A$:J+$'� gX$C9)1I5 J+#%h�=4J+� $C@s(X-:()pCg%1�ed#%gXe��)�)(�;<(.9)�8=41�r)1�*�! #%5)(+�H(�6?(X#�3?�';<�)(c$C9)� & 6 �8h¾9<(+J+6?( p�3?�';B�)(c$C9)� &)6 �3?�)(X!)(+*,(F�?9)�Ö3?�';B�)(c$C9)�)& 6 �l! #XJ+!)$,5)#o3?�)(X!)(+*,( pQ;B#%gX�j@s(X-�1KJj(+& 5)1feC(+*X�B=45 #,=4J+�
λ(Re, k)5 �j@A1i/2#%h�$'b 1Xp~@A1���=41
5 1��)(X=�! #+9B(+G�1a! #%J+� 1IgX5 �)/ Jj(+&)5 �)/�� 1I��1�-:1I/·ï21s@A5 #+9 ;<=4#%*X�?3`@A14;B5)(X& � 5)()u �Y(+J+#�=�1`5 1I! #XJ+!)$,5)��J+1�#%� �j@s=46)�� 1Ie�$C9)Jj(+JX�Fï21A@A5 #C;<=�#X*%�?38@A1v;B5B(+& � 5B(D/2#X�)(4@A$E;B#%! $'5 � J+�K#C;BhI#X*,(+�)(4@A$,b)�)/`1I6 ! 1��)�)/21�5 Jj(+9)5 �)/l! #C;<(Xr �)/d()u��8(>Jj(v@5)(X& � 5i=41�;B#%�x9)#�;B#E7�#%�)/2$C9B(>6'#4@A18=4$>;�(+J+18$DJj(XgX149B�Yu
ï�u gX� #�@ ��(X*X�)=�5 #%=�Jλ = λ(Re, k)
w g+9B(%=4JH! � �)/21I5 1 ¥ $'J+#%�
�Xuλ =
64
ReRe < Rek = 2320 ]
�?uλ = 0.0025Re0.333 2320 < Re < 4000
��(v@A& 1I5 6'#
¢ uλ = (1.8 logRe− 1.5)−2 4000 < Re < 3 · 106
��#X5)(X6'#X*
û)uλ = 0.3164Re−0.25 4000 < Re < 105
P�9<(+e��+@A$%=
ÿ?u 1√λ
= −2 log
(k
3.71+
2.51
Re√λ
) ��#X9)gX�)$,6 ] ðq(v@AJ23
k< Re <
560
k
� uλ = 0.1
(
1.46k +100
Re
) ¥ 9)J+��$C9
�'uλ =
(
1.74 + 2 log1
k
)−2 ���B(+5 JC9 ] ��� 6 $'�)(c;<e�1Re >
560
k
� uλ = 0.11k0.25 � �)7�� � 5B=4#X5
� uλ = a+ bRe−c
a = 0.0094 k0.225 + 0.53 k Re < 104ð.$I;
b = 88 k0.44 10−5 < k < 0.04
c = 1.62 k0.34
¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬xF¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ�
Ô\(+6'#j�'1XpC! #X� 14;x� e��)(XeI(�;<(XJ+�?3�$�Jj(XgX149B�YpI6'#%1I7�� r �+@A1�5 J�J+�)1�-�(�=41�/2#XG�1d#C;B�)1v;B� JX�8�86,#%� � ��b 1�-:1I/ � $I;B�j@A1I*X#%h;B�+@s(+h��B(X/d(Kk�*%�';B�K! � 14;<(+*,(+-�(%tvu
��eI(j;<(+r)�)/d(Hb 1�*X� 14;B5 #%=�J�6'#%1I7�� r �+@A1�5 Jj(�J+� 1�-�(>$'5)(+!)� 1v;igX� JX�feC(c;<(XJc()p'�?9)�Kb 1�gX� JX�"eC(c;<(XJj(�7�#%�)/2$C9B(�5B(#,=45 #%*X$E6'#4@A1x=�1:#%5"/2#XG�1x=��)(+& $'5)(XJ+�Yu¯ ¸YS:O2No¸ � R,|�{)|�{%¹d¸
�úh�#%J+#X*%#f=�*X�)/�eI(c;�(+r �)/d()p~(fJc(X6'#j�,1H��$F! �)(X6'=��Yp�$Fr 1I*X#%*X#C;B$Kb 1>gX�)J+�^$'hI�)(+�'1�5B(0! $?/2!)()u���J+� r)(4@:!)$'/2!)15)(0=4J+� $C@s(X-�1�b 1xgX�)J+�i#Xg%$%3?*,(+b 1I5f! �)1�6'# �CRX¹2TQU\T�}QUQ{�¡KPQN,¹�NaOQ¸YS:OQR p Yp [J/kg]
6'#4@A�F=41�/2#+G�1�=v3?*%(XJ+� J+�i6?(+#1I5 1�� hI�j@s(�! #�@A14;B� 5 �)r �"/d(%=41�6,#�@A$E! $?/2!)(�! � 1v;�(v@A1�7:9)$'�';B$�u#�x�)e�� & 6)�f! #,=�/d(XJ+�)(+5)#)p,! $?/2!)(��X#Xg%#Xh4(+b)$I@A1C�X7:9)$'�';! �)� J+�)=�5 #%/l1�5)1�� hI�j@A#%/qu
�n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n��n�n�
H
Yp
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Y1 + Yp = Y2 + Yg1−2⇐⇒ p1
ρ+ gz1 + α1
v21
2+ Yp = Yi =
p2
ρ+ gz2 + α2
v22
2+ Yg1−2
km�?u¡��û't�8(X6,#�@A1
z2 − z1 = Hpc@A1v;B5B(+& � 5B(ikm�?u¡��û't.=418=4*%#C;B�f5B( L
p1
ρ+ α1
v21
2+ Yp = Yi =
p2
ρ+ gH + α2
v22
2+ Yg1−2
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Pk ≡ Ph = ρ V̇ Ypkm�?u¡���,t
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tvL
P =Pkηp
=ρ V̇ Ypηp
km�?u¡�*�Xt
h�;B1.@A1ηp
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r 1�*%#X*%#C;B$xgX�)J+�8@A#X��$'*X1�6>9<(X/2� 5)(X� 5 #BuQ��� �)J+� & 5 �"ï21A@A5 #X9?;<=�#X*8gX�)#4@deI(x=�J+� $C@s(+-:1:$�r 1I*X�8@A1Rekr = 2320
u �(+J+�E=�$��D=v9B1v;B1Ib �>! #C;<(+r)�YL
ρ = 900 kg/m3pν = 4 ·10−5 m2/s
pD1 = 80 mm
pD2 = 100 mm
pL1 = 18 m
pL2 = 10 m
pζu = 0.1
pζv = 3.5
pζk = 0.2
pH = 4 m
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/d(+5 #,/21�Jj(+�B(f=I(DGE*X#,/�/21I� 1>!)(c;B#%*X�^! � � J+�B=�(+6?()u��8(f#%=45)#X*X$i!)#X6?(+eI� *%(X-:(f/d(+5 #,/21�Jj(+�)(f#C;B� 14;B� J+��6'#X1 ]7�� r �j@A1I5 J"9)#X6?(+9)5 #Xh�#XJ+!)#X�)("*X1I5 J+�?9<(ζv0
6'(X#f�_=45B(+h�$i$'hI�)(+�'1�5)1H!)$'/2!)1Xu £ J+� $C@s(+-:1�@A1>J+$'� gX$C9)1I5 J+5 #i$3?�';B�)(c$C9)� &)6 �ih¾9B(+JX6 �)/�r 1I*X�)/d()u. �(+J+�F=�$"=49)1v;B1Ib �F! #C;�(+r �YL
l = 3 mpl0 = 5 m
pL = 25 m
pD = 50 mm
ph = 11.8 mm
ph0 = 100 mm
pH = 2 m
pζu = ζk = 0.5
pζv = 2
pρ = 1000 kg/m3
pρm = 13600 kg/m3
ν = 1.006 · 10−6 m2/s�ηp = 0.8
u¢ u���(�r 1�*%#X*X#C;H5)(�=v9)� r)��#C;B�)1v;B� JX��! �)#XJ+#%6�*%#C;B1�6'#�@s(�=�1q=49)� *,(�� eq� 1Ie�1I� *X#,(+�)( ¥ $�� 1�eI1�� *%#%(+�Hð:uX��(�6'#X9)� 6'#
=41�! #%*X1�bB(�! �)#XJ+#%6f6 �)#Xe�r 1�*%#X*%#C;K(+6'#>=41�$D-:1�h4(H$'h��)(j;B�"! $?/2!)(H& �j@s(:@A186?(+�)(X6 J+1I� �)=4JX� 6?(x;�(+Jj(��)e��)(Xe�#%/qL
Yp = Y0
1 −(
V̇
V̇0
)2
��#XeI5)(+J+�H=4$�=49)1v;<1�b �>! #C;<(Xr �YLY0 = 60 J/kg
pV̇0 = 40 l/s
pH = 4 m
pL1 = 12 m
pL2 = 15 m
pD1 = 80 mm
pD2 = 100 m
pλ1 = 0.025
pλ2 = 0.027
pζu = 0.1
pζv = 7.5
pζk = 0.2
ppm = 0.2 bar
�pv = 0.1 bar
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Y1 = Y2 + Yg1−2 ⇐⇒ Y1 = Y2 + Yg1−R + YgR−2
p1
ρ+ gH =
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ρ+ C1
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2+ C2
v22
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ρ+ C1
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22 + v3D
23
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V̇3
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YgR2 = ζR2
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2
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v22
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ht = 5 muo �(+J+�Ö=4$
=v9)14;B1�b)�^! #C;�(+r �YLV̇B = 6 l/s
pd = 60 mm
pD = 150 mm
pl = 100 m
pl2 = 60 m
pH1 = 2 m
pH2 = 3 m
pλ = 0.03
kêeI(K=4*%1>r 1�*%�Btvpζk = 0.3
pζv = 3
pζv1 = 10
pζu = 0.5
pζm = 0.05
pζR = 0.25
ppv = 0.2 bar
�ηP = 0.75
u(,t w ;B�)1v;B� JX�i*X�)=�� 5 $
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l0 = l1 = 4 mpl2 = l3 = 4 m
pd0 = 60 mm
pd1 = 50 mm
pd2 = 30 mm
pd3 = 40 mm
ph1 = 5 m
ph2 = 3 m
pζs = 1
pζu = 0.5
pζk = 0.4
pζv1 = 2
pζv2 = 3
pζv3 = 4
pηP1 = ηP2 = 0.8
pρ = 1000 kg/m3
pλ = 0.028
u
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d = 50 mmp�6'#4@s(0*X#C;B�o6'(f! #%J+� #X�H(+& $K$'h��B(+�'1�5)(8@A1�gX9)1�5';�(D6?(+�)(X6 J+1I� �)=4JX� 6'1
K = 0.01242 m3/(s m1/2)6'#4@s(�@A1E! #X*%1�eC(+5)(f=�(f;B�)7�1I� 1�5)r �j@s(j9B5 �)/Ù/d(+5 #,/21�J+� #,/�$o6,#,/21�@A1>GE� *%()u>���)#XJ+#%6^6 �)#Xe>� 5)=�Jc(+9B(+r)�j@A$^=�1� 1�hI$C9)� ��1i!)#%/2#Xb)$�#X!)J+#X& 5)#XhK*X#C;<(lk�hI�)(+5B(a=�(_*X1I5 J+�?9B#%/
v2t4u w ;B� 14;B� J+��=�5)(+hI$`6'#4@s("@A1i! #%J+� 1IgX5)(�eI(
¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬xF¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ�5
! #XhI#X5F!)$'/2!)1Xu���#%e�5)(XJ+�^=�$F=v9)14;B1�b �^! #C;<(Xr �YLpm = 3 kPa
ppv = 800 Pa
pl0 = 4 m
pl1 = 2 m
pl2 = 3 m
pd0 = 65 mm
pd1 = 80 mm
pd2 = 50 mm
pH = 1 m
ph = 100 mm
pζu = 0.5
pζk = 0.5
pζk = 0.5
pζR0 = 0.5
pζR1 = 1
pζR2 = 0.5
pζv1 = ζv2 = 1.5
pλ = 0.025
�ηP = 0.8
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KpB! �)#XJ+#%6o7:9)$'�';<(06 � #Xe�r 1�*%#X*X#C;o$K6'#4@A�D@A1
$'h��)(X�,1I5)(�g+9)1I5';<(�@A18#C;B� 1I�,1I5f�)e��)(Xe�#%/V̇ = K
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V̇P = 4 l/spH = 7.5 m
pLu = 8 m
pLp = 12 m
pd = 50 mm
pζu = 0.5
pζv1 = 2
pρ = 750 kg/m3
pηP = 0.7
pd0 = 30 mm
pL0 = 2 m
pζv0 = 20
pζk = 0.5
�λ = 0.02
p'#C;B� 14;B� J+�YLkê(,t> �#%J+#X6"gX1I5 e�� 5B(�$f� 1�eI1�� *%#%(X�
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$H#%! J+#%& 5 #%/_*%#C;B$H! � �D6'#%/21�5 1I/d(8;B#XJX#X6?($f� 1�eI1�� *%#%(+�
Bp 6?(X#H�K=45B(+h�$0! $?/2! 18$0J+#,/Ö=49)$'&)(v@A$�u
k�*'t £ 5)(Xh�$D!)$'/2!)18$f=v9B$,&B(v@A$D6?(c;�(�@A18*%1�5 JX�?9i$D#X! JX#X& 5 #,/�*%#C;B$EeI(XJ+*X#%� 1�5~u
�6� ï21�eI1�� *%#%(+�)� ¥ �ið�=4$"=4! #�@A1�5 �i=v9B#+G�1I5 �)/�r 1�*%#X*X#C;<#%/qu2��#Xe�5B(+J+�K=4$0=49)1v;<1�b �K! #C;<(Xr �YL• 1I6 *X� *,(j9)1I5 J+5 �K#%J+! #X�)�D;B1I#X5 �)r)( L
C1 = 20pC2 = 12
pC3 = 10
�C4 = 15
• ! �)1�& 5 �)r �";B1�#%5 � r)()Ld1 = 50 mm
pd2 = d3 = d4 = d = 40 mm
• �)(Xe49)�)6'(H� eC/21��'$05)� *X#,(�$f� 1�eI1�� *%#%(+�)�)/d(H = 5 m
• pv = 0.1 bar�pm = 0.6 bar
w ;B� 14;B� J+�´eC(+! �)1I/2� 5)=�6'1`! � #%J+#X6'1�6)� #Xe�=�*X1_;<1�#X5)� r 1Xp�6?(+#Z��=�5)(+hI$�! $?/2! 1ÙkηP = 0.8
tK6'#�@A$�J+�)1�g%(! #%=�Jj(+*X�)J+�f$E;B1�#%5 � r $^��;�(�g%�i=41�! � #%J+#X6K6 � #%e�J+$E;B1I#X5 �)r $D$I;B*X#,=4J+�)$,&)� #)u
5�� �8(�=49)� r ��@A1�! �)� 6?(+eI(X50=v9B#+G�1I50r 1I*X#%*X#C;0$>6'#%/21:! $?/2!)(�=�5)(+gC;<1�*%(�*%#C;B#%/_�)1�e�1I� *X#,(+� £ � e��)1�e�1I� *X#,(+�)( ¥�Fð:uB��#XeI5)(+JX�F=4$K=v9)14;B1�b �o! #C;<(+r)�YL
L1 = 20 mpL2 = 15 m
pL3 = 20 m
pλ = 0.03
pH1 = 2 m
pH2 = 4 m�
pa = 1 baru ¥ 6'#f@A1i!)� #XJ+#%6
V̇1 = 7 l/s�l!)#X6?(+eI� *%(X-�1"#XJ+*%#X� 1I5 1
Ur 1�*%��=I(^G>�)*X#%/
h = 165 mmkH = 1 m
pρm = 13600 kg/m3
t.� eI�)(+& $'5)(XJ+�K6'#X� �)=�5 $f=�5)(+hI$D! $?/2! 1%p)6?(+#>�K! � � JX�)=�(X6i6,#�@A�K*+9B(j;<(H� e�5)(j;5 � *%#%(�J+1�& 5)#%=4JX�i$D� 1�eI1�� *%#%(X� $ ¥ uB�Y(+5 1C/d(+� � JX�K=�*X1:9)#%6'(+9)5 1�#XJX! #X� 1%u
¥ u)¦�#%b � bYL�§f¨f©^ª�«�¬xF¨i®i«¨°¯i±i²�¬f³´«�µ�¨i®i«·¶ N%¸,¹2TQ� {BP\U2RF|�R,º_»<R ñ�FòD��ò ° É�ÆiÃKÊ�Âià Á�À��^Ày«iÃ
�� ��! #,=4J+�)#4@A1�-.$E! � �)6'(XeI(X5 #%/�5)(x=49)� r �^�Xu ¢8! $?/2!)(�J+�)(+5B=4! #%� J+$C@A1�*%#C;B$��)e�� 1Ie�1��)*X#%(X�)(A�
B$>� 1Ie�1��)*X#%(X�
C6 � #Xe^r 1�*%#X*%#C;B1_�Xp��?p�¢ p�û`��ÿ u´��� *X#�*X#C;B1i$�� 1Ie�1��)*X#%(X� $ ¥ =41F#C;<�?GH(X*%(_6'#X5)=�Jj(+5 J+5)�)/ú=4Jj(j9)5)�)/;B#XJX� r)(+-:1I/�6,#%5)=4Jj(X5 J+5 #%h�! �)#XJ+#%6'(
V̇ = 20 l/s$K-:1�h4( p<;<#X6^=�$K� 1�eI1�� *%#%(+�)�
B�
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pm = 10 kPappv = 50 kPa
pd1 = d2 = 100 mm
pd3 = 120 mm
pd4 = d5 = 150 mm
pL1 = L2 = 2 m
pL3 = 1 m
pL4 = 24 m
pl5 = 15 m
pλ = 0.025
kêeI(D=�*X1xr 1I*X�Bt4pH = 5 m
ph = 2 m
pζs = 4
pζv1
= 5pζv2
= 8pζv4
= 4pζv5
= 10�ζk = 0.8
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i�·¸|n�fqº¹3»�¼ª R,}aR,ØfN6� v1 = 1.354 m/s
pv2 = 1.192 m/s
pv3 = 1.768 m/s
pv4 = 2.262 m/s
pv5 = 1.13 m/s
pYP =
143.4 J/kg�P = 7.263 kW
¼6� ��! #,=4J+�)#4@A1�-.$D! �)� 6?(+eI(X5 #%/�5)(�=v9)�)r �o�%u û�! $?/2!)(x! #%J+�)=�6 $C@A1�*%#C;B$��)e�� 1Ie�1��)*X#%(X�)( ¥ �0ð�$>� 1�eI1�� *%#%(X� 1 £�Du���#%e�5)(XJ+�i=4$K=v9)14;B1�b)�F! #C;<(Xr �YL
H = 5 mppm = 10 kPa
ppv = 5 kPa
pηP = 0.8
pζu = 0.5
pζR = 0.5
pλ = 0.03
k�eI(o=4*X1fr 1�*%�Bt��`!)#C;<(+r �a;<(+J+��$�Jj(XgX149B�Yu ¥ 6'#E@A1f$^;B1�#%5 � r �ÖÿF� eI/21I� 1�5B(igX� eI� 5)(^=4J+� $C@s(X-:(v5 = 1.8 m/s
p�#C;<� 1v;B�)J+��! �)#XJ+#%6,1"6 � #%ef=�*X10r)1�*X5)1D;B1I#X5 �)r 1Xp�6?(X#F�l=�5)(+hI$_6,#�@s(E@A1"! #XJX� 1�g%5)(FeC(o! #%h�#%5! $?/2! 1Xu
i Li [ m] di [ mm] ζvi� ¢ �"�v� ���� ¢)u � �"�v� �¢ ÿ �v�v� ¢û û �c�r� ûÿ ÿ �cÿr� ÿ
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i,·¸|���q�¹3»¾½ª R,}aR,ØfN6�• P.� eI� 5 1x=4JX� $C@s(+-�( L v1 = 1.377 m/s
pv2 = 0.532 m/s
pv3 = 1.546 m/s
pv4 = 1.483 m/s
pv5 = 1.8 m/s
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• ��(X! � 1C/2� 5)=46)�Z! � #XJX#Xr �YLV̇1 = 35.04 l/s
pV̇2 = 13.54 l/s
pV̇3 = 45.58 l/s
pV̇4 = 16.77 l/s
pV̇5 =
31.81 l/s
• �%1v;B�)5 � & 5 �i�B(c;i�K=�5)(+h4(�! $?/2! 1%LYP = 65.3 J/kg
�P = 3.965 kW
uñ�� �8(Z=v9)�)r �¤�Xu�ÿ^@A1`! � �)6'(XeI(X5�=49)#+GH1�5�r 1I*X#%*X#C;Ù$Ù6'#%/21`!)$'/2!B( px6 �)#Xear 1I*X5 1_;<1�#X5)� r 1Z�XpH�?px¢ pHÿl� �J+�)(X5)=4! #%� J+$C@A1�*X#C;B$l� eo*%149)�)6,#%ho#%J+*X#%� 1�5 #%hF� 1Ie�1��)*X#%(X�)( ¥ $l�)1�e�1I� *X#%� 1
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Dud��#XeI5)(+J+��=4$x=v9B1v;B1Ib ��! #C;�(+r �YL
H = 5 mpH1 = 1 m
pH2 = 2 m
ppm = 0.2 bar
ppv = 0.5 bar
pλ = 0.03
kêeI(D=�*X18r 1I*X�Bt4pηP = 0.8
p 6?(X#E�i! #C;<(+r)�0;�(+J+�F$"Jj(+g%149)�Yu w ;B� 14;B� J+�F=�5)(+hI$f!)#XJ+� 1IgX5 $feI(E!)#Xh�#%5! $?/2! 1Xp\$'6 $'!)(+5_eI(X! � 1C/2� 5)=46)��!)� #XJ+#%6_6,#�@A�^;B#XJ+�)& 1E$o� 1Ie�1I� *X#,(+�D
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i�·¸|n�fqº¹3»�¿ª R,}aR,ØfN6� v1 = 1.717 m/s
pv2 = v4 = 1.771 m/s
pv3 = 2.092 m/s
pv5 = 1.091 m/s
pv6 = 2.862 m/s
pv7 = 1.001 m/s
pV̇ = 0.121 m3/s
pYP = 132.56 J/kg
pP = 20 kW
�ζv7 = 125.6
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WY{<R'ÕKT � Tz�CR,UQ�FWY{)UQ�?P2N?W � Tn�IR Lψ =
AkA
< 1k�û)u¡�Ct
��#�@s(+*%(.6'#X5)J+�)(+6)r �j@A1�/Q9<(+eI(d@A1�� eI�)(jGH1�5 �j@s(�6,#C;�#%J+*X#%�)(q6'#4@A��=�1�5)(+9B(+eI1�g+9)� G�1�=49)#Xg%#C;B5 #4@�! #%*X� ���8J+1�& 5)#%=4JX�$f� 1�eI1�� *%#%(+�)$<p'Jv@4u�*X� 14;B5 #%=�J�6'#%1I7�� r �+@A1�5 Jj(H6'#X5 J+�B(+6 r �+@A18#X!)(j;<(�=I(H! #X*%1�b)(X-�1C/`*X�B=4� 5 1
Hu
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Ap ! #,=4Jj(+*Xn�(>=41yP.1��)5 $C9)�j@A1�*,(�@A1v;B5)(X& � 5)(>eC(H5 � *X#>$D�)1�e�1I� *X#,(+� $0�f6'#X5)J+�)(+6'#%*%(+5)�0! � 1C=41I6"6 ] 6YL
paρ
+ g H =paρ
+v2
2+ ζ
v2
2
k�ûBu �%t��#%1I7�� r �+@A1�5 J
ζ@A1�6'#%1I7�� r �+@A1�5 Jq9)#%6?(j9)5 #%h2h�$'gX� JX6'(.1I5 1�� hI�j@A1�6'#4@A��� 1I! � 1Ie�1�5)J+$C@A1�1�5)1�� hI�j@A$�! #%J+� 1IgX5 $�eI(�=4JX*%(+�B(+-:1
/d(X6 � #X*%� JC9)#Xh4(�6'#4@A�K5)(%=4Jj(v@A$D�)eI/21��'$D!)� #XJ+#%& 5 18eI#X5 1��Ke��';�(��)1�e�1I� *X#,(+�)()u���e.@A14;B5)(+&)� 5 1Dk�ûBu �%tq=41�;B#%gX�j@s(
v =1√
1 + ζ
√
2gH = ϕ√
2gH,h�;B1.@A1
ϕ =1√
1 + ζ
k�ûBu ¢,tð.149B� & � 5)(
ϕ=41H5)(+eI� *%( WY{BR'ÕKT � Tn�CR%UQ�a»)PQMXTQU2R u��ú=v9)$'&)(4@A$f;<(0=�1�h�$'g%� r �F$i� 1Ie�1I� *X#,(+� $K/2#XhI$"eC(+5 1C/d(+� �)J+�
kζ = 0
tvp?gX�)e�� 5)(H�)=�J+� r)(X-:(>$fJ+#%/l=v9B$,&B(v@A$8@A1 T~¹dR'N'[2U2N^»BP\M%T\U2N k�5)1I/d(>h�$'g%� Jj(+6?(�1I5 1��)h��j@A1ctvL
vi =√
2gHk�û)u û,t
£ J+*,(+� 5B(8gX� eI� 5)(Dkê� e��B(+e�ûBu �%t<@A1�$'*%1�6D/d(+-�(x#C;0�?;B1I(+9)5 1�g%� e�� 5)1XpI@A1��x@A1ϕ < 1
u2��� #XJ+#%6E6 � #%e�#XJX*X#X��@A1�#C;B� 1I�,1I5� eI�)(+eI#%/�k�gX�)e�� 5)(
v@A18g%� e��)5)(H$06'#X5 JX�)(+6'#X*,(+5 #,/_! � 1C=41�6)$?t4L
V̇ = v Ak = ϕAk√
2gHk�û)u�ÿXt
¥ 6,#`=�1F! #%*X� ��� 5)(Ak
� eI�)(+eI�Z6?(+#Ak = ψA
p�h�;B1>@A1A
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V̇ = µA√
2gHk�ûBu �,t
ð.149B� & � 5)(µ = ϕψ
=41_5)(Xe�� *,( WY{<R'ÕKT � Tz�CR,UQ��OQP2{)� {BWYN uÍ��#X1C7�� r �j@A1I5 J�! � #%J+#X6?(^@A1_$,*%1�6�/d(+-:��#C;@A14;B� 5 � r)1"�Ö#X5�=�1feC(F5 1I6 ��#%J+*X#%��/2#+GH10*%1�#%/d(D@A1v;B5)#%=4Jj(X*X5 #o/2#XG�1"#C;B� 14;B� J+�Ö1�6?=4! 1I� �)/21I5 Jj(j9)5 #BuaÔ��)! � & 5)(*%� 1v;B5)#%=4J>eC(
µ$f=49)$'&)(v@A$D6)� $,G>5)#Xh8#XJX*X#X�B( p'6?(c;<(�@A186'#X5 JX�)(+6 r)�j@s(H! #XJ+!)$,5B(:@A1
µ ≈ 0.62u
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V̇i = A√
2gHk�û)u �+t
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62%#C;K�?;B1I(+9)5 #Xh�! �)#XJ+#%6'(,tvu
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g (H1 −H2) = (ζ + 1)v2
2��#%1I7�� r �+@A1�5 Jζ� 1I! � 1�eI1�5 JX$I@A1oh�$'gX� Jj(X6l=�J+� $C@A5 1F1I5 1�� hI�j@A1i�)=�! � 1v;�#%J+*X#%�)( p�;B#X6�@A1ih�$'g%� Jj(+6l1�5 1I� h��j@A1i!)#%=v9B1
#%J+*X#%�)(H! � 1v;�=4Jj(+*Xn:1I5�@A14;B� 5 � r)#%/qp)J4@4ud&?9B(+5)#%/v2/2
5)(�;B1I=�5 #4@�=4J+�B(+5 �H@A1v;<5)(+& �)5 1Xu
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v =1√
1 + ζ
√
2g(H1 −H2) = ϕ√
2g(H1 −H2),h�;B1.@A1
ϕ =1√
1 + ζ
k�ûBu �,t
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V̇ = v Ak = ϕAk√
2g(H1 −H2) = ϕψA√
2g(H1 −H2)k�ûBu �,t
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bk(z)�
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dV̇ = v dAk = ϕ√
2gz bk(z) dz
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L
V̇ = ϕ√
2g
H+hk2∫
H−hk1
bk(z)√z dz
kêû)u¡�X�ct
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bk(z) =√
ψ · b(z), hk1 =√
ψ · h1, hk2 =√
ψ · h2
!)(:@A1�6'#X5B(+& 5 #>! �)#XJ+#%6�@A14;B5)(+6V̇ = ϕ
√
ψ√
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b(z)√z dz
kêû)u¡�C�%t
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2g
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H−h1
b(z)√z dz
k�û)u¡�I¢,t
ð.149B� & � 5)(µ = ϕ
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�� � � 1Ie�1��)*X#%(X� ¥ =4Jj(j9B5 #";B#XJX� & 1V̇ = 20 l/s
*%#C;B1XuKï21�eI1�� *%#%(+�)�A
�B
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d = 50 mmpζu = 0.5
pζv = 1.2
pλ = 0.03
pL = 1 m
�µ0 = 0.62
u¼6� w ;B� 14;B� J+�a3,#X� �)e�#X5)Jc(+9)5 $_=4J+�B(+5 � r)$
b*%149)�)6,#%hD#XJX*X#X�B(K#Xg+9B� 6?(i! �)(X*X#X$'h¾9)#%h>J+� #X$'h¾9B(oJj(+6'#";<(F!)� #XJ+#%6
6 � #%e:-:1�h4(Hg%$�;<1V̇ = 2 m3/s
u w =4Jj(+9)�f!)#Xe�5B(+J+�K! #C;<(Xr �K=4$�LH = 1 m
pa = 2 m
�µ = 0.62
u
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A(z) t
t = 0
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!)(�b 1�eI(a*%� 1I/21dt
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dt@A14;B5)(+6?(A(z) dz
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| dz | =
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dz,6?(c;�(�@A1
dz > 0 (6'#%#X�%;B� 5)(XJj(
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)
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µA0
√2gz
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µ = const.JX#X6'#%/o! �)(jGE-:1�-�(8� 1Ie�1��)*X#%(X�)( pj�)5 J+1�hI�)(jn�1�-:1I/^9)1I*X1���;B1C=45 1
=�J+�)(+5)1q@A1v;B5)(X& � 5 1%LT∫
0
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µA0
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0∫
H
A(z) dz√z
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µA0
√2g
H∫
0
A(z) dz√z
k�û)u¡���,t
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pB!)(>=41q@A14;B5)(+&)� 5)(Fkêû)u¡���%t.=�*X#C;B�5)()L
T =2A
√H
µA0
√2g
k�û)u¡���,t
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=4$0=49)1v;<1�b �K! #C;<(Xr �YLD = 1 m
pd = 20 mm
pH = 0.5 m
ph = 1 m
pµ = 0.62
u�?u w J+*X#%� 1�5_! �B(+eI(X5ar �?9)�)5';B� � &)5 �Ö=�$�;�=41D! $'5 �a*%#C;B#%/Ù� e0h�#X�)-�1IhD*X1�9)� 6'#XhD#%J+*X#%� 1�5)#XhE� 1Ie�1��)*X#%(X�)("6 � #%e
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ζu = 0.2pζv = 0.5
pζk = 0.5
pλ = 0.03
pl = 4 m
pd = 50 mm
pH = 5 mm
ph = 2 m
�D = 1 m
u¢ u�ï21�eI1�� *%#%(+�:! � �)6'(XeI(X5:5B(�=v9)�)r �x=41��B=4J+#%*X� 1C/21�5 #�!)$,5)�86,#%5)=4Jj(X5 J+5 �)/K!)� #XJ+#%6'#%/
V̇0
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Hu8��#XeI5)(+J+��=4$i=v9B1v;B1Ib �^!)#C;<(+r �~L
H = 2 mpD = 0.6 m
ph = 3 m
pl = l1 = 4 m
pl2 = 1 m
pd = 30 m
pd1 = 20 mm
pd2 = 15 mm
pζv = 4
pζv1 = 2
pζk = 0.5
pζu = 0.7
pλ = 0.02
�V̇0 = 2 l/s
u w ;B� 14;B� J+�Z!)#+9)#XGH(v@>�)(X*X5 #%J+14GE5 #%hi5 � *X#,(_$� 1�eI1�� *%#%(X� $�p,6?(X#��K*X�)1I/2186'#4@A1.@A18! #%J+� 1�g%5 #�;<(>=418! #%& 1�JX5 �K5 � *X#ik
H = 2 mt.! � #%/21I5 �KeI(
h0 = 1 mu
û)u ��e�hI#X� -:1�h>#%J+*X#%� 1�5 #%h�r)�?9)� 5';<� � & 5 #%hD=4$I;<()pY! $'5 #Xh>*%#C;B1"km=v9)�)6'(K¢,tvpY! �)1�& 5 �)6'(d = 1 m
�_*X�B=4� 5 1h =
1.5 mpd! � 1�Jj(X& 1"=�1f*%#C;<(F$^;B#%-��lr �?9B� 5';B� �)& 5 ��=4$I;Ö;<�)/21�5 eI�j@s(
D = H = 2 m$a6'#4@A1C/��)/d(^*X#C;B1D;B#
! #+9B#X*X�)5 1�*%�)=4�)5 1 ] 6)� #Xe>r 1I*f! �)1�& 5 �)6'(d0 = 25 mm
�^;B$,GE� 5 1L = 2.5 m
p�! #%e�5)(XJ+�?3o6'#X1C7�� r �j@A1I5)(+Jj(#XJ+!)#X�)()L
ζu = 0.5�ζv = 1.5
�"6'#X1I7��)r �j@A1�5)Jc(HJ+�)1�-�(λ = 0.03
u2ð.�)=�� 5)(�;B5)(HhI#X� -:1�h8=4$I;<(H�"5 � *%#%(�*X#C;B1$F;<#X-:1I/�=4$I;B$"@A1
a = 2 mu ��e��)(X& $'5)(+JX�a*X� 1C/21DeI(i6'#�@A1Db 1f=�1Dh�#%� -:��=4$I;�)=�! �)(Xe�5 � JX�Yuo�8(F=49)� r �i@A1
! � � 6?(XeI(+5"! #X9)#+G>(v@�$fJ+� 1�5)$,JX6 $t = 0 s
uÿ?u £ $I;B#X*%� ¥ �oð�=4!)#4@A1�5)�o=�(Er)1�*X�o! � 1�&)5 � 6?(
du��8(+6'#%5i#XJ+*,(+�)(X-:(>*%1�5 J+� 9B(D*X#C;�(>! #X& �)-�1�;<(E! �)1�J+� &)1�� e
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6'#4@A�F=�1x5)(+9B(+eI�"5B(! #X6 9)#X! r)$F=4$I;<( u w ;<� 1v;B�)J+�^*%� 1I/21HeI(f6'#4@A1Hb 1H5 � *X#%�o$F=�$�;<#X*X�B/d(D�?GH1v;B5)(X& � J+�~u���#C;<(Xr �YL
d = 50 mmp
d0 = 2 mmpl = 5 m
pH = 2 m
pA1 = 1 m2
pA2 = 2 m2
pA2 = 2 m2
pρ0 = 1.2 kg/m3
pζu = 0.5
pζv = 0.5
pλ = 0.02
pµ0 = 0.62
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