1 Allievis equations Equation of motion ("F = m a") Continuity: (mass in - mass in = accumulated mass) 0 = + + Q kQ t Q x H Ag ∂ ∂ ∂ ∂ 0 2 = + x Q Ag a t H ∂ ∂ ∂ ∂ K p L L ∆ = ∆ ρ K a = Hooks law: Propagation speed: { } pA dl l pA pA ∂ ∂ + ) ( dl Q=cA Torbjørn K. Nielsen 2005 Water hammer Joukowsky: [ ] mVs g V a H ∆ = ∆ ∆H t [sec] a L T L 2 ≤ If valve closing time: a L 2 2 2 2 L L L L a V L Q L a H if T g T gA T a ∆ ∆ = = > a L Torbjørn K. Nielsen 2005
1Allievisequations Equation of motion("F =m a")Continuity: (mass
in - mass in =accumulated mass) 0 = + + Q kQtQxHAg02= +xQAgatHKpLL
=Ka =Hooks law:Propagation speed:{}pAdllpApA+) (dlQ=cATorbjrnK.
Nielsen2005Water hammerJ oukowsky: [ ] mVsgV aH= Ht [sec]aLTL2 If
valve closing time:aL 2222LL LLa V L Q LaH if Tg T gA T a = =
>aLTorbjrnK. Nielsen20052Rigid theory:L ANewtons2. Law
gives:dtdQAAL H H gA1) (1 2 = 2LLL Q LH if TgAT a = >>LT dt
og Q Q Q dQ = = 0 00Q0H1H2TorbjrnK. Nielsen2005Methodeof
CharacteristicsMethodefor solving differential equationsThe wave
equation is "hyperbolic. Difference methods givessingular
matrixesMOC solves the wave equation along the characteristicsi.e.
along the singular pointsTorbjrnK. Nielsen20053011= + + = Q kQtQA
xHg L022= + =xQAgatHL02 1= + L L 012=++ ++xQgAatHQ kQtQA xHg
Determining the characteristic equationsTorbjrnK. Nielsen2005012=
++++Q kQtQgaxQA tH gxH Rearranging the
equations:tHdtdxxHdtdH+=Chain rule:tQdtdxxQdtdQ+=agga gdtdx = = =
2adtdx =TorbjrnK. Nielsen20054Results - two ordinary differential
equations:01= + + Q kQdtdQA dtdHagadtdx+ =01= + + Q kQdtdQA
dtdHagadtdx =}}C+C-TorbjrnK. Nielsen2005Two sets of equations, one
for pressure propagationIn the flow direction and one for the
opposite directiont+a-aA BPtoto+tMultiply with0 = + + dx Q
QgkdQgAadHPAPAPAgdxgadt=and integrates:QxTorbjrnK. Nielsen2005502)
(2=+ + A A A p A PQ QgDAx fQ QgAaH HDone for both C+and C-, gives
two eq. with two unknowns:02) (2= B B B p B PQ QgDAx fQ QgAaH
Htoto+tPA BtxxC+ C-to+ tTorbjrnK. Nielsen2005 The condition in time
t is known, find the condition int + For all pipes: Internal
points: Two equations, 2 unknowns: HPog QPa x t / = toto+tPA Btx In
the end points: One equation ( C+eller C-) + a border conditionC+
C-t6P PgH A Q 20 =P reservoirH H =toto+tPA BC+ C-Valve:Border
conditions:Reservoir:PGood book: Wylie&Streeter: Fluid
Transient
