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Hydrostatics
K141 HYAE Hydrostatics 1
HYDRAULICS
HYDROSTATICS HYDRODYNAMICS
HYDROSTATICSLiquid in equilibrium with regard
to Earthto moving
system(reservoir)
relativeequilibrium
K141 HYAE Hydrostatics 2
PRESSURE IN LIQUID- equilibrium: only normal stress → pressure (tangential stress τij ≈ 0)
pressure force:
pressure:
SF
ΔΔ
SF[ ]Pa
dSdFp =
- at motion: both normal and tangential stress
pressure (hydrodynamic) depends on velocity
∫ ∫===S S
dS pdFF dS pdF
In gravitational field:hydrostatic pressure: ph = ρ⋅g⋅hpressure in point M : p = pe+ ph = pe + ρ⋅g⋅h
pressure force at horizontal bottom: F = p⋅S
K141 HYAE Hydrostatics 3
21
21pghp
SpghSSp=⋅ρ⋅+
⋅=⋅ρ⋅⋅+⋅
21 FgV F =⋅ρ⋅+
hSVSpFSpF
22
11
⋅=⋅=⋅=
Requirement of force equilibrium in vertical direction:
h12 pghpp =⋅ρ⋅=−
Hydrostatic pressure in liquid in vertical tube closed by pistons(considering gravity acceleration)
K141 HYAE Hydrostatics 4
equilibrium of external forces(for ρ = const.):
( ) SdppcosadsSSp ⋅+=ϕ⋅⋅⋅⋅ρ+⋅
resulting acceleration in direction schange of pressure
on path ds
pressure force pressure forcebody force
dscosadp ⋅ϕ⋅⋅ρ= Euler hydrostatic equation(one-dimensional form)
CHANGE OF PRESSURE
K141 HYAE Hydrostatics 5
Euler hydrostatic equation(component form)
component form:
( )dzadyadxadp zyx ⋅+⋅+⋅ρ=
zyx dpdpdp dp ++=dxadp xx ⋅⋅ρ=
dyadp yy ⋅⋅ρ=
dzadp zz ⋅⋅ρ=
pppp zyx ===
Gravitational force field
dhds,ga == dhgdp ⋅⋅ρ=
∫ ∫ρ=p
p
h
0e
gdhdp hgpp e ⋅⋅ρ+= hhydrostatic pressure p g h= ρ ⋅ ⋅
K141 HYAE Hydrostatics 6
SURFACE AREA
h = const.p = const. 0dhgdp =⋅⋅ρ= dh = 0
22a11e hgphgp ⋅⋅ρ+=⋅⋅ρ+hgp
pppphgp
12
21
Δ⋅⋅ρ=Δ−=Δ
=Δ⋅⋅ρ+
Perpendicular to resulting accelerationin gravitational field horizontal plane level
=⋅⋅ρ+⋅⋅ρ+ 3111e hghgp3122a hghgp ⋅⋅ρ+⋅⋅ρ+=
differential pressure gauge(U-tube)
piezometer41ae hgpp ⋅⋅ρ=−
K141 HYAE Hydrostatics 7
normal atmospheric pressure pa = 101324.72 Pa ≅ 105 Patotal static pressure ps
ps > paoverprressure
pp = (ps – pa) > 0
ps < pa underpressurepva = (ps – pa) < 0
pe = pa hgpp as ⋅⋅ρ+=
1 a 1 p1 1
2 a 2 p2 2
3 a 3 va3 3
p p g h , p g h
p p g h , p g hp p g h , p g h
= + ρ ⋅ ⋅ = ρ ⋅ ⋅
= + ρ ⋅ ⋅ = ρ ⋅ ⋅
= − ρ ⋅ ⋅ = −ρ ⋅ ⋅
OVERPRESSURE, UNDERPRESSURE
K141 HYAE Hydrostatics 8
PASCAL’S LAWGradual pressure change Δp in small closed volume of liquid expends in all directions and passes on all points of liquid without any change.
Δp = const., p = const.SFp =
force [N]
area [m2]
22
2
1
11 p
SF
SFp ===
1
212 SSFF =
21 FpF Δ⇒Δ⇒Δ
Pressure head Suction head [m, m w.c.]g
ph⋅ρ
=g
ph vava ⋅ρ
=
sfor p 0= 5va amax
p p 10 Pa= − = va maxh 10mv.sl.≈
K141 HYAE Hydrostatics 9
HYDRAULIC PRESS
PRESSURE CONVERTER
21 1
2
Fth e o re tica lly F SS
= ⋅
1
221 S
Spp ⋅⋅η=
12
21 S
SFF ⋅⋅η=
η... efficiency(0,95 –1,0)
In practice → losses
K141 HYAE Hydrostatics 10
HYDROSTATIC FORCE
Hydrostatic force = force caused by hydrostatic pressure ph.dSpdF ⋅=
S S S
F pdS gzdS, for g const. : F g zdS= = ρ ρ ⋅ = = ρ ⋅ ⋅∫ ∫ ∫
F – passes through centre of pressure body ∫S
zdS- perpendicular to loaded area
z ... vertical depth (depth bellow level)
If overpressure pp (underpressure pva) on level → to enlarge (reduce) real depth z by pressure head
gp
,g
p vap
⋅ρ⋅ρ
total pressure force
K141 HYAE Hydrostatics 11
HORIZONTAL BOTTOM
ShdShzdSSS
⋅=⋅= ∫∫ ShgF ⋅⋅⋅ρ=
h·S – volume of pressure body
hydrostaticparadoxon
K141 HYAE Hydrostatics 12
tF g z S= ρ ⋅ ⋅ ⋅INCLINED PLANE SURFACE
SzzdS tS
⋅=∫
For prismatic areas with horizontal border – possible also
ω... area of pressure diagram [m2]Ω = ω⋅b ... volume of pressure body [m3]
Ω⋅⋅ρ=⋅ω⋅⋅ρ=⋅⋅+
⋅⋅ρ=⋅⋅⋅ρ= gbgba2
zzgSzgF 21t
baS,2
zzz 21t ⋅=
+=
– volume of pressure body
tz S⋅
tz S⋅
tz S⋅T - centre of area SC - centre of force F
K141 HYAE Hydrostatics 13
DETERMINATION OF POINT OF ACTION OF HYDROSTATICFORCE
Moment condition to x-axis:
I0 ... .second moment of loaded area S about gravity centre axis o
∫=⋅S
c ydFyF
Ix ..... second moment of loaded area S about x-axis
tt
o
t
2to
c yyS
IyS
ySIy +⋅
=⋅⋅+
=t
otc yS
Iyy⋅
=Δ=−
dSsinygdSzgdSpdF
,dSysingdSzgFSS
⋅α⋅⋅⋅ρ=⋅⋅⋅ρ=⋅=
⋅α⋅⋅ρ=⋅⋅ρ= ∫∫
∫∫ ⋅α⋅⋅ρ=⋅α⋅⋅ρ⋅S
2
Sc dSysingdSysingy
t
x
S
S
2
S
S
2
c ySI
dSy
dSy
dSysing
dSysingy
⋅==
⋅α⋅⋅ρ
⋅α⋅⋅ρ=
∫
∫
∫
∫
2tox ySII ⋅+=
K141 HYAE Hydrostatics 14
Determination of point of action of hydrostatic force F on rectangular area with horizontal upper edge corresponding with water level
baS
,ab121I
,2ay
3o
t
⋅=
⋅⋅=
=
a32
2aa
61
2a
2aba
ab121
yyS
Iy3
tt
oc =+=+
⋅⋅
⋅⋅=+
⋅=
centre of loaded area S→ on gravity centre axis o
point of action of hydrostatic
force F
K141 HYAE Hydrostatics 15
pressure diagram- components
pressure diagram- complex
RESOLUTION OF HYDROSTATIC FORCE IN COMPONENTS
z
x
FtgF
α =
K141 HYAE Hydrostatics 16
nihz
n2hz
n1hz
i
2
1
=
=
=
Effective distribution of horizontal beams
K141 HYAE Hydrostatics 17
HYDROSTATIC FORCE ACTING ON CURVED AREASTwo perpendicular horizontal (Fx, Fy), and vertical component (Fz):
xix dSzgcosdSzgdF ⋅⋅⋅ρ=α⋅⋅⋅⋅ρ=
yiy dSzgcosdSzgdF ⋅⋅⋅ρ=β⋅⋅⋅⋅ρ=
ziz dSzgcosdSzgdF ⋅⋅⋅ρ=γ⋅⋅⋅⋅ρ=
,SzgF xtxx ⋅⋅⋅ρ= ytyy SzgF ⋅⋅⋅ρ=
VgSzgF ztz ⋅⋅ρ=⋅⋅⋅ρ=
ztx (zty) ... vertical depth of projection Sx (Sy)Sx (Sy) ... .projection of area S at plane YZ (XZ)
V ... ……..volume of vertical column of liquid above area S
2z
2y
2x FFFF ++=,
FFcos,
FF
cos,FFcos zyx =γ=β=α
For prismatic areas also : 0F,bgF,bgF yzzxx =ω⋅⋅⋅ρ=ω⋅⋅⋅ρ=
K141 HYAE Hydrostatics 18
Course of hydrostatic pressure:
Hydrostatic force passes through centre of curvature of cylindrical area S
Solving hydrostatic force in components:
K141 HYAE Hydrostatics 19
FLOATING BODIESApplication of Archimedes principle:buoyancy force
ρk … density of liquid [kgm-3]W … volume of displaced liquid (displacement) [m3], W = W (tn)
Vertical cylinder submerged in liquid
External surface forces:ShgF,ShgF 2211 ⋅⋅⋅ρ=⋅⋅⋅ρ=
( )vzkk
12k
1k2k12
FWgShghhSg
ShgShgFF
=⋅⋅ρ=⋅⋅⋅ρ==−⋅⋅⋅ρ=
=⋅⋅⋅ρ−⋅⋅⋅ρ=−↑
WgF kvz ⋅⋅ρ= Archimedes principle
K141 HYAE Hydrostatics 20
Fvz < G ⇒ body gravitatesFvz > G ⇒ body moves up
till Fvz = GFvz = G ⇒ body in balance
V = W – body hoversV > W – body floats(V – body volume)
Fvz goes through centre C of displacement WG goes through centre of floating body
Resolution of immersion: G = Fvz W tn
K141 HYAE Hydrostatics 21
OVERVIEW OF MAIN TERMS AND TOPICS
overpressure, underpressure, static pressure
hydrostatic pressure
pressure head, suction head
Pascal´s law
hydrostatic force acting on plane and curved surface area
(dimension, direction, point of action)
pressure body - complex
- in components
floating bodies - Archimedes principle