Post on 01-Apr-2018
K141 HYAE Hydrostatics 1
HYDRAULICS
HYDROSTATICS HYDRODYNAMICS
HYDROSTATICS
Liquid in equilibrium with regard
to Earth
to moving
system
(reservoir)
relative
equilibrium
K141 HYAE Hydrostatics 2
PRESSURE IN LIQUID
- equilibrium: only normal stress pressure (tangential stress ij 0)
pressure force:
pressure:
S
F
S
F Pa
dS
dFp
- at motion: both normal and tangential stress
pressure (hydrodynamic) depends on velocity
S S
dS pdFF dS pdF
In gravitational field:
hydrostatic pressure: ph = gh
pressure in point M: p = pe+ ph = pe + gh
pressure force at horizontal bottom: F = pS
K141 HYAE Hydrostatics 3
21
21
pghpSpghSSp
21 FgV F
hSVSpFSpF
22
11
Requirement of force equilibrium
in vertical direction:
h12 pghpp
Hydrostatic pressure in liquid in vertical tube closed by pistons
(considering gravity acceleration)
K141 HYAE Hydrostatics 4
equilibrium of external forces
(for = const.):
SdppcosadsSSp
as → resulting acceleration in direction s change of pressure
on path ds
pressure force pressure force body force
dscosadp Euler hydrostatic equation
(one-dimensional form)
CHANGE OF PRESSURE
K141 HYAE Hydrostatics 5
Euler hydrostatic equation
(component form)
component form:
dzadyadxadp zyx
zyx dpdpdp dp
dxadp xx
dyadp yy
dzadp zz
pppp zyx
Gravitational force field
dhds,ga
dhgdp
p
p
h
0e
gdhdp hgpp e hhydrostatic pressure p g h
K141 HYAE Hydrostatics 6
SURFACE AREA
h = const. p = const. 0dhgdp dh = 0
22a11e hgphgp hgp
ppp
phgp
12
21
Perpendicular to resulting acceleration
in gravitational field horizontal plane level
3111e hghgp
3122a hghgp
differential pressure gauge
(U-tube)
piezometer
41ae hgpp
K141 HYAE Hydrostatics 7
normal atmospheric pressure pa = 101324.72 Pa 105 Pa total static pressure ps
ps > pa overpressure
pp = (ps – pa) > 0
ps < pa underpressure
pva = (ps – pa) < 0
pe = pa hgpp as
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 h
p p g h , p g h
OVERPRESSURE, UNDERPRESSURE
pva
ps=0
pa
pp
ps
ps
K141 HYAE Hydrostatics 8
PASCAL’S LAW
Gradual pressure change p in a small closed volume of liquid
expends in all directions and passes on all points of liquid without
any change.
p = const., p = const. S
Fp
force [N]
area [m2]
22
2
1
11 p
S
F
S
Fp
1
212
S
SFF
21 FpF
Pressure head Suction head [m, m w.c.] g
ph
g
ph va
va
sfor p 05
va amaxp p 10 Pa .c.wm10h maxva
K141 HYAE Hydrostatics 9
HYDRAULIC PRESS
PRESSURE CONVERTER
21 1
2
Ftheoretically F S
S
1
221
S
Spp
12
21 S
S
FF
... 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
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 g
p,
g
pvap
total pressure force
K141 HYAE Hydrostatics 11
HORIZONTAL BOTTOM
ShdShzdSSS
ShgF
h·S – volume of pressure body
hydrostatic
paradox
K141 HYAE Hydrostatics 12
tF g z S
INCLINED PLANE SURFACE
SzzdS t
S
For prismatic areas with
horizontal border – possible also
... area of pressure diagram [m2]
= b ... volume of pressure body [m3]
gbgba2
zzgSzgF 21
t
baS,2
zzz 21
t
– volume of pressure body
tz S
tz S
tz S
T - centre of area S
C – point of application of
force F
K141 HYAE Hydrostatics 13
DETERMINATION OF POINT OF APPLICATION
OF HYDROSTATIC FORCE
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
t
t
o
t
2
toc y
yS
I
yS
ySIy
t
otc
yS
Iyy
dSsinygdSzgdSpdF
,dSysingdSzgFSS
S
2
Sc dSysingdSysingy
t
x
S
S
2
S
S
2
cyS
I
dSy
dSy
dSysing
dSysing
y
2
tox ySII
K141 HYAE Hydrostatics 14
Determination of point of application of hydrostatic force F on
rectangular area with horizontal upper edge corresponding with
water level
baS
,ab12
1I
,2
ay
3
o
t
a3
2
2
aa
6
1
2
a
2
aba
ab12
1
yyS
Iy
3
t
t
oc
centre of loaded area S
on gravity centre axis o
point of application
of hydrostatic force F
K141 HYAE Hydrostatics 15
pressure diagram - components
pressure diagram
- complex
RESOLUTION OF HYDROSTATIC FORCE IN COMPONENTS
z
x
Ftg
F
K141 HYAE Hydrostatics 16 n
ihz
n
2hz
n
1hz
i
2
1
Effective distribution of horizontal beams
11 z3
2x
1i2ii2 z3
2ωz
3
22ωωx
1iiiiii z3
2ω1iz
3
2iωωx
From equation of moments:
K141 HYAE Hydrostatics 17
HYDROSTATIC FORCE ACTING ON CURVED AREAS
Two perpendicular horizontal (Fx, Fy), and vertical component (Fz):
xix dSzgcosdSzgdF
yiy dSzgcosdSzgdF
ziz dSzgcosdSzgdF
,SzgF xtxx ytyy SzgF
VgSzgF ztz
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 ,
F
Fcos,
F
Fcos,
F
Fcos zyx
For prismatic areas also : 0F,bgF,bgF yzzxx
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 BODIES
Application of Archimedes principle:
buoyancy force
k … 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
vzkk
12k
1k2k12
FWgShghhSg
ShgShgFF
WgF kvz Archimedes principle
K141 HYAE Hydrostatics 20
Fvz < G body gravitates
Fvz > G body moves up
till Fvz = G
Fvz = G body in balance
V = W – body hovers
V > W – body floats
(V – body volume)
Fvz goes through centre C of displacement W
G 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