Transcript 환경유체역학

환경유체역학
강사: 김 한승
A 1411
[email protected]
유체역학과 환경공학
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유체란?
유체역학이란?
Why is it an important subject in Environmental
Engineering?
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Basic media (air, water, groundwater, sludges, organic solvents-NAPLs)
물질전달 (macro/micro transport-advection, dispersion, diffusion)
반응공학 및 반응조 설계
차원해석
수계관리 - 수량, 수질
상수도공학 - 취수, 도수, 정수, 배수, 관망
하수도공학 - 배수로, 하수처리장, 배출
대기관리 - 오염배출, 확산, 배출정화시설, 실내공기정화, 덕트
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토양, 지하수 오염 관리 및 정화
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What is Fluids?
자연계 물질의 존재상태 – 고체, 액체, 기체
“유체”
Differences from the solids
 Solids – fixed distances btw. the component molecules → rigid body
(lattice structure) and resistant to shear stress
 Fluids – flexible (varied) mol.-mol. distance and structure → no
defined shape of body and continuously deformed by shear stress
(운동상태에서만 전단응력에 평형을 이룰 수 있다.)
Differences bwt. gases and liquids
Gases – large and varied mol.-mol. distance, subject to
compression/expansion → varied density (compressible fluids)
Liquids – relatively constant mol.-mol. distance, almost no
compression/expansion → no density changes (incompressible
fluids)
Fluids vs. Solids
Fluids
Solids
Gas or liquid
Solid
A substance can deform
A substance resists a shear
continuously under the action
stress in a static condition
of a shear stress
Irregular or relatively constant
spacing btw. component
molecules
Fixed spacing btw. component
molecules
Free or weakly limited
movement of molecules
Restricted movement of
molecules – lattice structured
Various shapes depending on
containers
Own shapes
Shear stress  time rate of
shear strain
Shear stress  shear strain
Differences btw. Ideal and Non-ideal (real) fluids
 Real – resistance (shear stress) generated in real fluids due to their
viscosity (viscous fluids)
 Ideal – no viscous effects (inviscid fluid-비점성 유체),
incompressible, useful for the theoretical analysis of fluids
소성체 – 고체와 액체의 특성을 모두 갖는다 (jellies, 도료, polymeric
solutions, paraffin)
What is “Fluid Mechanics”?
Science that describes physical actions and effects given by the forces
applied to the fluids in motion or no motion.
Classification of Fluid Mechanics
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1.
2.
3.
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1.
2.
3.
Upon target fluids
Hydrodynamics (동수역학) – incompressible fluids
Gas dynamics (기체동역학) – compressible fluids
Aerodynamics (항공역학) – gases (air) flowing over aircrafts,
rockets, etc.
Upon forces applied
Fluid statics (유체정역학) – fluids in no motion, no shear stress but
pressure
Fluid kinematics (유체운동학) – fluid elements in motion
Fluid dynamics (유체동역학) – fluids in motion
Fluids as a Continuum
유체유동의 수학적 해석을 위해서 개개의 분자로 구성되어 있는 유체를 하나의 가상
적인 균질한 연속체 (continuum)으로 취급하는 방법 - 여러 분자의 운동으로 야
기되는 효과의 평균값으로 전체 유체의 운동을 거시적으로 다루는 방법.
The number of molecules in the air at 1 atm and 0oC = ~ 107/mm3
 유체입자: 연속체로서 유동성질 등을 소유한 미소한 유체의 일부분, Not fluid
molecules
Rationale (e.g., gases)
Very short mol.-mol. distance (molecular mean free path ~
10-5cm)
Time scale for the mol.-mol. collisions << one for the system on which fluids
work (e.g., forces)
force
force
time
time
Mass, force, weight

Mass - a property of physical objects that measures the
amount of matter they contain, the property of a body that
causes it to have weight in a gravitational field. Constant
anywhere in space. (g, kg, slug, lbm)
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Force – a physical property that gives the movement of a
static object or changes in velocity or direction of an object
in motion (F = ma, N, kgf, dyne, lbf)
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Weight - the vertical force exerted by a mass as a result of
gravity (W = mg)
Class quiz

질량이 10kg인 물체를 저울로 달았더니 8.9kgf이었다. 이곳의 중력가속
도는?

지구상에서 100kgf인 물체를 중력가속도가 지구의 1/5인 위성으로 가져
가면 질량과 중량은 각각 얼마인가?

What is the weight of a pound mass (lbm) on the earth’s surface,
where the acceleration due to gravity is 32.2 ft/s2, and on the
moon’s surface, where the acceleration is 5.31 ft/s2?
Basic fluid properties
Mass density (중량밀도, 밀도,  ) = m/vol. kg/m3, lbm(or slug)/ft3
Specific weight (단위중량, 비중량, g ) = W/vol. =  g, N/m3, lbf/ft3
Specific volume (단위체적, 비체적, Vs) = 1/ , 1/ g (중력단위계)
Specific gravity (비중, S, dimensionless) =
for liquids,
g fluid  fluid

g water  water
for gases,
g gas  gas M gas


g air  air M air
where, gw : g of water at 4oC and ga: g of air at 1atm and 0oC.
Ideal gas law
pV = nRuT where, Ru: universal gas constant (8.31 kJ/kmol-K, 1545 ftlbf/lbmol-R)
→ In fluid mechanics, p= RT where, R: gas constant ( ~J/kg-K, ft-lbf/slug-oR)
Class quiz)
Density of air at standard sea-level pressure (atmospheric pressure)
and 0, 4, and 20oC?
Specific gravity and density of Helium?
Specific gravity of mercury?
101.3 KPa, 100oC하에 있는 탄산가스(CO2)의 비중량, 비체적, 밀도를 구하시오.
Specific heat (비열, c): , kJ/kg-oC
정적비열 (cv), 정압비열(cp), 비열비 (specific heat ratio, k) = cp / cv
Specific internal energy (비내부 에너지, u): J/kg
Elasticity (탄성) – compressibility (압축성)
Bulk modulus of elasticity (체적탄성계수, Ev) ~N/m2
= -dP/(dV/V) = dP/(d/)
Class quiz) 어떤 액체가 압축되고 있는 실린더에서 1000kgf/cm2 에서는
0.4m3의 체적을, 2000kgf/cm2에서는 0.396m3의 체적을 갖는다면 체적
탄성계수는?
Viscosity (점성)
외부에서 주어진 전단력(shear stress)으로 인해 유동하는 유체에 있어 유체층 사이
에 발생하는 상대운동을 저항하는 성질
Newton’s law → 유체층사이의 전단력 (shear stress, )  유체층사이의 상대적 변
형 (dV/dy, shear rate)
 
dV
dy
: dynamic/absolute viscosity (동역학적/절대 점성계수)
(=/): kinematic viscosity (동점성계수)
Viscosity of gases vs. liquids
Driving forces for viscosity – molecular cohesive force (분자응집력), molecular momentum
exchange (분자 운동량 교환)
In gasses – primarily controlled by molecular momentum exchange – viscosity  with
temperature
In liquids – by molecular cohesive force – viscosity  with temperature
For gases – Sutherland equation
3
2
  T  T0  S
  
0  T0  T  S
where, 0: dynamic viscosity at T0 S: Sutherland
constant
For liquids
 = Ceb/T
Where, C, b: empirical constants
Class quiz)
A board (1m 1m, 25N weight) slides down an inclined ramp (slope=20o) with a
velocity of 2 cm/s. The board is separated from the ramp by a thin film of oil
with a viscosity of 0.05Ns/m2. Calculate the spacing between the board and
ramp. (Neglect edge effects)
Newtonian vs. Non-Newtonian fluids
Newtonian fluids – shear stress is linearly related to shear rate (물, 공기, 저분자 액체)
Non-Newtonian fluids - shear stress is NOT linearly related to shear rate
Bingham – 캐첩, 치약
Shear-thinning (psedoplastic) – 고분자
용액, 슬러리 (슬러지), 펄프 용액
Shear-thickening (dilatent) – 수지, 고온
유리, 아스팔트
Surface tension
When a liquid is in contact with different phases (e.g., gas, solids), liquid
molecules at the surface exert “tension” on adjacent surface due to their
greater attraction btw. the molecules at the surface than those below the
surface (cohesive force, 응집력 > adhesive force, 부착력). → interfacial
tension (surface tension,  : liquid-gas contact)
Defined as “tension force per unit length” -  of water at room temp.= 0.073
N/m
  1/temp
The surface tension is typically ignored in most cases, but it must be taken into
account in very small scale flow (gas/liquid droplets present, small scale
models, etc.)
How to measure? – capillary rise technique, ring tensiometer (Du Nouy ring
method, etc.)
Capillary rise technique
Capillary tube (d < 1 cm)
Typically, =0o for water and clean glass
Vertical component of the surface tension, F, z =
d cos
Weight of water risen
W = g(h)(d2/4)
At equilibrium, F, z = W
h = 4 cos / gd
Contact angle ()
determines wetting/non-wetting phases
~ f (cohesive, adhesive forces)
<90o
air
water
glass
adhesive btw. liquid/solid > cohesive btw. liquid molecules
>90o
air
mercury
glass
adhesive btw. liquid/solid < cohesive btw. liquid molecules
F  L  pA
or
2r  pr 2
2
p
r
Wt  2 F  2L
p
4
r
F  F ,i  F ,o   ( Di  Do )
Vapor pressure (Pv, 증기압)
The pressure at which a liquid boils
Pv  temp (see Apendix Table A.5)
Ex) water boils at 100oC (212oF) and 1 atm (14.7 psia) and also
boils at 10oC (50oF) and 0.178 psia.
Important for cavitation (공동현상)
Fluid statics (유체정역학)
Deals with the fluids in no motion.
Forces applied on the fluids
1. Surface forces (표면력) – 압력 (수직방향), 점성전단력 (접선방향)
2. Body forces (체적력) – 접촉없이 유체에 가해지는 외부력 (e.g., 중력), 유체의 질
량,체적에 관계
→ In fluid statics, just consider pressure and gravity (no shear stress!)
Pressure? – dF/dA
Gravity? – W=gV
Pressure at a point in a static fluid acts
with the same magnitude in all directions.
Pn=Px=Py=Pz
Pascal’ law
A pressure change produced at one point in a closed system is transmitted
throughout the entire system. (p1 = p2 = p3 = …. = pn)
예제 3.1) If a force of 100N were
exerted on the handle of this hydraulic
jack, what load, F2, can the jack
support? Neglect lifter weight.
Abs. vs. Gage pressure (절대, 계기 압력)
“0” pressure (vacuum) is the absolute pressure.
→ atmospheric pressure at sea level = 101 kN/m2 (kPa), 14.6 psia.
Gage pressure = abs. pressure + atmospheric pressure
Negative gage pressure → vacuum pressure
Pressure variation with elevation
dp
 g
dz
Pressures are constant along a horizontal path, but vary along a vertical path
(gravity direction).
If no density change (g constant-incompressibie fluids),
P + gz = const. (piezometric pressure)
P/g + z = const. (piezometric head)
Class quiz)
Ex 3.2)
Ex 3.3)
Compressible fluids (기체)
Assumption – ideal gas
 = p/RT, g = pg/RT
→Pressure variation = f (z, temp)
In troposphere (대류권)
temp  with elevation
T = T0 - (z-z0)
In stratosphere (성층권)
constant temp with elevation
압력측정
Pressure gage
1. Manometer
Differential
manometer (시차액
주계)
2.
Bourden-tube gage
Class quiz)
탱크 상부 공기 층의 압력은? l1 = 40cm, l2=100cm l3=80cm
Class quiz)
280oK의 물이 흐르는 경사진 관의 두 압력측정지점에 시차액주계를 연결하였다. 수
은의 연직변위가 2.5cm라면 두 지점의 piezometric pressure와 piezometric
head 차이는?
Hydrostatic forces (정수력)
Forces given by the hydrostatic pressure (정수압, 정압) applied on a
submerged plate in a no-motion fluid
Note)
The first moment of area (단면 1차 모멘트) =
 ydA  y A
A
The second moment of area (단면 2차 모멘트)=
2
 y dA  I  y A
2
A
How much is the magnitude of hydrostatic force?
Fhydrostatic  p A
Where does the hydrostatic force act on a submerged plate?
ycp  y 
I
yA
Class quiz)
The end of pipe is closed by an elliptical
shape gate (54 m) and the gate is
fixed by a hinge at its top. How much
of normal force is required to open
the gate? Neglect the weight of the
gate.
곡면에 작용하는 정수력
Integrating pressure force along the curved surface
Easier way – use free-body diagram (자유물체도) and consider force equilibrium
in vertical/horizontal directions
Fh  FAC
Fv  FBC  W
F  Fh  Fv
2
  tan 1
Fv
Fh
2
Class quiz)
Find the magnitude and line of action of
the hydrostatic force acting on
surface AB (the thickness of the
circular AB is 1 m).
Buoyant force (부력)
A resultant hydrostatic force that acts on the surfaces of a body submerged or
floating in fluids
물체에 작용하는 수직력의 합력
1) Submerged body
FB = Fup - Fdown = g(Vb+ Va) - gVa = gVb = gVD
where, Va: vol. ABCEF,
Vb: vol. of the body,
VD: displaced volume (배수체적)
2) Floating body
0
FB = Fup - Fdown = gVD (Fdown = Patoms. A = 0)
Archimedes’ principle
1. 유체 속에 잠겨진 물체는 그 물체에 의해 배제
된 유체의 무게만큼 부력을 받는다.
2. 유체 위에 떠 있는 물체는 그 물체의 무게와
같은 양의 유체를 배제한다.
Apparent weight (겉보기 무게) of a body in a fluid
From the equilibrium of forces,
T
T + FB = W
Apparent weight, T = W – FB
W
FB
 (g body  g fluid )Vbody
Hydrometer (비중계)
측정하고자 하는 유체의 밀도 → 부력차 →
가라앉는 정도의 차이
 Specific weight (단위 중량), specific
gravity (비중)

Class quiz)
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어떤 물체가 대기중에서 무게가 60N이고 수중에서 11N이라면 이 물체의 체적과
비중은? (공기의 무게는 무시한다.)

비중이 1.03인 바닷물에 전 체적의 10%가 밖으로 나온 빙산이 떠 있다. 빙산의
비중은?

무게가 20g이고 지름이 6mm인 비중계를 측정하고자 하는 액체에 띄운 결과, 물
에 띄웠을 때 보다 6cm 더 가라앉았다. 이 액체의 비중은?
Class quiz)
The woodblock (505010 mm) has a specific gravity S1 = 0.3 and the volume
of the metal part is 6600 mm3. Mass of the metal part and the tension T of
the cord?
Free-body diagram