Please enable JavaScript.
Coggle requires JavaScript to display documents.
Fluid Static, FLUID DYNAMIC GROUP 2-1, DIFFERENTIAL MASS & MOMENTUM…
Fluid Static
system
open system (control volumes)
closed system (control mass)
fixed amount of mass, mass cannot cross the system boundary
both mass and energy can cross the system boundary
fluid statics
stationary fluids
pressure varies with elevation
fluid properties
density,
p
(kg/m^3)
pressure, P (pascal)
temperature, T (oC)
Internal energy, U (joule)
Thermodynamic Properties
viscosity
vapour pressure
specififc gravity
mass density
compressibility
TYPE OF TUBE
Simple U-tube manometer
Inverted U-tube manometer
U-tube with one leg enlarge
Two fluid U-tube manometer
Inclined U-tube manometer
fluids and the continuum
fluid is made up of independent molecules
discontinuous in nature
processing dimensions are much larger than molecular dimensions
processing duration much longer than time for molecular motion
TERMINOLOGIES
System
Control volume
Continuum
fluid pressure is a scalar field
directionless
Pressure in the fluid apply to all directions.
barometer
manometer
bent tube
pressure differs vertically but same horizontally
scalar field
pressure equal at same level
pascal law
p varies in space, but directionless
FUNDAMENTAL EQUATION
dP/dz = -pg
Pressure= Force/Area
P=F/A (Unit: Pa)
Laws Involved
Newton's Law
Pascal's Law
Archimedes' Law
manometer is for measure fluid pressure in two different point
barometer is for measure atmospheric pressure
bent tube consists fluids with different densities
continuum
Viscosity
Dynamic viscosity
kinematic vicosity
Liquid: When temperature high, viscosity decrease.
Gases: When temperature high, viscosity increases.
statics means stationary-zero shear stress-pressure varies vertically
barometer for atm-manometer for two diff fluid-bent tube for one/more liq. of diff density
archimedes' law- force equal to weight of fluid displaced by body(weight of fluid volume)
Related with Archimedes' Law (Buoyancy)
F=pgV
Pressures are equal at same level but vary in vertical direction due to gravity
Apparatus to measure pressure
Bent tube
Manometer
Barometer
Types of tube:
Simple U-tube manometer,
Inverted U-tube manometer,
U-tube with one leg enlarged,
Two fluid U-tube manometer,
Inclined U-tube manometer
PASCAL LAW
Properties
important property = pressure
studies of incompressible fluid at rest
Hydrostatic Pressure: Varies with depth
FLUID DYNAMIC
GROUP 2-1
NO SLIP CONDITION
F = ∫TdA
viscosity causing no-slip condition
Fluid in contact with solid “sticks”
to surface
VISCOUS vs INVISCID FLOW
VISCOUS
usually close to solid surfaces
example :
INVISCID FLOW
far away from surfaces
example :
FLOW VISUALIZATION
PATHLINES
ACTUAL PATH
traveled by an
INDIDUAL FLUID PARTICLE OVER SOME TIME PERIOD
X(a,b,c,t) = Xi(a,b,c,0) + ∫ v(a,b,c,t)dt
Pathlines in a Static Mixer
TIMELINES
Timeline of Turbulent Flow
Velocity development in a pipe
Set of
ADJACENT FLUID PARTICLES
that were marked at the
SAME (EARLIER) INSTANT IN TIME
STREAMLINES
Curves that are everywhere
TANGENT TO THE INSTANTANEOUS VELOCITY VECTOR
ds/v = dx/v = dy/v = dz/v
Streamline on a NASCAR
Lagrangian form-variable values for a particular
Eulerian form-variable values for every position and time
Law of Conservation of Mass
First Law of Thermodynamics
Newton's Second Law of Motion
Variable field – Variable defined as a function of position and time Velocity field – velocity as a function of position and time
BASIC CONCEPTS
LAGRANGIAN
A way of looking at fluid motion where observer follows an individual fluid parcel as it moves through space and time
mass moving with flow
EULERIAN
A way of looking at through fluid flows as time passes fluid motion that focuses on specific locations in space
control volume fixed in space
INTERNAL vs EXTERNAL FLOWS
INTERNAL FLOWS
dominated by viscosity throughout the flow field
EXTERNAL FLOWS
viscous effect are limited to boundary layer and wake
COMPRESSIBLE & INCOMPRESSIBLE FLOW
INCOMPRESSIBLE
happens if density remains constant
liquid flows : incompressible
COMPRESSIBLE
gas flows : compressible at HIGH SPEEDS
LAMINAR VS TURBULENT FLOWS
LAMINAR
highly ordered fluid motion with steamlines
TURBULENT
highly disordered fluid motion by velocity fluctuations and eddies
TRANSITIONAL
mixture laminar and turbulent flow
Reynolds number, Re = pvD/u
Re > 10^4 Turbulent
10^3<Re<10^4 Transition
Re < 10^3 Laminar
Classification of flows
Natural flows
cause by natural means
Forced flows
fluid is forced to flow over a surface
external means
pump or fans
steady flow
no change of field values at a point with time
unsteady flows
field values at a point change with time
1,2,3 dimensional flows
equation of motion are 3d vector equations
v= vx+vy+vz
Basis concept system and control volume
used in themodynamic mass and energy balances
applied on closed system but not on control volume
REYNOLD'S TRANSPORT THEOREM (RTT)
dBsystem/dt= dBcontrolvolume/dt = b1p1v1A1+b2p2v2A2
ARBITARY SYSTEM & CONTROL VOLUME
IN
dv=vcos θ dAdt = -vn dAdt
OUT
dv=vcos θ dAdt = vn dAdt
dB system/dt = d/dt (∫bpdV control volume) -
( ∫bpv cosθdA in control surface ) + ( ∫bpv cosθdA out control surface)
net flux = ∫bpv cosθdA out - ∫bpv cosθdA in = ∫bp(vn)dA
CONSERVATION OF MASS
∫ ∂p/∂dt dV control volume + ∫pvn dA control surface = 0
steady flow in fixed control volume
∫pvn dA control surface = 0
+incompressible fluid
∫vn dA control surface = 0
dBsystem/dt = d/dt (∫bpdV control volume) +
∫bpvn dA control surface)
DIFFERENTIAL MASS & MOMENTUM TRANSPORT (2-4)
Differential Conservation of Momentum
Integral net momentum flux of differential box
in the limit as volume shrink to ZERO.
Can be expanded using CHAIN RULE :
Rate of change of momentum in differential box :
Integral conservation of momentum
Components of integral momentum balance :
in the limit as differential volume shrink to ZERO.
Consider a differential rectangular box at (x,y,z)
Equation of motion :
substituting the material or substantive derivative
substituting Stokes' shear stress yield Navier-Stokes equation
for incompressible flow, Navier-Stokes equation becomes :
Stokes viscosity:
extension of Newton's Law of viscosity from 1-dimensional to 3-dimensional
Example: Shear stress, T(sub yx) has magnitude T, oriented to a surface which normal is the y-axis (xz plane), acting in the x direction.
1st subscript: direction of axis to which plane of shear stress action is normal
2nd subscript: direction of shear stress action
Navier-Stokes equation
rectangular coordinates
cylinderical coordinates
spherical coordinates
Application of Navier-Stokes equation: (flow in pipes & conducts)
Laminar flow in pipes
Eliminating vanishing terms : Continuity equation
Eliminate vanishing terms : Momentum equation
integrating wrt r
Hagen-Poiseuille Equation:
1 more item...
Types of time derivatives & differential operation
partial time derivative (flowing stream at fixed point, x, y, z)
total time derivative (moving about in the stream with velocities in the x, y, z direction)
substantial time derivative (observer floats along with velocity v of the flowing stream)
differential operations with scalars and vector
Normal stress & Stokes' viscosity in rectangular coordinates
Normal stress & Stokes' viscosity in cylindrical coordinates
Shear stress & Stokes' viscosity in rectangular coordinates
Shear stress & Stokes' viscosity in cylindrical coordinates
Shear stress & Stokes' viscosity in spherical coordinates
Normal stress & Stokes' viscosity in spherical coordinates
Rate of shear strain
Differential Conservation of Mass
Integral
Consider a differential rectangular box (delta)x (delta)y (delta)z at (x,y,z)
Integral mass balance for differential box is
Dividing by the volume and taking limit as volume goes to zero
Unsteady differential mass balance (Continuity equation for unsteady laminar flow)
If the flow is incompressible
The continuity equation expanded by chain rule
Substituting the material or substantive derivative
Continuity equation in rectangular coordinate
Continuity equation in cylindrical coordinate
Continuity equation in spherical coordinate
Inviscid Flow
Rectangular coordinates
Cylindrical coordinates
Spherical coordinates
Rotating fluids
Irrotational flows
Infinite cylinder
Assumptions
General solution
Velocity potential
Laplace equation in 2D
Viscous Flow
Limited effects of fluid friction, no changes in pressure across boundary
Reynold's number characterization
Boundary layer equation
Blasius's solution according to Bernoulli equation
Boundary layer
Vor Karman's suggestion
Requires velocity profile
substitution into Karman's integral equation
PARTICLES FLUIDIZATION:GROUP : 2-3
TYPES OF FLUIDIZATION
BUBLING FLUIDIZATION (AGGREGATIVE/HETEROGENEOUS FLUIDIZATION)
NON-BUBLING FLUIDIZATION (PARTICULATE/HOMOGENEOUS FLUIDIZATION)
TYPES OF POWDERS
GROUP A (AERATABLE)-EX:CRACKING CATALYST
GROUP B (BUBBLY)-EX-BUILDING SAND
GROUP C (COHESIVE)-EX-GRAVEL,COFEE BEANS
GROUP D (SPOUTABLE)-EX-FLOUR,CEMENT
CLASSIFICATION OF POWDERS
GELDART'S CLASSIFICATION
BUBBLING
SLUGGING
DESIGN APPROACH TO DILUTE PHASE :PNEUMATIC CONVERGING
introduction of the pneumatic converging
dense phase transport
the boundary
chocking velocity (vertical pipeline/transport)
saltation velocity (horizontal pipeline/transport)
pressure drop
component sum to total pressure drop
horizontal pipe
bends
acceleration
solid friction factor
vertical
top 10 tips for dense phase pneumatic conveying
step in dilute phase conveying design
top ten tips for reliable design and operation of pneumatic conveying system
Advantages of fluidized beds :
-liquid like behavior, easy to control
uniform temperature and concentration
resist rapid temp. changes, respond slowly to changes in operating condition and avoid temp. runaway.
Circulate solids between fluidized beds for heat exchange
Applicable for large or small scale operation
Achieving good heat and mass transfer rates
Disadvantage of fluidized beds
Difficult to predict and less efficient for a fine particles bubbling beds
Rapid mixing of solids causes non-uniform residence times for continuous flow reactors
Errosion of pipe and vessel wall due to particle collision
Particle comminution (breakup) is coomon
Fundamental
fluid passed upward through bed particle
pressure loss in fluid
due to frictional resistance increase with increasing fluid flow
bed become fluidized when
upward drag force = apparent weight of particle in bed
Force balance
Schematic diagram of fluidized bed: U increase, P increase due to frictional resistance
Ergun equation: Umf increases with particle size and particle density is affected by fluid properties
What is fluidization
Operation by which fine solids are transformed into a fluid like state through contact with gas or liquid
The terms fluidization and fluidized bed are used to described the condition of fully suspended particle, since the suspension behave as dense fluid
Refer to those gas-solids and liquid-solids system in which the solid phase is subjected to behave more or less like a fluid by the upwelling current of gas or liquid stream moving through the bed of solid particles.
INTEGRAL MASS, MOMENTUM : AND ENERGY TRANSPORT 2-2
REYNOLD TRANSPORT THEOREM (RTT)
Integral Transport of Momentum
Integral Transport of Mass
Mass is neither created nor destroyed
B = m
b = dm/dt = 1
dm/dt = 0
Integral Transport of Energy
3 Important physical laws in fluid dynamic
Law conservation of mass
Newton's second law of motion
First law of thermodynamics
Conservation of Mass in Control Volume
Steady flow in fixed control volume
Mass Balance
Steady flow in fixed control volume and incompressible fluid
density,p is not considered in equation.
Mass Balance
Conservation of Momentum
Newton Second Law
B = mv, b = d(mv)/dm = v
Momentum is the product of mass and velocity.
In fixed control Volume
Moving Velocity
Average velocity
Steady Flow
In one inlet and outlet
Reducing pipe bend
Control volume defined by pipe surface
Total force
Momentum involve component x,y and z direction.
Overall momentum balance:
X-component
Y-component
Resultant Force
Conservation of Momentum in Annular Shell
Steady lamina flow pipeline
Force Component
Rate of Change of momentum in control volume = 0
Net momentum flux when steady flow = 0
Fully developed flow, where pressure gradient is constant
Newtonian Fluids
Hagen-Poiseuille Equation
Conservation of energy
Energy
B = E = me ; b = d(me)/dm = e = u +v^2/2 +gz
Rate of energy received by the system
dBsys/dt = d(Esys)/dt = Q - W
WORK
Shaft work, Ws
Friction Work, Wt
Pressure work on surrounding, Wp
Energy balance for steady flow
Energy balance for steady flow of pump system
FRICTIONAL LOSS WORK
Frictional loss work Wt is caused by:
Fluid flowing through fittings such as valves experience
change in direction and friction at wall surface
Fluid in contact with inner wall surface of pipe experience
shear stress due to viscosity
Fluid negotiating elbow experience change in direction in
addition to friction at wall surface
Fluid undergoing sudden expansion and contraction due to
sudden change in pipe diameter
Friction pressure loss for laminar flow
Friction Loss work
Friction pressure loss for turbulent flow
Energy balance for steady frictionless flow
CALCULATION
CALCULATION OF PUMP POWER
CALCULATION OF NET POSITIVE SUCTION HEAD
•
MEASUREMENT OF FLUID FLOWRATE IN PIPES
Venturi meter
Impact tube or pitot tube
Orifice meter
COMPRESSIBLE FLOW IN PIPES & CONDUITS
TURBULENT FLOW (2-5)
Velocity profile of turbulent flow in pipe - can be expressed as a power law
Blob of fluid transported upward by fluctuating turbulent velocity, v through distance, L
in new position assuming the blob velocity remains unchanged
eddy viscosity
mix length theory to derived the nearly distribution near the wall
near the wall mixing length is propotional to y; L=Ky
use turbulance :
to mix & homogenize fluid mixtures
to accelerate chemical reaction rates in liquids and gases or in liquid-gas mixtures
flow variables vary with time
reynolds averaging
the mean of deviation is zero by definition
other properties presented in same way
velocity varying with time around a mean velocity
as the sume of mean velociy & a velocuty deviation
the mean kinetic energy per unit mass
can be measured by the turbulent intensity defines as the root mean square of the velocity deviation over the mean velocity of the flow.
types of turbulent flow
Homogenous turbulance
v= constant everywhere
Free shear layer turbulance
Turbulent free from effect of walls
Wall shear layer turbulance
in pipes, channels and boundary layer
in thin shear boundary layer flow & dimensional boundary eq:
Business Q theory
comparison viscous & Reynold's shear stress in shear layer
:
semi empirical theory
at constant pressure
eq of motion
System
Closed System
Open System
dilute phase transport
Continuum
CALCULATION OF NET POSITIVE SUCTION HEAD
•
correlation of Rizk
Puwani (1976)
phase diagram for horizontal pneumatic transport
substituting the continuity equation
based on cylindrical coordinates
because it follows the geometry
final equation
