Chapitre 3 Fluid Dynamics
1 Introduction
- In Molecular transport, we are concerned with the transfer or movement of a given
property by molecular movement through a system or medium which can be fluid or
solid.
- Property can be mass, heat or momentum.
- Each molecule of a system has a given quantity of this property.
- A transport of this property occurs when a difference of concentration of the
property exist.
- The difference in concentration is obtained from the following laws
* Heat transfer: Fourrier’s law
* Mass transfer: Fick’s law
* Momentum transfer: Newton’s law of viscosity.
Momentum:
-
When a fluid is flowing in the x direction, parallel to a solid surface, a velocity
gradient exists.
Vx = velocity in the x direction, decrease as we approach the surface in the z
direction.
The fluid has x-diercetd momentum and its concentration = vx
2 Viscosity of fluids
2.1 Newton’s law and viscosity
- When a fluid is flowing through a closed channel such as pipe or between two flats,
either of the two types of flow may occur, depending on the velocity of this fluid.
-
-
At low velocities, the fluid tends to flow without lateral mixing, and adjacent
layers slide past one another, like playing cards. This regime of type of flow is
called laminar flow.
At higher velocities, eddies form, leading to lateral mixing. This is called
turbulent flow.
-
A fluid is different from a solid in the discussion of viscosity by its behavior,
when subjected to a stress (force per unit area) or applied force.
-
An elastic solid deforms by an amount proportional to the applied stress
However a fluid when subjected to a similar applied stress will continue to
deform, i-e to flow at a velocity that increases with increasing stress.
A fluid exhibits resistance to this stress (viscosity).
Viscosity is that property of a fluid which gives rise to forces that resist the
relative movement of adjacent layers in the fluid.
These viscous forces arise from forces existing between the molecules in the
fluid and are similar character as the shear forces in solids.
-
-
Bottom plate moving parallel to the top plate at v=cte ( vz), faster than the top
plate because of F applied.
Force called viscous drag, it arises from the viscous forces in the fluid.
y= distance between plates, z direction
Layer above the bottom plate carried along at vz.
Layer just above it at a slightly slower velocity
Each layer at a slower velocity, as we go p in y direction.
Velocity profile linear
-
Experimentally for many fluids the force F has been found experimentally
F =( v , A,
y)
Newton’s law of viscosity
F/A =
vz /
= constant, called viscosity of fluid (Pa-s) or Kg/m-sec.
Units
y
0
F/A =
=
: shear stress or force per unit area
=
dvz/dy
(N/m2, dynes/ cm2, poundal/ft2)
/gc (dvz/ dy)
- The shear stress can also be interpreted as a flux of z directed momentum in the y
direction, which is the rate of flow of momentum per unit area.
2.2 Viscosities of Newtonian fluids
- Fluids that follow Newton’s law of viscosity are called Newtonian fluids.
- Linear relation between the shear stress
and velocity gradient
(rate of
shear).
- = constant and independent of rate of shear.
- For Non Newtonian fluids, the relation between
and (dvz/dy) is not linear, i.e,
is not constant, but is a function of the shear rate.
- Certain liquids do not obey Newton’s law (Paste, Slurries, high polymers).
- Science of the flow and deformation of fluids is called rheology.
- Viscosity of gases, which are Newtonian fluids increase with temperatures and is
independent of pressure.
3 Reynolds Experiment
- Types of flow in a channel is important.
- 2 types of flow can be observed:
* Velocity is slow, the flow patterns are smooth.
* Velocity is quite high, an instable pattern is observed in which eddies or small
packets of fluid particles are present, moving in all direction.
- First type of flow is called Laminar flow, and Newton’s law of viscosity holds.
- Second type of flow at higher velocities, is called Turbulent flow.
-
The existence of laminar and turbulent flow is most easily visualized by the
experiments of Reynolds.
Water flows at steady state through transparent pipe with the flow rate
controlled by a valve at the end of the pipe.
A fine steady stream of dye-colored water is introduced from a fine jet.
At low rates of water flow, the dye pattern is regular as no lateral mixing of
the fluid appeared.
This type of flow is laminar or viscous flow.
As the velocity increases, the thread of dye becomes dispersed and the pattern
is much disorganized. This type of flow is called turbulent flow.
Velocity at which the flow changes is the critical velocity.
Reynolds Number
- Transition from laminar to turbulent flow depends on velocity, density,
viscosity of the fluid and the tube diameter.
Re = D.v.
-
/
(Dimensionless number)
For a circular pipe, Re<2100 flow is laminar
‘’’’’’’’’’’’’’’’’’’’
Re>4000 flow is turbulent.
‘’’’’’’’’’ 2100<Re<4000, flow is in the transition region.
4. Overall mass balances.
4.1 Simple mass balances
- Consider an elementary balance on a simple geometry
Input = Output + Accumulation
- In fluid flow, usually at steady state, the rate of accumulation is zero, and we use the
rate of flow.
Rate of input = rate of output (steady state)
m1 =
1
A1 v1 = m2 =
2 A2.v2
( kg/sec)
- m1 and m2 masses flow rate at point 1 and 2 (kg/sec)
- v1, v2 velocities at point 1 and 2. (m/sec)
- A1, A2 cross sectional area of point1 and point 2.
Control volume for balances.
-
The laws for the conservation of mass, energy and momentum are all stated to
a defined system.
A system is defined as a collection of fluid of fixed identity in a well defined
space, which is called control volume.
4.2 Overall mass balances equations
Input = output + accumulation
Overall mass balance equations
{Rate of mass output } - {rate of mass input} + {rate of mass accumulation}=0 = rate of mass
{from control volume} {from control volume} {from control volume }
generation
{Rate of mass output } - {rate of mass input} = {Net mass efflux from}
{from control volume} {from control volume} { control volume }
= v2 2 A2 – v1
1
A1
{rate of mass accumulation in control volume} = dM/dt
v2 2 A2 – v1
-
1
A1 + dM/dt = 0
(Generally)
For a steady state
M2= v2 2 A2 = M1 = v1
-
1 A1= M
Overall mass balance for component I in a multi component system.
m2i –m1i +dM/dt= Ri
-
Average velocity to use in overall mass balance
At point1 v1=constant
At point 2 v2=constant
If velocity varies, use average bulk velocity
Vav= (1/A)
5
-
vdA
Overall Energy balance
Applying the principle of energy balance, as well as the first law of
thermodynamic to a control volune
e= q –w
e: Total energy per unit mass of fluid (J/kg)
q: Heat absorbed per unit mass of fluid (J/kg)
w: Work of all kinds per unit mass upon the surroundings (J/kg)
The overall energy balance give
Rate of entity output – rate of entity input + rate of entity accumulation =0
- The energy, e, can be classified into three ways;
a) Potential energy, ep, of a unit mass of fluid which is the energy present
because of the position of the mass in gravitational field (g)
ep=zg
z: relative height (m)
b) Kinetic energy, ek, energy present because of transitional or rotational motion
of the mass
ek= v2/2
v= velocity of the flow (m/sec)
c) Internal energy (u) which includes the energy due vibration energy in chemical
bonds
-
The total energy of the fluid/unit mass is,
e= u+ v2/2+ zg (SI units)
e= u + v2/gc + zg/gc (English lbf)
-
In addition energy is transferred when mass flows (open system), and the
internal energy is replaced by the enthalpy
h=u+Pv
Hence the total energy per unit mass at any point is
e= h+v2/2+zg (SI units)
Overall energy balance for steady state flow
e2-e1 = q-w
h2-h1 + ½ [v2 -v2 ]+g(z2-z1) = q-ws
=1/2 laminar flow
= 1 turbulent flow
-Overall energy balance for a given mass flow rate (m), for a steady state, one
dimension flow,
E= e.m
M(h2-h1 + ½ [v2 -v2 ]+g(z2-z1)) =Q – Ws
Q = m.q
Ws= m.ws
6
Overall Mechanical energy balance
-
Modified energy equation to deal with mechanical energy
Mechanical energy is a form of energy that is work or another form that can be
converted into work.
Mechanical energy terms can be converted almost completely into work.
Energy converted into heat or internal energy is lost work or a loss in
mechanical energy which is caused by frictional resistance to flow.
-
At steady flow for a unit mass,
W’ = P dV - F
First law of thermodynamic
And
U E
Q- W’
H = U +PV
H= U + P V + V P = U + PdV + VdP
Substituting for W’
U = Q - PdV + F
Substituting for U,
H = Q+ F + VdP
And =M/V
For M=1KG,
Overall energy balance
V=1/
(H2-H1) + ½ [v2 -v2 ]+g(z2-z1)) =Q – Ws
Substituting for H,
½ [v2 -v2 ]+g(z2-z1)) + dP/ + F + Ws=0
Overall mechanical energy balance
- Intergration depends on of state of fluid, for incompressible fluid, dP/
dP/ = (P2-P1), the overall mechanical energy equation becomes,
1/2 [v2 -v2 ]+g(z2-z1)) + (P2-P1)/
+ F + Ws=0
7
Bernoulli equations for Mechanical Energy balance
- Special case Ws=0
F=0, the equation becomes the Bernoulli’s equation.
- For turbulent flow which is of sufficient importance,
Z1g + v1/2 + P1/
= v2/2 + z2g + P2/
- This equation covers many situations of practical importance and is often used in
conjunction with the mass-balance equation for steady state.
8
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Shell Momentum balance and velocity profile
8.1 Shell momentum balance inside a pipe
Engineers often deal with the flow inside a circular pipe
Incompressible fluid flow inside a pipe.
Newton fluid, steady state, laminar flow, flow fully developed.
Velocity profile does not vary along the axis of the flow.
- Consider a control volume with an inside radius r, thickness r, length x.
-
At steady state, conservation of momentum gives,
Forces acting on control volume ( P/A)=
P Aix - PAI x+ = P (2 r r)x - P(2 r r) x+ x.
-
The shear force or drag force acting on the cylindrical surface at the radius r=
Shear stress (area) (2 r x)= rate of momentum flux.
Net Momentum efflux= 2 r x I r+ r -
2 r
r
Ir
Equating both equations
r
Ir+ r
-
r
Ir
=
r (PIx -
r
- In fully developed flow, the pressure gradient
P/ l
PIx+ x)
x
P/ x is constant and becomes
P= Pressure drop for a pipe length L.
Letting r approaches 0,
d (r
-
)/ dr =
(
P/L) r
Separating variables and integrating
= ( P/L) r/2 + c1/r
-
Constant of integration
C1=0 if momentum flux is not infinite at r=0
= (
P/2L) r =
(Po – Pl)r/2L
- This means that momentum flux varies linearly with the radius and the maximum
value occurs at r=R, at the wall
-
Newton’s law of viscosity
=
dvx/dr
= (Po – Pl)r/2L=
dvx/dr
dvx/dr = (Po – Pl)r/2L
-
Integrating
dvx= (Po – Pl)/2L
- Boundary conditions, vx=0, at r=R
dr
-
we obtain,
vx= (Po –Pl/4 l) R (1 – (r/R))
Velocity distribution is parabolic
-
Average velocity Vxav
Vxav = 1/A
da = 2 rdr and
VxdA
A = R2
Vxav = (1/
R2)
Vxav = (1/ R2)
-
Vx 2 rdr
(Po –Pl/4 l) R (1 – (r/R)) rdr
Integrating with boundary conditions
Vxav = (Po –Pl/8 l) R = (Po –Pl/32 l) D
This is known as the Hagen Poiseuille equation
With D= 2R
-
Maximum velocity occurs when r=0
Vmax = (Po –Pl/4
l) R2
8.2 Shell momentum for falling film
- Falling film describe various phenomena in mass transfer, coatings etc..
-
x=Thickness
- Momentum balance I direction over a system x thick, between z=0 and z=L (w:
distance in y direction)
Net efflux =LW
-
Ix+ x -LW
Ix
Gravity force acting on the fluid
( x W L) g
-Conservation of momentum at steady state
(W.l. x)
g = LW
Ix+ x
- LW
Ix
-Rearranging
=g
x
- Integrating using boundary conditions
x=0
=0
x=x
=gx
-
This means that the momentum flux profile is linear, the maximum value is at
the wall.
-
For a Newtonian fluid,
= vx/dx
(dvx/dx)= -( g/ )x
-
Separating variables and integrating,
Vx = -( g/2 ) x2 + C1
- At x=
Vz=0
C1 = ( g/ 2 ) 2
Vz= ( g /2 )( 1 – (x/ )2)
This means that the velocity profile is parabolic, the maximum velocity Vmax
occurs at x=0
Vzmax =
-
The average velocity Vzav
Vzav = 1/A VxdA = 1/W
-
Substituting for Vx and integrating
Vzav =
/3
-
The volumetric flow rate is,
q= Vav (cross sectional area)
q = Vav ( W)
q=
W
/3
- In falling films, the mass rate of flow per unit width of W, T (Kg/m-sec)
T=
-
Vzav
The Reynolds Number is
Re= 4T/
- Re< 1200, laminar flow.
=4
Vzav/