Momentum Heat Mass Transfer

1
Momentum Heat Mass Transfer
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Navier-Stokes equations Non-Newtonian fluids
Newtonian fluids and Navier Stokes equations.
Steady and transient flow between parallel
plates, flow in pipe, annular gap, hydraulic
diameter. Non-Newtonian fluids. RMW equation.
Thixotropic fluids.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
2
Unknowns / Equations
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There are 10 unknowns (assuming isothermal
flow) u,v,w, (3 velocities), p, ?xx, ?xy,(6
components of symmetric stress tensor) And the
same number of equations Continuity equation 3
Cauchys equations 6 Constitutive equations
3
Navier Stokes equations
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Substituting the constitutive equation for
viscous stresses (Generalised Newtonian Fluid)
into the divergency term of the Cauchys equation
gives, see the next slide
4
Navier Stokes equations
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Divergence of viscous stresses
This is the same, but written in the index
notation (you cannot make mistakes when
calculating derivatives)
These terms are small and will be replaced by a
parameter sm
These terms are ZERO for incompressible fluids
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Navier Stokes equations
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General form of Navier Stokes equations valid for
compressible/incompressible Non-Newtonian (with
the exception of viscoelastic or thixotropic)
fluids
Special case Newtonian fluids with constant
viscosity (compressible)
This term is zero for incompressible liquids
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Navier Stokes equations
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Special cases 2D flow (liquids) in Cartesian
coordinate system
2D flow in cylindrical coordinate system
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Navier Stokes equations
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Special cases 2D compressible flow formulated in
terms of stream function ? and vorticity ?
reduces number of equations (continuity equation
is automatically satisfied) and eliminates
pressure.
This term is zero for incompressible liquids
These equations follow from the Navier Stokes
equations using vorticity and stream function
according to the previously introduced
definitions
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Navier Stokes solutions
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The convective acceleration term makes
Navier Stokes nonlinear and therefore analytical
solutions can be found only when this term
disappears (flow in straight pipes) or is very
small comparing with the viscous term (Relt1,
creeping flow).
Modigliani
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Drag flow
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Laminar flow between parallel plates
Steady drag flow (no pressure gradient)
Shear stress (constant in the whole gap)
Transient drag flow (U-unit step of velocity of
plate)
Momentum of ?-layer
Stress in ?-layer
Stress is a flux of momentum
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Penetration depth
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Integration
yields expression for thickness of the
accelarated fluid layer (penetration depth)
This solution is only an approximation, because
the linear velocity profile with a turning point
at ? is not an exact solution of the Navier
Stokes equation. Exact solution exists in form of
an infinite series for a finite thickness of gap
H and for the case that H ?? is defined by error
function, giving more accurate prediction of the
penetration depth
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Extensional flow
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In the simple shear flow between parallel plates
velocities are defined in terms of the rate of
shear In the simple extensional flow (uniform
stretching of incompressible fluid in the
x-direction) velocities are defined in terms of
rate of elongation
y
x
Constitutive equation for Newtonian liquid gives
z
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Flow in a circular pipe
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Steady fully developed laminar flow only one
non-zero velocity component uz(r)
Balance of forces for control volume (ring dr x
dz)
Constitutive equation (Newtonian liquid)
Navier Stokes in z-direction
General solution
Boundary conditions r0,R
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Flow in a circular pipe
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Volumetric flowrate
Hagen Poisseuille law
Darcy Weisbach equation for calculation of
pressure drop in channels
Reynolds number is an indicator of
laminar/turbulent flow regime. Above the value
Re2300 (in US) or 2100 (in EU ?) the flow is
mostly turbulent. The complete profile including
the turbulent flow regime is presented in the
Moody diagram (see next page) or described by
using correlation Churchill S.W. Friction factor
equation spans all fluid-flow regimes. Chemical
Engineering, 1977, 84 pp.91-92.
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Flow in a circular pipe
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Pressure drop in a circular pipe Darcy Weisbach
equation
Frinction factor ?f depends upon Re and relative
roughness
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Flow in an annular gap
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Navier Stokes equation for flow between two
concentric pipes is the same and its general
solution is also the same as with the circular
pipe
uz(r)
Only the boundary conditions are different, giving
z
D12R1
D22R2
(use per partes for integration r lnr)?
96-parallel plates
64-circular pipe
verify that the limit for ??1 is 3/2 using
expansion
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Equivalent diameter
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General cross section of a channel can be
characterized by equivalent hydraulic diameter
Dh, that is used in definition of Reynolds
number.
Cross section surface
Volume of channel
Surface of wall
Perimeter of cross section
At turbulent flows the same correlations for
pressure drop (friction factor) can be used.
Correlations for circular pipe are usually used,
however the cross sections with sharp corners
(triangles, cusped ducts) lead to error up to 35
.
Equivalent diameter is used also in laminar
flows, but different correlations for different
cross sections must be used (from this point of
view the laminar regime is more
complicated). Modified definitions of equivalent
diameter exist for specific classes of cross
sections (e.g. average distance from the point of
maximum velocity in triangles, or square root of
the cross section area, see next slides).
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Equivalent diameter and ?fRe
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Equivalent diameter for rectangular cross section
Increased ?f when compared with circular pipe
b
a
Equivalent diameter for excentric inner tube is
independent of excentricity!
However ?fRe decreases with the increasing
excentricity!
?fRe varies from the values about 30 (corners) to
about 130 (bundle of pipes)
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Non-Newtonian fluid flow
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Non-Newtonian fluid flow
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How to calculate the volumetric flowrate in a
circular pipe as a function of pressure drop in
the case of non-Newtonian fluids?
In 1D case (fully developed unidirectional axial
flows) the constitutive equations for
incompressible generalised Newtonian fluids (GNF)
can be expressed as
for Newtonian fluid
shear rate
shear stress
for Power law fluid
for Bingham fluid
Volumetric flowrate for quite arbitrary radial
velocity profile is
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Non-Newtonian fluid flow
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Radial shear stress profile follows from the
equilibrium of forces
This linear shear stress profile holds for any
fluid and can be used for replacement of the
integration variable r
This is Rabinowitsch Mooney Weissenberg (RMW)
equation
giving volumetric flowrate regardless of specific
model as a function of wall shear stress
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Non-Newtonian fluid flow
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Application RMW equation for power law fluid
Friction factor
Modified Reynolds number (reduces to standard Re
for n1)
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Non-Newtonian fluid flow
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Application RMW equation for Bingham fluid
Introducing the friction factor and expression
for the wall shear stress
we obtain
Remark Given pressure drop (therefore ?w) it is
quite easy to calculate flowrate. Reversely
given flowrate the pressure drop must be
calculated by solution of algebraic equation of
the 4th order (but there exists graphical
representation of the previous equation).
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Non-Newtonian fluid flow
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Exercise Derive the RMW equation for the
Herschel Bulkley model
K-consistency, n-power law index, ?y-yield stress
.verify that the n1 reduces to the previously
derived Bingham model.
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Non-Newtonian fluid flow
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Practical importance of RMW equation is in the
fact that it enables generalization of
experiments with arbitrary liquid, without
necessity to identify a specific rheological
model.
Diagram of consistency variables
Experimentally determined curves ?w, ? are
independent of the pipe dimensions, therefore can
be used for design of pipelines (with the same
liquid and at the same temperature).
Remark different ?w, ? curves recorded at
different diameters of pipes indicate anomalies,
for example wall slip (Mooney analysis),
different flow regime (turbulent flow) or
experimental errors (insufficient stabilization
length of pipe, Bagley correction).
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Thixotropic fluid in a pipe
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Research described in the following pages was
motivated by experimentally determined strange
behaviour of a secret fluid X How is it
possible that one and the same fluid at the same
temperature and at the same flowrate exhibits
different pressure drops ?
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Thixotropic fluid in a pipe
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Thixotropic fluids are characterised by
viscosity, which depends upon the deformation
history (structure and consistency is affected by
shear rate at previous times). Example of
constitutive equation of a thixotropic liquid was
presented as the HZS model (see previous
lecture). Problem of pressure drop of a
thixotropic fluid in laminar flow in a circular
pipe is usually solved numerically, see the list
of relevant papers
Ahmadpour A., Sadeghy K. An exact solution for
laminar, unidirectional flow of Houska
thixotropic fluids in a circular pipe. J. of
Non-Newtonian Fluid Mechanics, 194 (2013),
pp.23-31 Corvisier P., Nouar C., Devienne R.,
Lebouché M. Development of a thixotropic fluid
flow in a pipe. Experiments in Fluids, 31 (2001),
pp.579-587 Schmitt L., Ghnassia G., Bimbenet
J.J., Cuvelier G. Flow properties of stirred
Youghurt Calculation of the pressure drop for a
thixotropic fluid. J.Food Eng. 37 (1998),
pp.367-388 Escudier M.P., Presti F. Pipe flow of
a thixotropic liquid. J. Non-Newtonian Fluid
Mech., 62 (1996), pp.291-306 Billingham J.,
Fergusson J.W.J. Laminar unidirectional flow of
a thixotropic fluid in a circular pipe. J. of
Non-Newtonian Fluid Mech., 47 (1993),
pp.21-55 Kemblowski Z., Petera J. Memory effects
during the flow of thixotropic fluids in pipes.
Rheol.Acta 20, (1981), pp. 311-323
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Thixotropic fluid in a pipe
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The HZS model of thixotropic fluids is
represented by the Herschel Bulkley constitutive
equation (a combination of power law liquid with
a consistency coefficient K and power law index n
and Bingham liquid with a yield stress ?y )
Structural parameter ?1 describes fully
recovered inner structure (and high consistency
of liquid), while ?0 corresponds to completely
destroyed structure (and minimum consistency K
and yield stress). Time changes of ? are
described by
There exist many different modifications and
interpretations, for example the latest work,
Ahmadpour (2013), assumes only partial and not
the material time derivative on the left side,
the diffusion term on the right side is
considered only by Billingham and Fergusson
(1993) in a generalized Moores model of
thixotropy.
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Thixotropic fluid in a pipe
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The tensorial form of the HZS model can be
simplified for the special case of unidirectional
simple shear flow to scalar equations for the
shear stress ? and the shear rate
Complete solution of a creeping laminar flow in a
pipe should calculate axial as well as radial
profiles of velocity and structure parameter
based upon linearity of radial shear stress
profile
A great simplification would be assumption that
the ? depends only upon the axial coordinate and
time, ?(t,x). This assumption can be accepted
only if the flow is so slow that the diffusion in
the radial direction has enough time to equalize
the radial profile of ? and that the problem of
?-transport with a nonuniform radial velocity
profile can be substituted by a model with plug
flow (constant velocity) and modified diffusion
in the axial direction (model of axial
dispersion, which will be discussed later
lecture on mass transport)
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Thixotropic fluid in a pipe
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For ? independent of radial coordinate it is
possible to apply RMW (Rabinowitsch, Mooney,
Weissenberg) equation with
We need to calculate pressure drop (dp/dx2?w/R)
for given flowrate therefore it is necessary to
invert the previous equation
Remark Iterative evaluation is necessary, but is
fast and convergent for arbitrary n,K,?y
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Thixotropic fluid in a pipe
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If the structural parameter depends only upon x
and time and if we neglect the axial dispersion
term the evolution of ? is described by
hyperbolic partial differential equation
which can be integrated analytically along
characteristic dxu.dt as soon as the velocity
and shear rate are constant
?(t) is value of structural parameter of a fluid
particle having value ?0 at time t0 (assuming
that the particle is under action of constant
shear rate).
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Thixotropic fluid in a pipe
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In our experiment the fresh fluid at the inlet to
pipe has (pressumably) fully recovered structure,
therefore ?01. At a distance x from inlet
depends ? of a fluid particle upon the time of
action (tx/u) and upon the intensity of action
(?m). The time of action decreases with the
increasing flowrate, while the intensity of
decomposition increases with flowrate
fluid X
flow meter
differential pressure transducer
2R
x
?1
??
It is interesting that there exists a flowrate
when ? is minimum, see graph. This extreme
(maximum of thixotropy effect) is determined by
equation
that must be solved numerically.
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Thixotropic fluid in a pipe
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Resulting expressions (for example expressions
for friction factor, for structural parameter,
etc) are usually formulated in terms of
dimensionless parameters, Debora number,
thixotropy number, Bingham number and other, see
for example Billingham, Fergusson (1993) .
and for the special case m1, the Debora number
at maximum tixotropy goes to infinity. Nontrivial
optimum exists only for mlt1.
Example ab0.01, m0.9, R0.01, x1 optimum at
De1, Tx5.3 (therefore flowrate 2.10-5 m3/s)
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Thixotropic fluid in a pipe
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The following MATLAB program calculates pressures
?p corresponding to the stepwise increase and
decrease of flowrate
fluid X
VmaxVnstep/2
flow meter
VminV1
duration of time steps ?tistep is determined
according to flowrate (velocity u) and constant ?x
differential pressure transducer
time
t10
tnstep
2R
?1
L
characteristics
t
x
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Thixotropic fluid in a pipe
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function dt,lamlamnew(a,b,m,vdot,r,lamx,nx,dx)
vector of structural parameters lam(1..nx) at
time tnew inlet value is always lam(1)1 (fresh
fluid) a,b,m model parameters (see model
HZS) uvdot/(3.141r2)gammavdot/(3.141r3) la
m(1)1dtdx/ugamgammam for
i1nx-1 lam(i1)(a-(a-(abgam)lamx(i))exp(-(a
bgam)dt))/(abgam) end
simulation (flowrate up and down) l3 length
of pipe r0.01 radius nx50nstep150 k50
dk500 ty3000 dty3000 n0.8 Herschel
Bulkley a0.000001 b0.01 m0.9 HZS
thixotropy model vmin1e-5 vmax1e-4
min,max.flowrate xlinspace(0,l,nx)dxl/(nx-1)
time sequence of flowrate and corresponding
times nstep2nstep/2dv(vmax-vmin)/nstep2
v(1)vmintime(1)0 for i2nstep2
v(i)v(i-1)dv time(i)time(i-1)dx3.141r2/
v(i-1) end for instep21nstep
v(i)v(i-1)-dv time(i)time(i-1)dx3.141r2/
v(i-1) end lamx(1nx)1 dp(1),tauxdpa(k,d
k,ty,dty,n,v(1),r,lamx,nx,dx) dt,lamlamnew(a,b
,m,v(1),r,lamx,nx,dx) figure(3) hold off for
istep2nstep lamx(1nx)lam(1nx)
dp(istep),tauxdpa(k,dk,ty,dty,n,v(istep),r,lamx
,nx,dx) dt,lamlamnew(a,b,m,v(istep),r,lamx
,nx,dx) if mod(istep,40)0
plot(x(1nx),lam(1nx)) hold on
end end figure(1) plot(time(1nstep),v(1nstep)) f
igure(2) hold off plot(v(1nstep),dp(1nstep)) hol
d on plot(v(1),dp(1),'rd')
function tauwhb(ty,k,n,vdot,r) ty-yield
stress, k-consistency, n-flow index,
vdot-flowrate, r-radius kappan/(3n1) eps1e10
iter0 while (iterlt50 epsgt1e-4)
iteriter1 tauwtyk(vdot/(kappa3.141r3)
)n tty/tauw kappann/(3n1)(1-t/(2n
1)-2n/((n1)(2n1))(t2nt3))
epsabs(kappa-kappan) kappakappan end
function dp,tauxdpa(k,dk,ty,dty,n,vdot,r,lamx,n
x,dx) pipe length (nx-1).dx , radius r,
flowrate vdot, strict parameter
lam(1),...lam(nx) result pressure drop dp,
vector of wall stresses taux(1)....taux(nx) for
i1nx taux(i)hb(tylamx(i)dty,klamx(i)dk,
n,vdot,r) end dp0 for i1nx-1
dpdptaux(i)2dx/r end
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Thixotropic fluid in a pipe
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k50 dk500 ty3000 dty3000 n0.8 a0.000001
b0.01 m0.9
Very slow (180 s) Medium (70 s)
Fast changes (25 s)
red diamond is starting point (zero time, minimum
flowrate)
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Thixotropic fluid in a pipe
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k50 dk500 ty3000 dty3000 n0.8 medium rate
of flowrate changes
a0.01 b0.01
a0.1 b0.01
a1 b0.01
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Thixotropic fluid in a pipe
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k50 dk500 ty3000 dty3000 n0.8 a0.001
b0.01 m0.9
m0.9
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EXAM
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Navier Stokes equations
39
What is important (at least for exam)
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Navier Stokes equations (2D incompressible)
Formulation with vorticity and stream function
40
What is important (at least for exam)
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Drag flow between parallel plates (steady and
transient) Penetration depth
Flow in a circular pipe
Darcy Weisbach, Reynolds number and Mooney diagram
41
What is important (at least for exam)
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Non Newtonian flows RMW equation