Momentum Heat Mass Transfer

1
Momentum Heat Mass Transfer
MHMT14
Mass transfer
Ficks law. Molecular mass transfer. Mass
transfer with chemical reactions. Unsteady mass
transfer. Convective mass transfer. Axial
dispersion model.
Rudolf Žitný, Ústav procesní a zpracovatelské
techniky CVUT FS 2010
2
Mass Transfer - diffusion
MHMT14
General transport equation for property P can be
applied for the mass transport too
In case of mixture of several components (we
shall consider as an example binary mixture
consisting in components A and B) the transported
properties P are mass concentrations ?A of
components proportional to mass fraction ?A and
density ?
Remark components can be for example water
vapour and air, or chemical species, CH4, O2, N2,
Mass flux of a component A is directly
proportional to the driving potential gradient
of concentration (or mass fraction), this is
Ficks law
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Mass Transfer transport eqs.
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Substituting into the general transport equation
we obtain transport equations for each component
separately
Using identity
it is possible to write the transport equation as
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Mass Transfer transport eqs.
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So far everything seems to be as usual. For
example exactly the same equation (Fourier
Kirchhoff) holds if you substitute T for the mass
fraction ?A. However, what does it mean the
velocity u in the case of mixture? Individual
components are moving with different velocities
and resulting mass flux (kg/(m2.s)) is the sum of
the component fluxes
Especially at gases the concentrations are
expressed in terms of molar concentrations and
molar fractions
5
1D steady diffusion
MHMT14
General transport equation for steady state,
constant density and DAB, without source term
reduces to
For gases it is more suitable to assume constant
overall pressure and to use molar fractions (yA)
instead mass fractions
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1D steady diffusion
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Let us consider the case that u(m)0, i.e. the
same number of molecules A is moving in one
direction as the number of B molecules in the
opposite direction. This is so called equimolar
diffusion and concentration profile is linear
Different case is for example evaporation of
water vapors (component A) into air (component
B). Air cannot be absorbed in water and therefore
mass or molar flux is zero (uB0) and mean molar
velocity is determined by the velocity of vapors
uA
(how to calculate the molar flux NzA will be
shown in the next slide)
Equation of transport (steady state, without
sources and unidirectional diffusion)
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1D steady diffusion
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The mean molar velocity (convective velocity) for
uB0 and the resulting molar flux NA follow from
the definition of molar flux and the Ficks law
Substituting for yA the previously calculated
exponential yA profile gives
and the molar fraction profile
and you see that this profile is also
independent of DAB
8
Analogy heat and mass transfer
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We could continue the lecture by unsteady
diffusion, in a similar way and with similar
results as in the heat transfer. For example the
principle of penetration depth remains
and thus it is possible to estimate the range of
concentration changes corresponding to duration
of a concentration disturbance (typical problem
calculate the depth of soil filled by petrol
spilled on surface at a specified time after
accident). Problem of mass transfer from a
surface to flowing fluid (convection) is also
solved in a similar way. It is only necessary to
use appropriate variables
Heat transfer Mass transfer




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Analogy heat and mass transfer
MHMT14
Analogical criteria
Heat transfer Mass transfer




Analogical correlations (valid for low
concentrations, close to equimolar diffusion)
Heat transfer Mass transfer
Parallel flow around a plate
Flow around a sphere
Flow around cylinder
10
RTD axial dispersion
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Do you remember lecture 8 (RTD)? The case of
injection a tracer into a stream of fluid flowing
through an apparatus was analyzed with the aim to
identify the RTD Residence Time Distribution of
particles.
This is an example of transient convective
diffusion problem (distribution of tracer
concentration).
Weyden
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RTD axial dispersion
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The problem can be formulated as follows a
liquid flows in a long pipe with a fully
developed velocity profile, for example
in the laminar regime,
where r is dimensionless radial coordinate and U
is mean velocity. At inlet (x0) a small amount
of tracer is injected at an infinitely short
time. The injection should simulate the situation
when all fluid particles passing through x0 are
labeled during a short time interval dt. These
labeled molecules are component A and their mean
concentration cmA(t) is recorded by a detector
located at a distance L behind the injection
point
cA
cA(t) is impulse response
t
L
The mean concentration in a cross-section cmA can
be defined either as the area average
or the mass average
(both definitions are the same for the case when
velocity and/or concentrations are uniform at
cross section of pipe)
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RTD axial dispersion
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Distribution of tracer concentration is described
by the transport equation
this term, axial diffusion, is in fact negligibly
small when compared with the radial diffusion
The solution cA(r,x,t) obtained in the lecture 8
assumed zero diffusion (DAB0), therefore purely
convective transport of tracer giving impulse
response
Remark this definition of mean concentration
ensures that the recorded impulse response is
identical with the residence time distribution
Validity of this convective solution is
restricted to very short times, so short that the
penetration depth of tracer is less than the
radius of pipe
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RTD axial dispersion
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Transport equation taking into account radial
velocity profile is very difficult to solve (see
later). However, for example at turbulent flow
regime the velocity profile is almost uniform and
the transport equation is simplified
And this is called axial dispersion model ADM

• The parabolic PDE should be completed by boundary
  conditions
• Open/Open problem cA?0 for x??, x?-?
• Closed/closed pipe of a finite length (DAB0 for
  xlt0 and xgtL)

Remark The closed end means that a labeled
particle A once entering the inlet of pipe at x0
cannot move back due to random migration and also
a particle once leaving the outlet cannot be
returned back into the system (in our case pipe).
Initial condition at time t0 There is Zero
concentration everywhere with the exception of
origin where a unit amount of tracer was injected
Dirac delta function
14
RTD axial dispersion
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It is not necessary but useful to simplify the
diffusion equation by using the transformation to
a convected coordinate system (?,t) moving with
fluid
?
This is a linear parabolic equation, exactly the
same as the equation for unsteady heat transfer
(distribution of temperature in a plate). Only
the boundary and initial conditions are
different. The analytical solution based upon
infinite series of terms Fi(t)Gi(?) is suitable
for the case of bounded regions (e.g. a plate of
finite thickness), while in infinite regions
??(-?,?) integral transforms are preferred. In
our case we try to use the Laplace transform of
time t?s
which transforms the time derivative to
multiplication by Laplace variable s
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RTD axial dispersion
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Application of Laplace transform to partial
differential diffusion equation gives
and this is only an ordinary differential
equation with respect spatial coordinate ?
Solution of this equation for ??0 (in this region
the delta function is zero) is easy
This a continuous function of ? with
discontinuous first derivative at ?0
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RTD axial dispersion
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Thus determined coefficient A(s) completes the
solution (expressed in the Laplace domain),
satisfying open/open boundary conditions and also
initial condition
Next step must be the inverse Laplace transform
usually performed by using tables of Laplace
transforms. In this reference you find
and this is almost our case, the only difference
is scaling of the s-variable by a constant DAB.
There is simple rule for scaling
(Prove!)
Using the formula for inverse transform we obtain
final result, concentration at a distance x and
time t
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RTD axial dispersion
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Comparison of impulse response of the convective
model (parabolic velocity profile) and the axial
diffusion model (constant velocity U) for
open/open case
DAB0.05 m2/s U1 m/s L1 m
Please, note the fact, that the value of
diffusion coefficient 0.05 is absolutely
unrealistic, even at gases the molecular
diffusion is of the order 10-5. Much greater
values (e.g. 0.05) are effective dispersion
coefficients (see later)
Gaussian concentration distribution along a pipe
at a fixed time
Width of concentration pulse is very well
characterised by the penetration depth
A small amount of tracer diffused before the
injection cross section (open/open case)
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RTD axial dispersion
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The closed/closed problem can be solved in a
similar way, using Laplace transform and the
inverse transform
giving similar result
Both models of axial dispersion (open/open,
closed/closed) are frequently used in practice
for modeling RTD in apparatuses like tubular
reactors, packed beds, extruders, fluidised beds,
bubble columns and many others. However to match
experimental results it is necessary to use
experimentally determined coefficient De which is
usually much greater than the molecular diffusion
coefficient DAB evaluated from tables or
correlations. This is the same situation like
with the turbulent viscosity which is much
greater than the molecular viscosity. And the
same reason effective diffusion coefficient
(called dispersion coefficient) is not a material
parameter and its value is affected by
macroscopic motion - convection. Laminar flow in
pipe with nonuniform velocity profile is a good
example axial dispersion is determined first of
all by convection (by the parabolic velocity
profile). This problem was first solved by
G.I.Taylor, see next slides
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Axial dispersion model ADM
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G.I.Taylor (do you remember his analysis of large
bubbles?) performed experiment with injection of
colour tracer into laminar flow in pipe.
Taylor G.I. Dispersion of soluble matter in
solvent flowing slowly through a tube.
Proc.Roy.Soc. A, 219, pp.186-203 (1953)
Experimental setup consists in a long glass tube
with small boring (alternatively 0.5 and 1 mm).
Water flows inside the tube very slowly (U?1
mm/s) and thus perfect fully stabilised parabolic
velocity profile exists in the whole tube.
Diluted potassium permanganate was used as a
tracer and its concentration was evaluated
visually, comparing colour in the test tube A
with color of prepared samples with precisely
determined concentrations in the tube B. Flowrate
was controlled by the valve N and measured from
the motion of meniscus it the tube T.
During flow an expanded blob of tracer, moving
with the mean liquid velocity, was observed. When
the flow was stopped the expansion of blob was
stopped also. The axial dispersion of tracer was
observed only during flow. Theoretical
explanation presented by Taylor (1953, and 1954)
gives very surprising result Dispersion
increases with the decreasing diffusion
coefficient!!!
Capillary d1 mm L152 cm t11000 s
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ADM laminar/turbulent flow
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Theoretical models for axial dispersion in a pipe
developed by Taylor 1953 (very slow laminar
flow), 1954 (turbulent flow) can be summarized in
this way
m2/s
laminar
turbulent
Example (corresponding to the Taylors
experiment) R0.0005 m, U0.001 m/s, DAB10-9
m2/s. De5e-6 (dispersion coefficient is 500
times greater than diffusion coefficient), Re1
(laminar flow, stabilisation of parabolic
velocity profile almost immediately, at a
distance from inlet less than 0.1 mm), minimum
time corresponding to equilibrations of radial
concentration profile according to penetration
theory 80 s (therefore axial dispersion model can
be used only for times longer than 80 seconds)..
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ADM - restrictions
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Axial dispersion model can be applied either at
very high flowrates (at turbulent flow regime) or
at very small flowrates, when radial diffusion
has got enough time to equilibrate transversal
concentration profile. There is a gap for
intermediate flowrates, where a numerical
solution is still necessary.
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ADM - restrictions
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The following cA(r,x,t) profiles were calculated
numerically for different values of diffusion
coefficient. Solution for DAB1e-7 can be
approximated quite well by convective model, but
the ADM is not a very good approximation in both
cases.
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18861975
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Diffraction-quant. m.(24 years old) Motion of
shocks (25 years old)Instabilities T.C. (38
years old) Statistical theory of turbulence
Drops and bubbles ... Taylor dispersion (68
years old)... Electrohydrodynamics (83 years old)
Classical Physics Through the Work of GI Taylor
MIT Course
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ADM - derivation
MHMT14
Taylor (1953) demonstrated and experimentally
verified that the transport equation
can be substituted by the axial dispersion
equation (U-mean velocity)
where cmA is area average of concentration, and
De is dispersion coefficient
Taylor used the area average because it
corresponded to the used experimental technique
(area averaged colour of tracer). The solution
also assumed impermeable wall, therefore zero
concentration gradient at wall was applied as a
boundary condition.
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ADM - derivation
MHMT14
In the following I shall try to follow the
Taylors derivation with only slight
changes Instead of the area average the mass
average will be used. The reason why is in the
fact that only then it is possible to interpret
the impulse response (response to infinitely
short injection of tracer) as the RTD residence
time distribution. The second modification
concerns the boundary condition at wall instead
of impermeable wall (zero gradient) Newtons
boundary condition will be used. Why? Exactly the
same transport equation holds also for
temperature and the only difference is
temperature diffusivity a replacing diffusion
coefficient DAB. Similar stimulus response
experiments are carried out with a temperature
marking instead of the tracer injection (the
temperature marking can be realised by short
heating of incoming fluid by an ohmic or
dielectric heater, and response can be easily
recorded by thermocouples). However, in the case
that the tube will not be perfectly insulated,
the dispersion of temperature T differs from the
dispersion of a component cA. In the following
equations the symbol for temperature T(r,x,t)
will be used as an alternative of concentration
cA(r,x,t).
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ADM - derivation
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Transport equation for T (temperature or
concentration) and boundary condition at wall can
be written in terms of dimensionless variables
where r-radius/R, x-distance/R. Pe is Peclet
number defined as
? is dimensionless time (Fourier number)
The coefficient k can be interpreted as the Biot
number
where ? represents heat transfer coefficient from
the pipe wall to environment.
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ADM - derivation
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Transport equation can be transformed to the
moving coordinate system (moving right with the
mean fluid velocity U)
(1)
Mass average Tm for parabolic velocity profile is
the integral
(2)
Approximation of solution of PDE is suggested in
the form
(3)
based upon assumption that a radial profile
exists only if there are some changes in the
axial direction. The functions h(r) and g(r) are
to be specified.
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ADM - derivation
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Substituting approximation (3) into (1)
(4)
The function h(r) should be an even function h(r)
h1h2r2h3r4
The coefficient h1 follows from definition of
average temperature
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ADM - derivation
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In a similar way the function g(r) is derived
The five coefficients A,B,C,D,E are selected so
that the radial dependence will be eliminated (3
equations for coefficients at r2, r4, r6), then
the normalisation condition
and finally required boundary
condition at wall
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ADM - derivation
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After the suggested (little bit tedious)
manipulations the following transport equation
can be obtained
and returning back to the fixed coordinate system
If we repeat the whole procedure but now using
the area average Tm
The both equations reduces to the ADM for
insulated (impermeable) wall (k0)
and this is the result obtained by Taylor
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ADM - derivation
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Frankly, I am not sure, if the derivation is
correct isnt it surprising that the ADM model
with insulated walls is the same when using area
and mass averaged concentrations (temperatures)?
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Mass transfer - reactions
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Diffusion plays a dominant role in chemical
reactions and combustion, because species react
only in the case that they are sufficiently mixed
to a molecular level (micromixing).
Hockney
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Mass transfer - reactions
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?i mass fraction of specie i in mixture kg
of i/kg of mixture ??i mass concentration of
specie kg of i/m3 Mass balance of species
(for each specie one transport equation)
Because only micromixed reactants can react
Rate of production of specie i kg/m3s

• Production of species is controlled by
• Diffusion of reactants (micromixing) tdiffusion
  (diffusion time constant)
• Chemistry (rate equation for perfectly mixed
  reactants) treaction (reaction constant)

Damkohler number
Daltlt1
Reaction controlled by kinetics (Arrhenius)
Dagtgt1
Turbulent diffusion controlled combustion
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Species transport - Fluent
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This is example of 2 pages in Fluents manual
(Fluent is the most frequently used program for
Computer Fluid Dynamics modelling)
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EXAM
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Mass transfer
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What is important (at least for exam)
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Transport equation, written either in
concentrations (mass or molar) or in fractions
Fick law
Penetration depth
37
What is important (at least for exam)
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Analogy and corresponding dimensionless criteria
Axial dispersion model and relationship between
diffusion and dispersion coefficients