Momentum Heat Mass Transfer MHMT14 Fick´s law. Molecular mass transfer. Mass transfer with chemical...
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Momentum Heat Mass Transfer MHMT14 Fick´s 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 ČVUT FS 2010 Mass transfer source Dt D
Momentum Heat Mass Transfer MHMT14 Fick´s law. Molecular mass transfer. Mass transfer with chemical reactions. Unsteady mass transfer. Convective mass
Momentum Heat Mass Transfer MHMT14 Ficks law. Molecular mass
transfer. Mass transfer with chemical reactions. Unsteady mass
transfer. Convective mass transfer. Axial dispersion model. Rudolf
itn, stav procesn a zpracovatelsk techniky VUT FS 2010 Mass
transfer
Slide 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, CH 4, O 2, N 2,
Mass flux of a component A is directly proportional to the driving
potential gradient of concentration (or mass fraction), this is
Ficks law
Slide 3
Mass Transfer transport eqs. MHMT14 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
Slide 4
MHMT14 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/(m
2.s)) is the sum of the component fluxes Especially at gases the
concentrations are expressed in terms of molar concentrations and
molar fractions Mass Transfer transport eqs.
Slide 5
MHMT14 1D steady diffusion General transport equation for
steady state, constant density and D AB, without source term
reduces to For gases it is more suitable to assume constant overall
pressure and to use molar fractions (y A ) instead mass fractions z
A1 y A1 A2 y A2 L
Slide 6
MHMT14 1D steady diffusion 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 (u B =0) and mean
molar velocity is determined by the velocity of vapors u A Equation
of transport (steady state, without sources and unidirectional
diffusion) L z y A1 y A2 y A (z) z L y A1 y A2 y A (z) (how to
calculate the molar flux N zA will be shown in the next slide)
Slide 7
MHMT14 1D steady diffusion The mean molar velocity (convective
velocity) for u B =0 and the resulting molar flux N A follow from
the definition of molar flux and the Ficks law Substituting for y A
the previously calculated exponential y A profile gives and the
molar fraction profile and you see that this profile is also
independent of D AB
Slide 8
MHMT14 Analogy heat and mass transfer 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 transferMass transfer
Slide 9
MHMT14 Analogy heat and mass transfer Analogical criteria Heat
transferMass transfer Analogical correlations (valid for low
concentrations, close to equimolar diffusion) Heat transferMass
transfer Parallel flow around a plate Flow around a sphere Flow
around cylinder
Slide 10
RTD axial dispersion MHMT14 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.lecture 8 Weyden This is
an example of transient convective diffusion problem (distribution
of tracer concentration).
Slide 11
RTD axial dispersion MHMT14 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
(x=0) a small amount of tracer is injected at an infinitely short
time. The injection should simulate the situation when all fluid
particles passing through x=0 are labeled during a short time
interval dt. These labeled molecules are component A and their mean
concentration c mA (t) is recorded by a detector located at a
distance L behind the injection point L t cAcA c A (t) is impulse
response The mean concentration in a cross-section c mA 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)
Slide 12
RTD axial dispersion MHMT14 Distribution of tracer
concentration is described by the transport equation The solution c
A (r,x,t) obtained in the lecture 8 assumed zero diffusion (D AB
=0), therefore purely convective transport of tracerlecture 8
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 this term, axial diffusion, is in fact negligibly small when
compared with the radial diffusion
Slide 13
RTD axial dispersion MHMT14 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
Ut x Colour injection R c A (x,t) The parabolic PDE should be
completed by boundary conditions A)Open/Open problem c A 0 for x ,
x - B)Closed/closed pipe of a finite length (D AB =0 for x L) L x
Remark: The closed end means that a labeled particle A once
entering the inlet of pipe at x=0 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 t=0: There is Zero concentration everywhere with the
exception of origin where a unit amount of tracer was injected
Dirac delta function And this is called axial dispersion model
ADM
Slide 14
RTD axial dispersion MHMT14 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 F i (t)G i
( ) 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 sLaplace transform which transforms the time
derivative to multiplication by Laplace variable s
Slide 15
RTD axial dispersion MHMT14 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
Slide 16
RTD axial dispersion MHMT14 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
findthis reference and this is almost our case, the only difference
is scaling of the s-variable by a constant D AB. 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
Slide 17
RTD axial dispersion MHMT14 t cAcA D AB =0.05 m 2 /s U=1 m/s
L=1 m t=1 x cAcA Comparison of impulse response of the convective
model (parabolic velocity profile) and the axial diffusion model
(constant velocity U) for open/open case 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) 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)
Slide 18
RTD axial dispersion MHMT14 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 D e which is usually much greater than the
molecular diffusion coefficient D AB 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
Slide 19
Axial dispersion model ADM MHMT14 Comparison tube Illuminated
glass plate Distilled water KMnO 4 Meniscus velocity 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) Capillary
d=1 mm L=152 cm t=11000 s 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!!!
Slide 20
ADM laminar/turbulent flow MHMT14 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 UtUt x
tracer injection R c A (x,t) [m 2 /s] laminar turbulent Example
(corresponding to the Taylors experiment) R=0.0005 m, U=0.001 m/s,
D AB =10 -9 m 2 /s. D e =5e-6 (dispersion coefficient is 500 times
greater than diffusion coefficient), Re=1 (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)..
Slide 21
ADM - restrictions MHMT14 L [m] Q [m 3 /s] Q Q3600 R D AB
-diffusion coefficient, D e -dispersion coefficient [m 2 /s] Very
small flowrates Very large flowrates Turbulent flow (R-radius,
-kinematic viscosity) Q>100 D AB L Purely convective transport
Taylors dispersion Hic sunt leones (see next slide) 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.
Slide 22
MHMT14 ADM - restrictions D AB =10 -7 m 2 /s 2s 4s 6s 8s 10s R
Pipe axis D AB =10 -6 m 2 /s 2s 4s 6s 8s 10s =1e-7 can be
approximated quite well by convective model, but the ADM is not a
very good approximation in both cases. The following c A (r,x,t)
profiles were calculated numerically for different values of
diffusion coefficient. Solution for D AB =1e-7 can be approximated
quite well by convective model, but the ADM is not a very good
approximation in both cases. ADM (Taylor) D AB =10 -6 m 2 /s
Numerical solution t D AB =10 -7 m 2 /s Numerical solution t ADM
(Taylor) L=1m, R=0.01m, U=0.1 m/s cAcA
Slide 23
Classical Physics Through the Work of GI Taylor MIT Course
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) 18861975 MHMT14
Slide 24
ADM - derivation Taylor (1953) demonstrated and experimentally
verified that the transport equation can be substituted by the
axial dispersion equation (U-mean velocity) where c mA is area
average of concentration, and D e 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.
Slide 25
MHMT14 ADM - derivation 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 D AB. 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 c A. In the following equations
the symbol for temperature T(r,x,t) will be used as an alternative
of concentration c A (r,x,t).
Slide 26
MHMT14 ADM - derivation 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.
Slide 27
MHMT14 ADM - derivation Transport equation can be transformed
to the moving coordinate system (moving right with the mean fluid
velocity U) Mass average T m for parabolic velocity profile is the
integral Approximation of solution of PDE is suggested in the form
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. (1) (2) (3)
Slide 28
MHMT14 ADM - derivation Substituting approximation (3) into (1)
The function h(r) should be an even function h(r) =h 1 +h 2 r 2 +h
3 r 4 The coefficient h 1 follows from definition of average
temperature (4)
Slide 29
The five coefficients A,B,C,D,E are selected so that the radial
dependence will be eliminated (3 equations for coefficients at r 2,
r 4, r 6 ), then the normalisation condition and finally required
boundary condition at wall: MHMT14 ADM - derivation In a similar
way the function g(r) is derived
Slide 30
MHMT14 ADM - derivation 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 T m The
both equations reduces to the ADM for insulated (impermeable) wall
(k=0) and this is the result obtained by Taylor
Slide 31
MHMT14 ADM - derivation 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)?
Slide 32
Mass transfer - reactions MHMT14 Hockney 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).
Slide 33
Mass transfer - reactions MHMT14 i mass fraction of specie i in
mixture [kg of i]/[kg of mixture] i mass concentration of specie
[kg of i]/[m 3 ] Mass balance of species ( for each specie one
transport equation) Rate of production of specie i [kg/m 3 s]
Production of species is controlled by Diffusion of reactants
(micromixing) t diffusion (diffusion time constant) Chemistry (rate
equation for perfectly mixed reactants) t reaction (reaction
constant) Damkohler number Da1 Reaction controlled by kinetics
(Arrhenius) Turbulent diffusion controlled combustion Because only
micromixed reactants can react
Slide 34
Species transport - Fluent MHMT14 This is example of 2 pages in
Fluents manual (Fluent is the most frequently used program for
Computer Fluid Dynamics modelling)
Slide 35
EXAM MHMT14 Mass transfer
Slide 36
What is important (at least for exam) MHMT14 Transport
equation, written either in concentrations (mass or molar) or in
fractions Fick law Penetration depth
Slide 37
What is important (at least for exam) MHMT14 Analogy and
corresponding dimensionless criteria Axial dispersion model and
relationship between diffusion and dispersion coefficients