????? — ?? Lecture 8 Forced convection

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????? ??Lecture 8Forced convection

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Forced convection

• Forced convection
• The flow is determined by factors other than
  diffusion, factors like pressure gradients and
  wetted area
• exist whether or not diffusion occurs
• Analyzing tools
• simple physical models
• elaborate analytical mathematics

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The film theory (for interfacial mass transfer)

• Assuming a stagnant film exists near interface, a
  solute present at high dilution is slowly
  diffusing across this film.
• At steady-state

or
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?
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Example
Carbon dioxide is being scrubbed out of a gas
using water flowing through a packed bed of 1 cm
Berl saddles. The carbon dioxide is absorbed at a
rate of 2.3 x 10-6 mol/cm2 sec. The carbon
dioxide is present at a partial pressure of 10
atm, the Henrys law coefficient H is 600 atm,
and the diffusion coefficient of carbon dioxide
in water is 1.9 x 10-5 cm2/sec. Find the film
thickness.
9.3 x 10-4 mol/cm3
The interfacial concentration
1/18 mol/cm3
10 atm
600 atm
Mass transfer
k 2.5 x 10-3 cm/sec
typical order 10-2 cm
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Penetration theory (Higbie, 1935)

• The falling film is very thick. In the z
  direction, diffusion is much more important than
  convection, and in the x direction, diffusion is
  much less important then convection.
• Flux at the interface

The flux averaged over x is
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Sherwood number
Peclet number
Reynolds number
Schmidt number
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Surface-renewal theory (Dankwerts, 1951)

• It consists of two regions
• Interfacial region mass transfer occurs by
  penetration theory.
• Renewal region constantly exchanged with new
  elements from a second bulk region.

p1
Bulk liquid
The length of time that small fluid elements
spend in the interfacial region is the key.
c1i
Gas
c1
Residence time distribution
z 0
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Residence time distribution
The fraction of surface elements ? remaining at
time t
The residence time distribution of surface
element
In the interfacial region, the flux is that for
diffusion into a infinite slab
The average flux is
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Comparison of the three theories

• The film theory
• The penetration theory
• The surface-renewal theory

Film thickness
Contact time
Surface residence time
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Boundary layer theory

• A more complete description of mass transfer
• Based on parallel with earlier studies of fluid
  mechanics and heat transfer

• The sharp-edged plate made of a sparingly
  soluble solute is immersed in a rapid flowing
  solvent.
• A boundary layer is formed.
• The boundary layer is usually defined as the
  locus of distance over which 99 of the
  disruptive effect occurs.
• When the flow pattern develops, the solute
  dissolves off the plate.

turbulent region
laminar region
turbulent region
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Layers caused by flow and by mass transfer

• The distance that the solute penetrates produces
  a new concentration boundary layer ?c , but this
  layer is not the same as that observed for flow
  ?. The two layers influence each other.
• When the dissolving solute is only sparingly
  soluble, the boundary layer caused by the flow is
  unaffected.

Assuming flow varies as a power series in the
boundary layer thickness
?
find ?c ?
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Find the boundary layer for flow first
Assuming the fluid flowing parallel to the flat
plate follows
The boundary conditions are
The fluid sticks to the plate. The plate is solid
and the stress on it is constant.
Far from the plate, the plate has no effect.
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turbulent region
Mass balance on the control volume of the width
W, the thickness ?x and the height l
Dividing W?x ?x ? 0
x-momentum balance gives
Dividing W?x ?x ? 0
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We have ? now and use
to find vx.
How about ?c ?
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Find the boundary layer for concentration
Assuming the concentration profile parallel to
the flat plate follows
The boundary conditions are
The concentration and flux are constant at the
plate.
Far from the plate, the plate has no effect.
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turbulent region
Mass balance on the control volume of the width
W, the thickness ?x and the height l
Dividing W?x ?x ? 0
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Mass transfer coefficient?
Similar to film theory
Schmidt number
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Averaged over length L

• Valid for a flat plate when the boundary layer is
  laminar (I.e., Re lt 300,000)
• ,between the prediction of the
  film theory and the penetration/surface-renewal
  theories.

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Water flows at 10 cm/sec over a sharp-edged plate
of benzoic acid. The dissolution of benzoic acid
is diffusion-controlled, with a diffusion
coefficient of 1.0 x 10-5 cm2/sec. Find (a) the
distance at which the laminar boundary layer
ends, (b) the thickness of the flow and
concentration boundary layers at that point, and
(c) the local mass transfer coefficients at the
leading edge and at the position of transition,
as well as the average mass transfer coefficient
over this length.
(a) the length before the turbulent region begins
x 300 cm
x 300 cm
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Graetz-Nusselt problem

• Mass transfer across the walls of a pipe
  containing fluid in laminar flow.
• Find the dissolution rate as a function of
  quantities like Reynolds and Schmidt numbers

Flow conditions
Diffusion conditions
Sparing soluble solute
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Fixed solute concentration at the wall of a short
tube
Mass balance for the solute in a constant-density
fluid on a washer-shaped region
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Averaged over length L
incomplete gamma function
How to find the mass transfer coefficient?
Schmidt
1.62
Sherwood
Reynolds
diameter length
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Water is flowing at 6.1 cm/sec through a pipe of
2.3 cm in diameter. The walls of a 14-cm section
of this pipe are made of benzoic acid, whose
diffusion coefficient in water is 1.0 x 10-5
cm2/sec. Find the average mass transfer
coefficient over this section.
Check before ending the question!
Laminar flow?
Short pipe?
OK!
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Concentrated solutions

• In most application, correlations for dilute
  solutions can also be applied to concentrated
  solutions.
• In a few cases, k is a function of the driving
  force

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mass transfer coefficient for rapid mass transfer
dilute system
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A mass balance on a thin film shell ?z thick
shows that the total flux is a constant
or
The relation between the dilute mass transfer
coefficient k0 and the concentrated mass transfer
coefficient k.
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Benzene is evaporating from a flat porous plate
into pure flowing air. Using the film theory,
find N1/k0c1i and k/k0 as a function of the
concentration of benzene at the surface of the
plate.
The benzene evaporates off the plate into air
flowing parallel to the plate
In dilute solution
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Summary of forced convection

• Table 13.6-1
• all predictions cluster around experimentally
  observed values
• In most cases. Sh ?Re1/2 and Sc1/3
• Recommend film and penetration theories