Lecture 8 forced convection
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高等輸送二 — 質傳 Lecture 8 Forced convection. 郭修伯 助理教授. 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

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高等輸送二 — 質傳 Lecture 8 Forced convection

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Lecture 8 forced convection

高等輸送二 — 質傳Lecture 8Forced convection

郭修伯 助理教授


Forced convection

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


The film theory for interfacial mass transfer

Liquid film

p1

c1i

Gas

Bulk liquid

c1

z = 0

z = l

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


Lecture 8 forced convection

?


Lecture 8 forced convection

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 Henry’s 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


Penetration theory higbie 1935

p1

c1i

Gas

Bulk liquid

c1

z = 

z = 0

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:


Lecture 8 forced convection

Sherwood number

Peclet number

Reynolds number

Schmidt number


Lecture 8 forced convection

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


Lecture 8 forced convection

The probability that a given surface element will be at the surface for time t

E(t)dt =

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:


Comparison of the three theories

Comparison of the three theories

  • The film theory

  • The penetration theory

  • The surface-renewal theory

Film thickness

Contact time

Surface residence time


Boundary layer theory

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


Layers caused by flow and by mass transfer

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 ?


Lecture 8 forced convection

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.


Lecture 8 forced convection

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


Lecture 8 forced convection

We have  now and use to find vx.

How about c ?


Lecture 8 forced convection

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.


Lecture 8 forced convection

turbulent region

Mass balance on the control volume of the width W, the thickness x and the height l:

Dividing Wx

x  0


Lecture 8 forced convection

Mass transfer coefficient?

Similar to film theory

Schmidt number:


Lecture 8 forced convection

Averaged over length L

  • Valid for a flat plate when the boundary layer is laminar (I.e., Re < 300,000)

  • ,between the prediction of the film theory and the penetration/surface-renewal theories.


Lecture 8 forced convection

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


Graetz nusselt problem

r

z

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


Lecture 8 forced convection

Solute accumulation by convection

Solute accumulation by diffusion

Solute accumulation

=

r

z

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:


Lecture 8 forced convection

Averaged over length L:

incomplete gamma function

How to find the mass transfer coefficient?

Schmidt

1.62

Sherwood

Reynolds

diameter length


Lecture 8 forced convection

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!


Concentrated solutions

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:

?

mass transfer coefficient for rapid mass transfer

dilute system


Lecture 8 forced convection

Bulk liquid

c1i

p1

Gas

c1

z = l

z = 0

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.


Lecture 8 forced convection

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:


Summary of forced convection

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


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