mass transfer in laminar & turbulent flow · pdf file“convective” heat &...

Mass Transfer in Laminar & Turbulent Flow

Mass Transfer Coefficients

25 MassTransfer.key - January 31, 2014

“Convective” Heat & Mass TransferFourier’s law requires us to resolve ∇T. This, in turn, requires detailed knowledge of u since it will cause non-trival T profiles.ℓ

T∞

Tw

T∞ in “bulk” and Tw near “wall,” with a complicated T profile in between!

Concept: if we only care about an overall q, and not about T(z), then we can approximate:

Heat transfer coefficient, h, is empirical, and contains a lot of physics (must be “correlated” for different situations).

q = ��rT

Note: h=λ∕ℓ only if T is linear.

Fick’s law requires us to resolve ∇xA. This, in turn, requires detailed knowledge of u since it will cause non-trival x profiles.

Concept: if we only care about an overall JA, and not about xA(z), then we can approximate:

JA = �cADABrxA

qz = ��dT

dz⇡ ��

T1 � Tw

`= h (Tw � T1)

q ⇡ h (Tw � T1)

JAz ⇡ �cDABx1 � xw

`

= kc(cAw � cA1)

Mass transfer coefficient, kc, is empirical, and contains a lot of physics (must be “correlated” for different situations).

JA ⇡ kc(cAw � cA1)Note: kc=DAB∕ℓ only if cA is linear.

ℓ

x∞

xw


Mass vs. Heat TransferFick’s “Law” for a binary mixture of A and B:

Fourier’s “Law”:

• Only really valid for binary mixtures (2 species).

• There are several diffusive fluxes, depending on the frame of reference that you choose (molar, mass)!

• Need an appropriate choice for DAB. Dependent on temperature, pressure and (sometimes) composition.

• ∇xA is not the only possible driving force for JA. Could include other species (for more than 2 components) or even ∇T and ∇p! See SHR §3.8 for more information. Lots of cool physics here...

• need a model for k. Dependent on temperature, pressure & composition.

• Heat can be transferred by radiation! No analogous mechanism for mass (until you invent teleportation).

• Heat can diffuse by other mechanisms as well. ∇x can also cause heat diffusion!

JA = �cDABrxAmolar diffusive flux relative to a molar-averaged velocity q = �krT

JA = kc�cA“Convective” mass transfer: “Convective” heat transfer:

q = h�T


Analogies in Diffusive Fluxes

x

zux

• Only gradients are in z-direction • No velocity (bulk flow) in z-direction • Diffusion in z-direction • Constant properties (c, ρ, μ, k, DAB, etc.)

qz = �kdT

dz

��z=0

⇡ h(Ti � T )

⌧zx

= �µdu

x

dz

��z=0

⇡ f⇢u2

2

Reynolds’ analogy:

Chilton-Colburn analogy:

f

2= NStH = NStM

NStH =NNu

NReNPr=

h

⇢Cp

ux

=h

Cp

G

NStM =NSh

NReNSc=

kc

ux

=kc

⇢

G

(valid only when NPr = NSc = 1)

SHR §3.5

x-momentum diffusive flux

diffusive heat flux

diffusive mass flux

See equations (3-166)-(3-171) in SHR

Other analogies also exist. See SHR §3.5.3-3.5.4

JAz = �DABdcAdz

��z=0

⇡ kc(cA � cA)

G = ux

⇢“mass velocity”

f = Fanning friction factor

See SHR Table 3.13 for summary of some

dimensionless groups

jM ⌘ f

2| {z }momentum

= jH ⌘ NStHN2/3Pr| {z }

heat

= jD ⌘ NStMN2/3Sc| {z }

mass


Mass Transfer Coefficients

cA1 cA2

z

JA = �cDABdxA

dzFick’s law:

If kc is chosen “just right” then:

The “mass-transfer coefficient,” kc

• Units of length/time

• Analogous to the heat transfer coefficient: q = hΔT • Useful when we don’t know (or want to know) xA(z). • Need to choose kc “just right” to get the correct flux.

• kc=DAB/Δz is usually not even close (because xA is not linear in general and Δz is typically not known).

• Usually evaluated at one location (e.g. phase interface) since JA might vary over z.

Approaches to get kc: • If we can get an analytic solution

for xA(z) then we can get an exact form for kc using Fick’s law. (frequently not possible)

• Often, kc is correlated with non-dimensional groups like Reynolds number, Schmidt number and Peclet number.

• Analogies with heat transfer.

NSh =`kcDAB

ℓ is an appropriate length scale

dxA

dz⇡ xA2 � xA1

`

JA ⇡ �cDABxA2 � xA1

`

⇡ ckc(xA1 � xA2)

JA = kc�cA = �DABdcAdz


Accounting for Bulk FlowExamples:

• Evaporation • Condensation • Absorption • Stripping

NA =JA

1� xA

NB ≪ NA

Typically, correlations give you k, not k′.

NA = xAN + JA

= xA(NA +NB) + JA

For C > 2 components, this gets much more complicated.

Note: k′ is just a convenience definition.

JA = kc�cA = �cDAB@c

@z

k

0c =

kc

1� xA=

kc

xB

NA = k0c�cA


Two-Phase Mass Transfer

• Phase equilibrium at gas-liquid interface to relate pA to cA (discontinuity in mole fractions)

• diffusion through a liquid “film” of thickness δ • well-mixed “bulk” region (fluid motion)

• we want NA.

NA = JA = kc�cAgas

bulk liquid

pA

cAi

cAb

liquid film

z=0 z=δ(no bulk flow)

Models for kc...

SHR §3.6


Film TheorySHR §3.6.1

gasbulk liquid

pA

cAi

cAb

liquid film

z=0 z=δ

• c is constant • NA is constant with z. • NB=0

NA = xA (NA +NB)| {z }N

+JA

= � cDAB

1� xA

dxA

dz

NA = JA = �cDABdxA

dz

Separate this ODE & integrate it to find xA(z).

NA =cDAB

�

ln

1� xAb

1� xAi

�

=cDAB

�(1� xA)LM(xAi � xAb)

NA =DAB

�

(cAi � cAb)

=cDAB

�

(xAi � xAb)

Force fit into form to find kc.NA = kc(cAi � cAb)Film theory is often not

very accurate, but is widely used nonetheless.

kc =DAB

�

Accounting for bulk flowNeglecting bulk flow

Since δ is unknown, kc is typically replaced by an empirical correlation.

k

0c =

DAB

�(1� xA)LM=

kc

(1� xA)LM

(1� x

A

)LM =x

Ai � x

Ab

ln [(1�xAb)/(1�xAi )]

= (xB

)LM


Other modelsConcept:

• A pocket of fresh fluid arrives from the bulk to the film and stays there for some period of time, whereafter it is replaced by a new packet of fresh fluid.

• Essentially diffusion into an infinite slab (but only for a short time tc). We solved this problem already!

gasbulk liquid

pA

cAi

cAb

liquid film

z=0 z=δHigbie Model: fluid parcels are

replaced at uniform interval tc.

Surface Renewal Model: “old” fluid parcels are more likely to be replaced than “young” parcels

kc = 2

rDAB

⇡tc

kc =pDAB s

s - rate of surface renewal (1/sec) (fraction of surface area replaced by fresh fluid in unit time)

Choosing tc: • Bubbles: ratio of bubble diameter to its relative velocity in the fluid • Droplets: residence time (assuming no internal circulation) • Packed tower: typically around 1s. • Correlations...?

SHR §3.6.2-3.6.3

It is difficult to determine s...

NA = kc(cAi � cAb)


Two-Film TheoryOverall Mass Transfer Coefficients

SHR §3.7

idealized picture more realistic picture


Two-Film Theory Assumptions & Formulation

No reaction. Phase equilibrium at the interface & Henry’s law applies. • This has the effect of being a “contact resistance” to mass

transfer (analog to heat transfer). • Henry’s law implies a linear relationship between the gas &

liquid phases compositions. Typically only valid for small ranges of xA or yA.

Bulk flow is negligible (NA = JA). • “trace” species diffusing (NA ≈ 0, NB = 0)

• equimolar counterdiffusion (NA = -NB).

Constant total molar concentration, c.

NA = kp(pAb � pAi)

NA = kc(cAi � cAb)

cAi = HA pAi

pA = HAxA

pA = cA/HA

yA = HAxA

Several forms of Henry’s law exist:

distinguish by units of HA.

Gas phase:

Liquid phase:

Phase Equil.:

Combine to eliminate

interface compositions.

NA =pAbHA � cAb

(HA/kp) + (1/kc)

c⇤A = HA pAb

NA = KL(c⇤A � cAb)

1

KL⌘ HA

kp+

1

kc“fictitious” cA in equilibrium with pAb.

“overall” mass transfer coefficient based on liquid phase.


Variations on the Themec⇤A = HA pAbNA = KL(c

⇤A � cAb)

1

KL⌘ HA

kp+

1

kc

NA = KG(pAb � p⇤A)1

KG=

1

kp+

1

HAkC

N

A

= K

x

(x⇤A

� x

Ab)

= K

y

(yAb � y

⇤A

)

1

Kx

=1

KA

ky

+1

kx

1

Ky

=1

ky

+K

A

kx

KA is the K-value from equilibrium thermo:

x

⇤A =

yAb

KA

Liquid phase concentration

Gas phase concentration

p⇤A =cAb

HA

Liquid mole fraction

Gas mole fraction y

⇤A = xAbKA

KA =yAi

xAi

SI AE

k m/s ft/h

k kmol/(s-m lbmol/(h-ft

k kmol/(s-m lbmol/(h-ft


“Large” Driving ForcesFo

r “la

rge”

var

iatio

n in

xA

(or

y A)

acro

ss a

pha

se, t

he

chan

ge in

KA

is im

port

ant.

Henry’s law is not valid here!

SHR §3.7.3

1

Ky

=1

ky

+m

x

kx

1

Kx

=1

kx

+1

my

ky

N

A

= k

y

(yAb � y

Ai) = K

y

(yAb � y

⇤A

)

= k

x

(xAi � x

Ab) = K

x

(x⇤A

� x

Ab)

See derivation in SHR §3.7.3

mx evaluate on the liquid side.

my evaluate on the gas side.

m =dyAdxA


mass transfer in laminar & turbulent flow · pdf file“convective” heat &...

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