pm3125: lectures 7 to 9

R. Shanthini 24 May 2010

Content of Lectures 7 to 9:

Mass transfer: concept and theory

PM3125: Lectures 7 to 9


Mass transfer occurs when a component in a mixture goes from one point to another.

Mass Transfer

Mass transfer can occur by either diffusion or convection.

Diffusion is the mass transfer in a stationary solid or fluid under a concentration gradient.

Convection is the mass transfer between a boundary surface and a moving fluid or between

relatively immiscible moving fluids.


Stirring the water with a spoon creates forced convection.

That helps the sugar molecules to transfer to the bulk water much faster.

Mass transfer can occur by either diffusion or by convection.

Diffusion(slower)

Example of Mass Transfer


Stirring the water with a spoon creates forced convection.

That helps the sugar molecules to transfer to the bulk water much faster.

Mass transfer can occur by either diffusion or by convection.

Diffusion(slower)

Convection(faster)

Example of Mass Transfer


Example of Mass TransferAt the surface of the lung:

Air BloodOxygen

Carbon dioxide

High oxygen concentrationLow carbon dioxide concentration

Low oxygen concentrationHigh carbon dioxide concentration


Diffusion (also known as molecular diffusion)is a net transport of molecules

from a region of higher concentrationto a region of lower concentration

by random molecular motion.

Diffusion


A B

A B

Liquids A and B are separated from each other.

Separation removed.

A goes from high concentration of A to low concentration of A. B goes from high concentration of B to low concentration of B.

Molecules of A and B are uniformly distributed everywhere in the vessel purely due to the DIFFUSION.

Diffusion


• Scale of mixing: Mixing on a molecular scale relies on diffusion as the final step in mixing process because of the smallest eddy size

• Solid-phase reaction: The only mechanism for intra particle mass transfer is molecular diffusion

• Mass transfer across a phase boundary:Oxygen transfer from gas bubble to fermentation broth; Penicillin recovery from aqueous to organic liquid

Examples of Diffusion


Fick’s Law of Diffusion

JA

CA

CA + ΔCA

Δx

JA = DABΔCA

Δx

A & B

JA = -DABΔCA

Δx


JA = -DABΔCA

Δx

Fick’s Law of Diffusion

diffusion coefficient (or diffusivity) of A in B

What is the unit of diffusivity?

concentration gradient (mass per volume per

distance)

diffusion flux of A in relation to the bulk motion in x-direction

(mass per area per time)


= -k ΔTΔx

Fourier’s Law of Heat Conduction

Thermal conductivity

Temperature gradient (temperature per

distance)

Heat flux (Energy per area per time)

Q.

A

Describe the similarities between

Fick’s Law and Fourier’s Law


For ions (dissolved matter) in dilute aqueous solution at room temperature:

D ≈ 0.6 to 2 x10-9 m2/s

For biological molecules in dilute aqueous solution at room temperature:

D ≈ 10-11 to 10-10 m2/s

For gases in air at 1 atm and at room temperature: D ≈ 10-6 to x10-5 m2/s

Diffusivity

Diffusivity depends on the type of solute, type of solvent, temperature, pressure, solution phase (gas, liquid or solid) and other characteristics.


Prediction of Binary Gas Diffusivity

DAB - diffusivity in cm2/sP - absolute pressure in atmMi - molecular weightT - temperature in KVi - sum of the diffusion volume for component i

DAB is proportional to 1/P and T1.75


Prediction of Binary Gas Diffusivity


Prediction of Diffusivity in Liquids

DAB - diffusivity in cm2/sT - temperature in Kμ - viscosity of solution in kg/m sVA - solute molar volume at its normal boiling point

in m3/kmol

DAB is proportional to 1/μ and T

DAB =9.96 x 10-12 T

μ VA1/3

For very large spherical molecules (A) of 1000 molecular weight or greater diffusing in a liquid solvent (B) of small molecules:

applicable for biological

solutes such as proteins


Prediction of Diffusivity in Liquids

DAB - diffusivity in cm2/sMB - molecular weight of solvent BT - temperature in Kμ - viscosity of solvent B in kg/m sVA - solute molar volume at its normal boiling point in m3/kmolΦ - association parameter of the solvent, which 2.6 for water,

1.9 for methanol, 1.5 for ethanol, and so onDAB is proportional to 1/μB and T

DAB =1.173 x 10-12 (Φ MB)1/2 T

μB VA0.6

For smaller molecules (A) diffusing in a dilute liquid solution of solvent (B):

applicable for biological solutes


Prediction of Diffusivity of Electrolytes in Liquids

DoAB is diffusivity in cm2/s

n+ is the valence of cationn- is the valence of anionλ+ and λ- are the limiting ionic conductances in very dilute solutions T is 298.2 when using the above at 25oC

DAB is proportional to T

DoAB =

8.928 x 10-10 T (1/n+ + 1/n-)

For smaller molecules (A) diffusing in a dilute liquid solution of solvent (B):

(1/λ+ + 1/ λ-)


JA = - DAB∆CA

∆x

Fick’s First Law of Diffusion (again)

JA is the diffusion flux of A in relation to the bulk motion in x-direction

where ED is the eddy diffusivity, and is dependent on the flow pattern

If circulating currents or eddies are present (which will always be present), then

NA = - (D + ED)∆CA

∆x


Microscopic (or Fick’s Law) approach:

JA = - D∆CA

∆x

Macroscopic (or mass transfer coefficient) approach:

NA = - k ΔCA

where k is known as the mass transfer coefficient


NA = - k ΔCAis used when the mass transfer is caused by molecular diffusion plus other mechanisms such as convection.



NA = - k ΔCA

mass transfer coefficient

concentration difference

(mass per volume)

net mass flux of A (mass per area per time)

What is the unit of k?



Newton’s Law of Cooling in Convective Heat Transfer

Q.

conv. = h (Tsurface – Tfluid)

Heat transfer coefficient

Heated surface at Tsurface

Flowing fluid at Tfluid

A

temperature difference

convective heat flux (energy per area per time)


Describe the similarities between the convective heat transfer equation and the macroscopic approach to mass transfer.


NA -k ΔCA=

NA

CA1

CA2

A & B

k (CA1 – C A2 )=



NA -k ΔCA=

NA

CA1

CA2

A & B

k (CA1 – C A2 )=

CA1 = PA1 / RTCA2 = PA2 / RT



NA k (PA1 – P A2 ) / R T=

NA

PA1

PA2

A & B



Other Driving Forces

Mass transfer is driven by concentration gradient as well as by pressure gradient as we have just seen.

In pharmaceutical sciences, we also must consider mass transfer driven by electric potential gradient (as in the transport of ions) and temperature gradient.

Transport Processes in Pharmaceutical Systems (Drugs and the Pharmaceutical Sciences, vol. 102),

edited by G.L. Amidon, P.I. Lee, and E.M. Topp(Nov 1999)


Oxygen transfer from gas bubble to cell1. Transfer from the interior of the bubble to the gas-liquid interface2. Movement across the gas film at the gas-liquid interface3. Diffusion through the relatively stagnant liquid film surrounding the

bubble4. Transport through the bulk liquid5. Diffusion through the relatively stagnant liquid film surrounding the

cells6. Movement across the liquid-cell interface7. If the cells are in floc, clump or solid particle, diffusion through the

solid of the individual cell8. Transport through the cytoplasm to the site of reaction.


1. Transfer through the bulk phase in the bubble is relatively fast2. The gas-liquid interface itself contributes negligible resistance3. The liquid film around the bubble is a major resistance to oxygen transfer4. In a well mixed fermenter, concentration gradients in the bulk liquid are

minimized and mass transfer resistance in this region is small, except for viscous liquid.

5. The size of single cell <<< gas bubble, thus the liquid film around cell is thinner than that around the bubble. The mass transfer resistance is negligible, except the cells form large clumps.

6. Resistance at the cell-liquid interface is generally neglected7. The mass transfer resistance is small, except the cells form large clumps or flocs.8. Intracellular oxygen transfer resistance is negligible because of the small distance

involved


Interfacial Mass Transfer

air-waterinterface

air

water

Cw = concentration of solute in water

Ca = concentration of solute in airPa = partial pressure of solute in air

volatilization

absorption

Transport of a volatile chemical across the air/water interface.

Pa = Ca RT



air-waterinterface

air

water

δa and δw are boundary layer zones offering much resistance to mass transfer.

δa

δw

Cw,i

Pa,i

Pa = partial pressure of solute in air

Cw = concentration of solute in water

Pa,i vs Cw,i?



air-waterinterface

air

water

δa

δw

Cw,i

Pa,i

Pa

Cw

Henry’s Law:Pa,i = H Cw,i at equilibrium, where H is Henry’s constant

δa and δw are boundary layer zones offering much resistance to mass transfer.


Henry’s LawPa,i = H Cw,i at equilibrium, where H is Henry’s constantUnit of H = [Pressure]/[concentration]

= bar / (kg.m3)

Pa,i = Ca,i RT is the ideal gas equation

Therefore, Ca,i = (H/RT) Cw,i at equilibrium,

where (H/RT) is known as the dimensionless Henry’s constant

H depends on the solute, solvent and temperature


Gas-Liquid Equilibrium Partitioning Curve

Cw,i

Pa,i

Pa

Cw

P*a

C*w Cw

Pa

Pa,i = H Cw,i

P*a = H’ Cw

Pa = H’’ C*w

H = H’ = H’’if the partitioning

curve is linear



air-waterinterface

air

water

δa

δw

Cw,i

Pa,i

Pa

CwP*a

C*w

NA = KG (Pa – Pa,i)

KG = gas phase mass transfer coefficientNA = KL (Cw,i – Cw)

KL = liquid phase mass transfer coefficient



air-waterinterface

air

water

δa

δw

Cw,i

Pa,i

Cw

C*w

KOG = overall gas phase mass transfer coefficientKOL = overall liquid phase mass transfer coefficient


NA = KL (Cw,i – Cw)

NA = KOG (Pa – P*a)

NA = KOL (C*w – Cw)P*

a

Pa



Cw,i

Pa,i

KOG = overall gas phase mass transfer coefficient

KOL = overall liquid phase mass transfer coefficient



= KOG (Pa – P*a)

= KOL (C*w – Cw)

KG = gas phase mass transfer coefficient

KL = liquid phase mass transfer coefficient

C*w

Pa

CwP*a


Relating KOL to KL

Pa - Pa,i = H (C*w - Cw,i)

C*w – Cw = C*w – Cw,i + Cw,i – Cw

NA / KOL = C*w – Cw,i + NA /KL

If the equilibrium partitioning curve is linear over the concentration range C*w to Cw,i, then

NA / KG = H (C*w – Cw,i)

(1)

(2)

Combining (1) and (2), we get

1KOL

1H KG

= +1

KL


Relating KOG to KG

Pa,i – P*a = H (Cw,i – Cw)

Pa - P*a = Pa – Pa,i + Pa,i – P*a

NA / KOG = NA /KG + Pa,i – P*a

If the equilibrium partitioning curve is linear over the concentration range Pa,i to P*a then

(3)

(4)

Combining (3) and (4), we get

1KOG

1KG

= +HKL

Pa,i – P*a = H NA / KL


Summary: Interfacial Mass Transfer



= KOG (Pa – P*a)

= KOL (C*w – Cw)

1KOL

1H KG

= +1

KL

1KOG

1KG

= +HKL

H = P*a / Cw = Pa,i / Cw,i = Pa / C*w

Two-film Theory

1KOG

=HKOL


Gas & Liquid-side Resistances in Interfacial Mass Transfer

1KOL

1H KG

= +1

KL

1KOG

1KG

= +HKL

fG = fraction of gas-side resistance

=1/KOG

1/KG

1/KG

1/KG=+ H/KL KL

KL=+ H KG

fL = fraction of liquid-side resistance

=1/KOL

1/KL

1/HKG

1/KL=+ 1/KL + KL/H

KG=KG


Gas & Liquid-side Resistances in Interfacial Mass Transfer

fG = 1/KOG

1/KG

1/KG

1/KG=+ H/KL KL

KL=+ H KG

=1/KOL

1/KL

1/HKG

1/KL=+ 1/KL + KL/H

KG=KG

If fG > fL, use the overall gas-side mass transfer coefficient and the overall gas-side driving force.

If fL > fG use the overall liquid-side mass transfer coefficient and the overall liquid-side driving force.

fL


For a very soluble gas

Gas-liquidinterface

gas

liquid

δa

δw

Cw,i

Pa,i

C*w

Cw ≈ Cw,i

Pa

NA = KG (Pa – Pa,i) = KOG (Pa – P*a)

P*a ≈ Pa,i

KOG ≈ KG

fG > fL

CwP*a


Gas-liquidinterface

gas

liquid

δa

δw

Cw,i

Pa,i

Cw

C*wPa ≈ Pa,i

P*a

PaC*w ≈ Cw,i

KOL ≈ KL

NA = KL (Cw,i – Cw) = KOL (C*w – Cw)

For an almost insoluble gas fL > fG


Encyclopedia of Pharmaceutical Technology (Hardcover)by James Swarbrick (Author)

Transport Processes in Pharmaceutical Systems (Drugs and the Pharmaceutical Sciences, vol. 102), edited by G.L. Amidon, P.I. Lee, and E.M. Topp, Nov 1999

pm3125: lectures 7 to 9

Documents

theory mass transfer

mass transfermass transfer

concentration gradient

oxygen transfer

intra particle mass

diffusion volume

diffusion scale of mixing

room temperature