1. PM3125: Lectures 7 to 9 Content of Lectures 7 to 9:
Mass transfer: concept and theory
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2. Mass Transfer
Mass transfer occurs when a component in a mixture goes from one point to another.
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.
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3. Example of Mass Transfer
Mass transfer can occur by either diffusion or by convection.
Stirring the water with a spoon creates forced convection.
That helps the sugar molecules to transfer to the bulk water much faster.
Diffusion
(slower)
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4. Example of Mass Transfer
Mass transfer can occur by either diffusion or by convection.
Stirring the water with a spoon creates forced convection.
That helps the sugar molecules to transfer to the bulk water much faster.
Convection
(faster)
Diffusion
(slower)
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5. Example of Mass Transfer
At the surface of the lung:
Air
Blood
Oxygen
High oxygen concentration
Low carbon dioxide concentration
Low oxygen concentration
High carbon dioxide concentration
Carbon dioxide
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6. Diffusion Diffusion (also known as molecular diffusion)
is a net transport of molecules
from a region of higher concentration
to a region of lower concentration
by random molecular motion.
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7. 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. A B
Molecules of A and B are uniformly distributed everywhere in the vessel purely due to the DIFFUSION.
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8. Examples of Diffusion Scale of mixing: Solid-phase reaction:
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
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9. Fick’s Law of Diffusion
JA = -DAB
ΔCA Δx
JA = DAB
ΔCA Δx CA
A & B JA
CA + ΔCA Δx
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10. ΔCA JA = -DAB Δx Fick’s Law of Diffusion
concentration gradient
(mass per volume per distance)
diffusion coefficient
(or diffusivity) of A in B
diffusion flux of A in relation to the bulk motion in x-direction
(mass per area per time)
What is the unit of diffusivity?
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11. . Q ΔT = -k A Δx Fourier’s Law of Heat Conduction
Temperature gradient
(temperature per distance) A Δx
Thermal conductivity
Describe the similarities between Fick’s Law and Fourier’s Law
Heat flux
(Energy per area per time)
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12. Diffusivity
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 ≈ to m2/s
For gases in air at 1 atm and at room temperature:
D ≈ 10-6 to x10-5 m2/s
Diffusivity depends on the type of solute, type of solvent, temperature, pressure, solution phase (gas, liquid or solid) and other characteristics.
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13. DAB is proportional to 1/P and T1.75
Prediction of Binary Gas Diffusivity
DAB - diffusivity in cm2/s
P - absolute pressure in atm
Mi - molecular weight
T - temperature in K
Vi - sum of the diffusion volume for component i
DAB is proportional to 1/P and T1.75
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14. Prediction of Binary Gas Diffusivity
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15. DAB is proportional to 1/μ and T
Prediction of Diffusivity in Liquids
For very large spherical molecules (A) of 1000 molecular weight or greater diffusing in a liquid solvent (B) of small molecules:
9.96 x T
DAB =
applicable for biological solutes such as proteins
μ VA1/3
DAB - diffusivity in cm2/s
T - temperature in K
μ - viscosity of solution in kg/m s
VA - solute molar volume at its normal boiling point
in m3/kmol
DAB is proportional to 1/μ and T
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16. DAB is proportional to 1/μB and T
Prediction of Diffusivity in Liquids
For smaller molecules (A) diffusing in a dilute liquid solution of solvent (B):
1.173 x (Φ MB)1/2 T
DAB =
μB VA0.6
applicable for biological solutes
DAB - diffusivity in cm2/s
MB - molecular weight of solvent B
T - temperature in K
μ - viscosity of solvent B in kg/m s
VA - solute molar volume at its normal boiling point in m3/kmol
Φ - association parameter of the solvent, which 2.6 for water,
1.9 for methanol, 1.5 for ethanol, and so on
DAB is proportional to 1/μB and T
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17. DAB is proportional to T
Prediction of Diffusivity of Electrolytes in Liquids
For smaller molecules (A) diffusing in a dilute liquid solution of solvent (B):
8.928 x T (1/n+ + 1/n-)
DoAB =
(1/λ+ + 1/ λ-)
DoAB is diffusivity in cm2/s
n+ is the valence of cation
n- is the valence of anion
λ+ and λ- are the limiting ionic conductances in very dilute solutions
T is when using the above at 25oC
DAB is proportional to T
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18. Fick’s First Law of Diffusion (again)
∆CA
JA is the diffusion flux of A in relation to the bulk motion in x-direction
JA = - DAB ∆x
If circulating currents or eddies are present (which will always be present), then
∆CA
NA = - (D + ED) ∆x
where ED is the eddy diffusivity, and is dependent on the flow pattern
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19. Microscopic (or Fick’s Law) approach:
∆CA
JA = - D ∆x
Macroscopic (or mass transfer coefficient) approach:
NA = - k
ΔCA
where k is known as the mass transfer coefficient
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20. NA = - k ΔCA Macroscopic (or mass transfer coefficient) approach:
is used when the mass transfer is caused by molecular diffusion plus other mechanisms such as convection.
NA = - k
ΔCA
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21. NA = - k ΔCA Macroscopic (or mass transfer coefficient) approach:
concentration difference
(mass per volume)
mass transfer coefficient
net mass flux of A
(mass per area per time)
What is the unit of k?
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22. . Newton’s Law of Cooling in Convective Heat Transfer
Flowing fluid at Tfluid
Heated surface at Tsurface Q .
conv.
= h (Tsurface – Tfluid) A
temperature difference
convective heat flux
(energy per area per time)
Heat transfer coefficient
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23. Describe the similarities between the convective heat transfer equation and the macroscopic approach to mass transfer.
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24. NA -k ΔCA = k (CA1 – C A2 ) = CA1 NA CA2
Macroscopic (or mass transfer coefficient) approach: NA -k
ΔCA = k
(CA1 – C A2 ) =
CA1
A & B NA
CA2
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25. NA -k ΔCA = k (CA1 – C A2 ) = CA1 = PA1 / RT CA1 CA2 = PA2 / RT NA CA2
Macroscopic (or mass transfer coefficient) approach: NA -k
ΔCA = k
(CA1 – C A2 ) =
CA1 = PA1 / RT
CA1
CA2 = PA2 / RT
A & B NA
CA2
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26. NA = k (PA1 – P A2 ) / R T PA1 NA PA2
Macroscopic (or mass transfer coefficient) approach: NA k
(PA1 – P A2 ) / R T =
PA1
A & B NA
PA2
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27. 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)
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28. Oxygen transfer from gas bubble to cell
Transfer from the interior of the bubble to the gas-liquid interface
Movement across the gas film at the gas-liquid interface
Diffusion through the relatively stagnant liquid film surrounding the bubble
Transport through the bulk liquid
Diffusion through the relatively stagnant liquid film surrounding the cells
Movement across the liquid-cell interface
If the cells are in floc, clump or solid particle, diffusion through the solid of the individual cell
Transport through the cytoplasm to the site of reaction.
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29. Transfer through the bulk phase in the bubble is relatively fast
The gas-liquid interface itself contributes negligible resistance
The liquid film around the bubble is a major resistance to oxygen transfer
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.
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.
Resistance at the cell-liquid interface is generally neglected
The mass transfer resistance is small, except the cells form large clumps or flocs.
Intracellular oxygen transfer resistance is negligible because of the small distance involved
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30. Interfacial Mass Transfer
Pa = partial pressure of solute in air
Ca = concentration of solute in air
air
volatilization
Pa = Ca RT
air-water
interface
water
absorption
Cw = concentration of solute in water
Transport of a volatile chemical across the air/water interface.
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31. Interfacial Mass Transfer
Pa = partial pressure of solute in air
air δa
Pa,i
air-water
interface
Cw,i δw
water
Pa,i vs Cw,i?
Cw = concentration of solute in water
δa and δw are boundary layer zones offering much resistance to mass transfer.
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32. Interfacial Mass TransferPa
air δa
Pa,i
air-water
interface
Cw,i δw
water
Henry’s Law:
Pa,i = H Cw,i at equilibrium,
where H is Henry’s constant Cw
δa and δw are boundary layer zones offering much resistance to mass transfer.
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33. Henry’s Law Pa,i = H Cw,i at equilibrium, where H is Henry’s constant
Unit 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
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34. Gas-Liquid Equilibrium Partitioning Curve
Pa = H’’ C*w Pa
Pa,i = H Cw,i
Pa,i
H = H’ = H’’
if the partitioning curve is linear
P*a = H’ Cw
P*a Cw
Cw,i
C*w Cw
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35. Interfacial Mass Transfer
NA = KG (Pa – Pa,i) Pa
C*w
air δa
Pa,i
air-water
interface
Cw,i δw
water
P*a Cw
NA = KL (Cw,i – Cw)
KG = gas phase mass transfer coefficient
KL = liquid phase mass transfer coefficient
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36. Interfacial Mass Transfer
NA = KG (Pa – Pa,i) Pa
C*w
NA = KOG (Pa – P*a)
air δa
Pa,i
air-water
interface
Cw,i δw
water
NA = KOL (C*w – Cw)
P*a Cw
NA = KL (Cw,i – Cw)
KOG = overall gas phase mass transfer coefficient
KOL = overall liquid phase mass transfer coefficient
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37. Interfacial Mass Transfer
NA = KG (Pa – Pa,i)
= KOG (Pa – P*a) Pa
C*w
KG = gas phase mass transfer coefficient
KOG = overall gas phase mass transfer coefficient
Pa,i
Cw,i
NA = KL (Cw,i – Cw)
= KOL (C*w – Cw)
P*a Cw
KL = liquid phase mass transfer coefficient
KOL = overall liquid phase mass transfer coefficient
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38. Relating KOL to KL C*w – Cw = C*w – Cw,i + Cw,i – Cw
NA / KOL = C*w – Cw,i + NA /KL
(1)
If the equilibrium partitioning curve is linear over the concentration range C*w to Cw,i, then
Pa - Pa,i = H (C*w - Cw,i)
NA / KG = H (C*w – Cw,i)
(2)
Combining (1) and (2), we get 1
KOL
H KG = + KL
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39. Relating KOG to KG Pa - P*a = Pa – Pa,i + Pa,i – P*a
NA / KOG = NA /KG + Pa,i – P*a
(3)
If the equilibrium partitioning curve is linear over the concentration range Pa,i to P*a then
Pa,i – P*a = H (Cw,i – Cw)
Pa,i – P*a = H NA / KL
(4)
Combining (3) and (4), we get 1
KOG KG = + H KL
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40. Summary: Interfacial Mass Transfer
NA = KG (Pa – Pa,i)
= KOG (Pa – P*a)
NA = KL (Cw,i – Cw)
= KOL (C*w – Cw) 1
KOL
H KG = + KL 1
KOG = H
KOL 1
KOG KG = + H KL
H = P*a / Cw = Pa,i / Cw,i = Pa / C*w
Two-film Theory
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41. Gas & Liquid-side Resistances in Interfacial Mass Transfer1
KOG KG = + H KL
fG = fraction of gas-side resistance =
1/KOG
1/KG
1/KG =
+ H/KL KL =
+ H KG 1
KOL
H KG = + KL
fL = fraction of liquid-side resistance =
1/KOL
1/KL
1/HKG
1/KL =
+ 1/KL
+ KL/H KG =
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42. Gas & Liquid-side Resistances in Interfacial Mass Transfer
1/KOG
1/KG
1/KG =
+ H/KL KL =
+ H KG
fG = =
1/KOL
1/KL
1/HKG
1/KL =
+ 1/KL
+ KL/H KG = fL
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.
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43. For a very soluble gas C*w Cw,i KOG ≈ KG fG > fL Pa δa Pa,i δw
Gas-liquid
interface
Cw,i δw
liquid
Cw ≈ Cw,i
P*a ≈ Pa,i
P*a Cw
NA = KG (Pa – Pa,i)
= KOG (Pa – P*a)
KOG ≈ KG
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44. For an almost insoluble gas
fL > fG Pa
C*w
Pa ≈ Pa,i
C*w ≈ Cw,i
gas δa
Pa,i
Gas-liquid
interface
Cw,i δw
liquid
P*a Cw
NA = KL (Cw,i – Cw)
= KOL (C*w – Cw)
KOL ≈ KL
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45. 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
Encyclopedia of Pharmaceutical Technology (Hardcover)
by James Swarbrick (Author)
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