PM3125: Lectures 7 to 9

1. PM3125: Lectures 7 to 9 Content of Lectures 7 to 9: Mass transfer: concept and theory

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. Diffusionis the mass transfer in a stationary solid or fluid under a concentration gradient. Convectionis the mass transfer between a boundary surface and a moving fluid or between relatively immiscible moving fluids.

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)

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)

5. Example of Mass Transfer At the surface of the lung: Blood Air Oxygen High oxygen concentration Low carbon dioxide concentration Low oxygen concentration High carbon dioxide concentration Carbon dioxide

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.

7. A B A B Diffusion 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.

8. Examples of 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

9. ΔCA ΔCA JA = DAB JA = -DAB Δx Δx Fick’s Law of Diffusion CA A & B JA CA + ΔCA Δx

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?

11. Fourier’s Law of Heat Conduction . Q ΔT = -k 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)

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

13. 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

14. Prediction of Binary Gas Diffusivity

15. 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 10-12 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

16. Prediction of Diffusivity in Liquids For smaller molecules (A) diffusing in a dilute liquid solution of solvent (B): 1.173 x 10-12 (Φ MB)1/2T 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

17. Prediction of Diffusivity of Electrolytes in Liquids For smaller molecules (A) diffusing in a dilute liquid solution of solvent (B): 8.928 x 10-10 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 298.2 when using the above at 25oC DAB is proportional to T

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

19. Microscopic (or Fick’s Law) approach: ∆CA JA = - D ∆x Macroscopic (or mass transfer coefficient) approach: ΔCA NA = - k where k is known as the mass transfer coefficient

20. 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

21. Macroscopic (or mass transfer coefficient) approach: NA = - k ΔCA concentration difference (mass per volume) mass transfer coefficient net mass flux of A (mass per area per time) What is the unit of k?

22. conv. . Q Newton’s Law of Cooling in Convective Heat Transfer Flowing fluid at Tfluid Heated surface at Tsurface = h (Tsurface – Tfluid) A temperature difference convective heat flux (energy per area per time) Heat transfer coefficient

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

24. -k k = = ΔCA (CA1 – C A2 ) Macroscopic (or mass transfer coefficient) approach: NA CA1 A & B NA CA2

25. -k k = = ΔCA (CA1 – C A2 ) Macroscopic (or mass transfer coefficient) approach: NA CA1 = PA1 / RT CA1 A & B CA2 = PA2 / RT NA CA2

26. k = (PA1 – P A2 ) / R T Macroscopic (or mass transfer coefficient) approach: NA PA1 A & B NA PA2

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)

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.

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

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.

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 δware boundary layer zones offering much resistance to mass transfer.

32. Interfacial Mass Transfer Pa 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 δware boundary layer zones offering much resistance to mass transfer.

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

34. Gas-Liquid Equilibrium Partitioning Curve Pa 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

35. Interfacial Mass Transfer NA = KG (Pa – Pa,i) Pa C*w air δa Pa,i air-water interface Cw,i δw water Cw P*a NA = KL (Cw,i – Cw) KG = gas phase mass transfer coefficient KL = liquid phase mass transfer coefficient

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) Cw P*a NA = KL (Cw,i – Cw) KOG = overall gas phase mass transfer coefficient KOL = overall liquid phase mass transfer coefficient

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) Cw P*a KL = liquid phase mass transfer coefficient KOL = overall liquid phase mass transfer coefficient

38. Relating KOL to KL 1 1 1 = + KOL H KG 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) (2) NA / KG = H (C*w – Cw,i) Combining (1) and (2), we get

39. Relating KOG to KG 1 1 H = + KOG KG KL 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

40. Summary: Interfacial Mass Transfer 1 H = KOG KOL 1 1 1 1 H 1 = = + + KOG KOL H KG KG KL KL NA = KG (Pa – Pa,i) = KOG (Pa – P*a) NA = KL (Cw,i – Cw) = KOL (C*w – Cw) H = P*a / Cw = Pa,i / Cw,i = Pa / C*w Two-film Theory

41. 1/KG 1/KG KL = = = 1/KOG 1/KG + H/KL KL + H KG 1/KL 1/KL KG 1 1 = = 1 1 H 1 = 1/KOL 1/HKG + 1/KL KG + KL/H = = + + KOG KOL H KG KG KL KL Gas & Liquid-side Resistances in Interfacial Mass Transfer fG = fraction of gas-side resistance fL = fraction of liquid-side resistance

42. 1/KG 1/KG KL = = 1/KOG 1/KG + H/KL KL + H KG 1/KL 1/KL KG = = = 1/KOL 1/HKG + 1/KL KG + KL/H Gas & Liquid-side Resistances in Interfacial Mass Transfer fG = 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.

43. For a very soluble gas fG > fL Pa C*w gas δa Pa,i Gas-liquid interface Cw,i δw liquid Cw≈ Cw,i P*a ≈ Pa,i Cw P*a NA = KG (Pa – Pa,i) = KOG (Pa – P*a) KOG≈ KG

44. For an almost insoluble gas fL > fG Pa C*w C*w ≈ Cw,i Pa ≈ Pa,i gas δa Pa,i Gas-liquid interface Cw,i δw liquid Cw P*a NA = KL (Cw,i – Cw) = KOL (C*w – Cw) KOL≈ KL

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)