Lecture 20: Reaction and Diffusion in Porous Catalyst

10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. K. Dane Wittrup Lecture 20: Reaction and Diffusion in Porous Catalyst This lecture covers: Effective diffusivity, internal and overall effectiveness factor, Thiele modulus, and apparent reaction rates Reaction & Diffusion -Diffusion in a porous solid phase Ex. Precious metals on ceramic supports or drug/nutrient delivery through tissues -Derive steady state material balance accounting for diffusion and reaction in a spherical geometry -Thiele modulus (φ) r R [S]0= surface concentration of a growth substrate (ex. glucose and O2) Figure 1. Sphere of Cells Assume pseudo-homogeneous medium and Fick’s Law describes diffusion Flux = − D d [S ] dr , where Flux [=] # molecules length 2 and D [=] time area ⋅ time − rs = knCsn V C − rs = max s K s + Cs Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. -rs n=1 n=0 Cs Figure 2. Rate of reaction versus species concentration Cs  K s ⇒ 1st order rs ≈ Vmax Cs Ks Cs  K s ⇒ 0th order rs ≈ Vmax Steady-state Shell Balance In some cases, S still hasn’t penetrated to the center t=0 Æ tÆ ∞ Figure 3. Time progression as species, S, enters the sphere Thin shell r to (r+Δr) Sin by diffusion –Sout by diffusion –Scons by reaction=0 Flux ⋅ 4π r 2 r + Δr − Flux ⋅ 4π r 2 − kn Csn 4π r 2 Δr = 0 r Divide through by 4πΔr and take the limit as ΔrÆ0 ( d Flux ⋅ r 2 dr ) −k C r n n 2 s =0 dC d ⎛ ⎞ − D s ⋅ r 2 ⎟ − knCsn r 2 = 0 ⎜ dr ⎝ dr ⎠ ⎫ d 2Cs 2 dCs kn n + − Cs = 0 ⎬ 2nd order ODE 2 dr r dr D ⎭ 2 Boundary conditions: 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. K. Dane Wittrup Lecture 20 Page 2 of 5 Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Cs r=R = Cs ,0 Cs is finite everywhere, or dCs dr =0 r =0 Nondimensionalize C r S= s R Cs ,0 ρ= d 2 S 2 dS + − φ 2S n = 0 dρ2 ρ dρ Boundary conditions: S=1 @ ρ=1 S is finite everywhere φ = 2 kn R 2Csn,0−1 D = (R 2 D ) (1 k C ) n n −1 s ,0 = characteristic diffusion time characteristic reaction time If diffusion is slow Æ diffusion dominates If reaction is slow Æ reaction dominates If φ2<<1 Æ reaction limited regime If φ2>>1 Æ diffusion limited regime φ2<<1 substrate throughout φ2>>1 substrate can’t make it to the center Figure 4. Reaction and diffusion limited regimes S= 1 sinh(φρ ) ρ sinh(φ ) sinh( z ) = e z − e− z 2 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. K. Dane Wittrup Lecture 20 Page 3 of 5 Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. 1 φ = 0.1 S φ =1 φ↑ φ = 10 ρ 1 Figure 5. S versus ρ for various values of φ Define “effectiveness factor” η overall rate of reaction η= rate if Cs =CS,0 everywhere overall reaction rate in sphere at steady state = [inward flux @ r=R (ρ=1)]*Area =D dCs dr 4π R 2 r=R = 4π RDCs ,0 η= 3 φ2 dS dρ = 4π RDCs ,0 (φ coth φ − 1) ρ =1 (φ coth φ − 1) 1 η 0.1 η∝ 1 φ 0.01 .01 0.1 1 10 100 φ Figure 6. Log-log plot of effectiveness factor versus thiele modulus Higher values of Thiele modulus Æ effectiveness goes down *For a variety of reaction kinetics, geometries and rate laws, plots of η vs φ all look the same. 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. K. Dane Wittrup Lecture 20 Page 4 of 5 Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY]. Shrinking Core Model In cases with noncatalytic and irreversible reaction, diffusion limit is describable by the “shrinking core model”. Figure 7. Shrinking core model Rapid, irreversible reaction limited by rate of diffusion of a reactant from the surface The following must be written down for the shell balance: 1. Rate of reaction 2. Rate of diffusion 3. Rate of movement of the core 10.37 Chemical and Biological Reaction Engineering, Spring 2007 Prof. K. Dane Wittrup Lecture 20 Page 5 of 5 Cite as: K. Dane Wittrup, course materials for 10.37 Chemical and Biological Reaction Engineering, Spring 2007. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].