L 29-Heterogeneous Catalysis and Reactor Design Prof. K.K.Pant Department of Chemical Engineering IIT Delhi. kkpant@chemical.iitd.ac.in

Diffusion and Reaction in a spherical pellet (Reading assignment: Fogler, Ch 12)) In – out – disappearance =0 (spherical shell balance) WAr (4 πr2)r - WAr (4 πr2)r+∆ r - rA’ (4 πr2 c ∆r) =0 Dividing by -4 π ∆ r =r+Δr Moles = WAr (4 πr2)r Molar flux r=0, CA finite, r=R, CA=CAS -rA=c(-r’A) Boundary conditions

consider 1st order What about n-th order ?  c(-r’A) =-rA -rA=kCA Differentiation & Divide by –r2De What about n-th order ?  Differentiation & Divide by –r2De

Dimensionless eq. – 1st order Thiele Module About for n-th order ? Thiele Module

Y =  , , =>= y/  d/ d = 1/ (dy/d)- y/ 2 d2/ d2 = 1/ (d2y/d2)- 2/ 2 dy/d + 2y/3 d2y/d 2 - ø2y=0 y= A Cosh ø  + B Sinh ø  A=0 as φ must be finite at the centre, (B. C. =0, coshø 1; and Sinh ø 0. And at =1, =1,=> B= 1/Sinh ø Thus , = CA/CAs = 1/  [Sinh ø  / Sinh ø]

Thiele Modulus, n If n is large – internal diffusion limits the overall rate If n is small – the surface reaction limits the

Internal Effectiveness Factor Internal effectiveness Factor,  is: ranged 0 – 1  for a first-order reaction in a spherical catalyst pellet

Calculation of Catalytic Effectiveness Factor η = Actual overall rate(RA /Rate in the absence of diffusion resistance (RAs ) Global rate RA = 4 πR2 De (dCA/dr) at r=R Or RA = 4 πR De (d  /d ) at  =1 ((d  /d  )at  =1 = (ø cot h ø-1) RA = 4 πR De CAS (ø cot h ø-1) Global Rate. Thus η = [4 πR De CAS (ø cot h ø-1)] / k’ ρc CAS 4/3 πR 3 η = 3 (ø cot h ø-1)/ k’ ρc R2/De R η = 3 (ø cot h ø-1)/ ø2 for ø> 20, η= 3/ ø strong pore diffusion resistance

Calculation of Catalytic Effectiveness Factor where - Thiele Modulus 1st order reaction rate: Spherical Pellet Cylindrical Pellet Slab Pellet

Internal Effectiveness Factor

Weisz – Prater Criterion for internal diffusion Uses the measured values of the rate of reaction to determine if Internal diffusion controls the rate. Weisz-Prater Parameter CWP ηø2 = 3(Ø Coth Ø-1) ηø2 = (observed rate/rate cal. at CAS) x (rate calculated at CAS) / diffusion Rate) η = (-r’A(obs)/ -r’As Ø2= -r”AS Sa ρp R2/De CAs = -r’AS ρp R2/De CAs

CWP= (-r’A(obs)/ -r’As )(-r’AS ρp R2/De CAS) CWP = (-r’A(obs) (ρp R2/De CAS) These are measured or known terms. if CWP << 1, No diffusion limitations and no concentration gradient exists in the pellet. CWP >> 1, Internal diffusion limits the rate.

Non isothermal pellet Energy balance Mass Balance

Non isothermal pellet effectiveness factor

Falsified Kinetics Measurement of the apparent reaction order and activation energy results primarily when internal diffusion limitations are present. This becomes serious if the catalyst pellet shape and size between lab (apparent) and real reactor (true) regime were Too different. Smaller catalyst pellet  reduces the diffusion limitation  higher activation energy  more temperature sensitive  RUNAWAY REACTION CONDITIONS!!!!

Falsified Kinetics With the same rate of production, reaction order and activation energy to be measured (apparent rate)

Overall effectiveness factor (Both internal and external diffusion are important

Overall Effectiveness Factor At Steady state, Moles transported from bulk fluid to the external surface of the Catalyst (WAr Ap )= Net Rate of reaction with in and on the pellet, MA = WAr Ap = -r”A(As+ Ap),=( Molar Flux x Ext. Surface Area of pellet) For a single spherical pellet of Radius R, AP= 4π R2, and As= SA x mass of pellet,(As >>Ap) AP = (ext. SA/reactor volume) (reactor volume) = ac ∆V As= (int. SA/.mass of catalyst) (mass cat./vol. cat) (vol cat/reactor vol.) . Rect vol.) => AS= SA ρ c (1- ø) ∆V