Ch E 542 – Intermediate Reactor Analysis & Design

Ch E 542 – Intermediate Reactor Analysis & Design Gas-Liquid Reactions

Two-film model pA pAi gas CAi liquid CAb y y+dy yG yL film boundaries

Single irreversible reaction In the case for which chemical reaction is very slow relative to mass transfer, the amount of A that reacts during transfer through the liquid film is negligible. Reaction occurs completely in the bulk. Rate of transfer of A from liquid interface to bulk is therefore:

Single irreversible reaction When reaction cannot be neglected relative to mass transfer in the liquid film, Gradients are limited to the film for mass balances on species A and B with boundary conditions as

1st-order (or pseudo 1st order) kinetics pseudo 1st-order • In this case, mass balance on A becomes • integrated: • where  is called the Hatta number, and is similar to the modulus used in the effectiveness factor approach

1st-order (or pseudo 1st order) kinetics Accounting for BCs Which allows determination of rate as

1st-order (or pseudo 1st order) kinetics Must be combined with mass balance in bulk to define CAb and CBb (contained in ) - Chapter 14. CBb is constant at a fixed location in the reactor unit, but not necessarily everywhere in the unit

1st-order (or pseudo 1st order) kinetics  is a utilization factor analogous to the catalysis effectiveness factor Compare this flux with that obtained when there is no resistance to mass transfer (i.e., when the concentration of A in the liquid is uniformly CAi)

1st-order (or pseudo 1st order) kinetics For rapid reactions ( > 3) While for  > 5

1st-order (or pseudo 1st order) kinetics at large , reaction is complete in film this result easily derived by starting with the assumption that CAb=0 Eliminating the interfacial concentration using Henry's law and the gas film flux equation

1st-order (or pseudo 1st order) kinetics substitute obtain define leads to Similar result can be obtained based on a utilization factor for the gas-phase composition

1st-order (or pseudo 1st order) kinetics In region of extremely rapid reaction, utilization factor approach (observed to maximum possible rates) requires accurate knowledge of k values. Alternatively, refer to the physical liquid-phase mass transfer rate, which is increased (enhanced) by the chemical reaction. Enhancement factor, FA, obtained as:

1st-order (or pseudo 1st order) kinetics Under the assumption of no A in the bulk,

Single, instantaneous, irreversible pA pAi CAb CAi gas liquid yL y1 yG 0

Single, instantaneous, irreversible in the region y = 0 to y = y1 after one integration gives rise to concentration profiles 0 0 summed

Single, instantaneous, irreversible define a utilization factor as express in terms of Shm eliminate CAi summed

Single, instantaneous, irreversible define a utilization factor as express in terms of Shm eliminate CAi L increases with CBb to the point where the reaction planeoccurs at y = y1 takes place at the interface, y1 = 0, when and the corresponding [B] is

Single, instantaneous, irreversible L increases with CBb to the point where the reaction planeoccurs at y = y1 takes place at the interface, y1 = 0, when and the corresponding [B] is For this value of C'Bb, interfacial concentrations CAi and CBi 0. Thus , rate will not increase beyond this value, since now the rate is completely determined by transport rate from the gas phase.

Single, instantaneous, irreversible pA CAb gas liquid yL yG 0

Single, instantaneous, irreversible Alternatively, define an enhancement factor that defines how mass transfer is enhanced by chemical reaction as:

Surface Renewal Theory Liquid is assumed to be a mosaic of elements with differing age distributions at the surface. Rate of absorption at the surface is an average of the rates of absorption into each element, weighted by a distribution function, (t). Higbie's penetration theory is a surface renewal model for which each liquid element is exposed to the gas for the same length of time before being replaced.

Surface Renewal Theory Danckwerts' approach (1951) uses a distribution function for completely random replacement of surface elements. The fraction of the surface exposed to the gas between t and t+dt is given by The distribution is normalized as

Surface Renewal Theory Danckwerts' distribution function arises when the rate of replacement of surface elements is proportional to the number of elements (fraction of surface) having age t: Integrating, where s is a rate coefficient for surface element replacement,

Surface Renewal Theory (1 instantaneous reaction) Reaction plane no longer fixed, as with two-film model. Elements have finite capacity, thus transient behavior. unsteady A balance in unsteady B balance inzone from y=0 to y1(t) zone from y=y1(t) to 

Surface Renewal Theory (1 instantaneous reaction) by inspection of result for profile of CA, and BCs, movement of reaction plane: Solution by Laplace transform

Surface Renewal Theory (1 instantaneous reaction) Application of boundary conditions In the reaction planestoichiometry of the reaction requires Using Fick's law, allows evaluation of 

Surface Renewal Theory (1 instantaneous reaction) Using Fick's law allows evaluation of flux at any time: Surface absorption rate using Higbie's uniform age,

Surface Renewal Theory (1 instantaneous reaction) Using Fick's law allows evaluation of flux at any time: Application of Danckwerts' age distribution function

Surface Renewal Theory (1 instantaneous reaction) Expressing in terms of an enhancement factor, FA: Application of Danckwerts' age distribution function

Surface Renewal Theory (first-order reaction) unsteady A balance in a unit volume element of liquid Transform into Laplace space

Surface Renewal Theory (first-order reaction) Which is easily solved as Applying BCs… Transform into Laplace space

Surface Renewal Theory (first-order reaction) Which is easily solved as Inverse transform yields

Surface Renewal Theory (first-order reaction) Which simplifies at large kt to Therefore, a steady state is reached Inverse transform yields

Surface Renewal Theory (first-order reaction) Which simplifies at large kt to Therefore, a steady state is reached At time t, the instantaneous absorption rate of an element of surface age t is given as The rate forall elements is

Surface Renewal Theory (first-order reaction) Film theory predicted flux to be proportional to DA, while surface renewal predicts a square root relationship. In the absence of reaction, Using this definitions of mass transfer coefficient,

Ch E 542 – Intermediate Reactor Analysis & Design