Ozone Reaction Kinetics for Water and Wastewater Systems - Chapter 4 ppt

If it is assumed that one component A of a gas phase is transferred to theliquid phase, the rate of mass transfer or absorption rate of A is as follows: 4.1 where SI units of N A are in

4 Fundamentals

of Gas–Liquid Reaction Kinetics

The kinetics of heterogeneous reactions is governed by absorption theories of gases

in liquids accompanied by chemical reactions The fundamentals of these theoriesare necessary to understand the phenomena developing during the ozonation ofcompounds in water The necessary steps to study the kinetics of gas–liquid reactionsare described below Since ozone reactions can be considered irreversible, isothermic,and of second order1 (for a general ozone–B reaction) or pseudo first-order (a case

of ozone decomposition reaction), the discussion that follows mainly refers to thistype of gas–liquid reactions Nonetheless, some fundamental aspects on series-parallel reactions are also given Notice that ozonation of compounds in water yields

a series of by-products that also react with ozone Therefore, the kinetics of parallel gas–liquid reactions constitutes another part of this study As a first step,the physical absorption of a gas in a liquid is treated as commented below

series-4.1 PHYSICAL ABSORPTION

In a gas–liquid reacting system, diffusion, convection, and chemical reaction proceedsimultaneously, and the behavior of the system can be predicted with the use ofmodels that simulate the situation for practical purposes These models are based

on those describing the gas physical absorption phenomena, that is, they are based

on gas absorption theories In a general case, when gas and liquid phases are incontact, the components of one phase can transfer to the other to reach the equilib-rium If it is assumed that one component A of a gas phase is transferred to theliquid phase, the rate of mass transfer or absorption rate of A is as follows:

(4.1)

where SI units of N A are in molm–2s–1, k G and k L are the individual mass transfercoefficients for the gas and liquid phases, respectively; P Ab and P i the partial pres-sures of A in the bulk gas and at the interface, respectively; and C A*and C Ab, arethe concentrations of A at the interface and in the bulk of the liquid, respectively(see Figure 4.1) One of the two problems of the rate law is to find some mathematicalexpression for the mass transfer coefficients The other one is to know the interfacialconcentrations, P i or C A* Theoretical expressions for mass transfer coefficients

N A=k P G( Ab−P i)=k C L( A*−C Ab)

can be found from the solution of the microscopic mass balance equation of thetransferred component A that, applied to the liquid phase, is as follows:

(4.2)

where the term on the left side represents the molecular and turbulent transport rate

of A and the first and second ones on the right side represent the terms of convectionand accumulation rates of A, respectively Equation (4.2) is conveniently simplifiedaccording to the hypothesis of the absorption theories The most applied absorptiontheories are: the film and surface renewal theories

4.1.1 T HE F ILM T HEORY

Lewis and Withman2 proposed that when two nonmiscible phases are in contact, themain resistance to mass transfer is located in a stationary layer of width δ closed tothe interface, called the film layer It is also assumed that mass transfer through thefilm is only due to diffusion and that the concentration profiles with distance to theinterface are reached instantaneously It is then called a theory of the pseudo sta-tionary state

In a gas–liquid system there will be two films, one for each phase In mostcommon situations the gas is bubbled into the liquid phase, so the interfacial surface

is due to the external surface of bubbles Concentration profiles of the gas componentbeing absorbed for both the gas and liquid films are as shown in Figure 4.2 Thefilm theory assumes a plane interfacial surface when the bubble radius is much lowerthan the film thick, δ, a situation fulfilled in most of the gas–liquid systems Accord-ing to these statements (diffusion, stationary state, and one direction for masstransfer), Equation (4.2) reduces to:

x

A A

∂

2

Equation (4.3) can be solved with the following conditions:

(4.7)

Thus, δL, the width of film, is the characteristic parameter of film theory

FIGURE 4.2 Concentration profile of a gas component A with the distance to the interface during its physical absorption according to the film theory.

0 x

Bulk gas Gas film Liquid film Bulk liquid

=δ

4.1.2 S URFACE R ENEWAL T HEORIES

In these models, the liquid is assumed to be formed by elements of infinite widththat are exposed to the interface for a given time and then are replaced by otherelements coming from the bulk liquid While the liquid elements are at the interface,mass transfer occurs by diffusion in a nonstationary way The most-used surfacerenewal theory is that proposed by Danckwerts3 which assumes a distribution func-tion of exposition times for the liquid elements For this surface renewal theory,Equation (4.2) reduces to the following one:

(4.8)with the boundary limits:

(4.9)

After application of Fick’s law,4 once the concentration profile of A with time andposition [C A = f(x,t)] has been determined from the solution of Equation (4.8), themean absorption rate of A is as follows:

x

C t

This is the case of the ozonation reactions Determination of the gas absorption rate(or ozonation rate for ozone processes) also requires following the steps shown abovefor the case of physical absorption, that is:

• Solving the microscopic differential mass balance equation to find out theconcentration profile of the gas being absorbed with distance to the interface

• The application of Fick’s law to yield the gas absorption rate

Solution of the mass balance Equation (4.13) depends on the absorption theoryapplied as explained below for the film and Danckwerts theories The cases that aretreated correspond to irreversible first- (or pseudo first-) and second-order reactionswhich are usually the case of simple ozonation reactions in water

4.2.1 F ILM T HEORY

4.2.1.1 Irreversible First-Order or Pseudo First-Order Reactions

This is the case of reactions with the following stoichiometry:

1 First-order reaction:

(4.14)with

(4.15)

2 Pseudo first-order reaction:

(4.16)with

∂

∂2 2 = 1

being solved with the boundary conditions given in Equation (4.4) Solution of this

system leads to the following concentration profile of A with the distance to the

interface:

(4.19)

where Ha1 is called the dimensionless number of Hatta for an irreversible first-order

reaction defined as follows:

(4.20)

The square of Ha1 represents the ratio between the maximum chemical reaction rate

through the film layer and the maximum physical absorption rate:

(4.21)

where a is the specific interfacial area and k L a the volumetric mass transfer coefficient

in the liquid phase Notice that the product aδL is the ratio between the liquid film

and liquid total volumes Thus, Ha1 indicates the relative importance of chemical

reaction and mass transfer rates in the gas liquid system

Application of Fick’s law at the gas–liquid interface leads to the gas absorption

rate equation, or to the rate or kinetic law for this type of reaction:

(4.22)

where M1 is the maximum physical absorption rate, k L C A*is expressed per unit of

interfacial surface Notice that in Equation (4.21), the maximum physical absorption

rate is expressed per unit of volume

In Equation (4.22), it is convenient to express C Ab as a function of chemical and

mass transfer parameters Then, the mass transfer rate at the other edge of the film

layer, (x = δL ), N Aδ, and the chemical reaction rate in the bulk of the liquid, R b1, are

x Ha

sinhsinh

11

where units of both rates are given per surface of interfacial area, β being the liquidhold-up, it is defined as the ratio of liquid to total (gas plus liquid) volumes

By equalizing Equation (4.23) and Equation (4.24), the ratio between

concen-trations of A in the bulk of the liquid and at the interface, C Ab /C A*, can be obtained.Then, after substitution in Equation (4.22), Equation (4.25) is obtained:

(4.25)

Also, if C Ab /C A* is expressed as a function of β/aδ L, a dimensionless number thatrepresents the ratio between the volumes of the total liquid (film plus bulk liquids)and film layer, the following alternative equation is obtained5:

Equation (4.25) or Equation (4.26) constitute the general kinetic equations for order (or pseudo first-order) gas–liquid reactions

first-As can be deduced from Equation (4.25), the absorption rate is a function ofthree maximum rates:

• R b1 maximum chemical reaction rate in the bulk liquid = k1C Ab β/a

• M1 maximum physical absorption rate at the interface = k L C A*

• R F maximum chemical reaction rate through the film layer = k 1 C A*aδL

Also, depending on the values of Ha1, the absorption rate develops in different kineticregimes6:

• The fast kinetic regime when Ha1 > 3, then C A0 = 0 with:

(4.26)

N Ao=M Ha1 1

• The diffusional kinetic regime when Ha1 < 0.3 and C Ab = 0 with:

(4.30)

Since the general Equation (4.25) is rather complex, the absorption rate is usuallyexpressed as a function of another dimensionless number called reaction factor E:

(4.31)

As deduced from Equation (4.31), the reaction factor can be defined as the number

of times the maximum physical absorption rate increases due to the chemical tion Notice that this definition has only physical meaning when the kinetic regime

reac-is fast or moderate, (for C Ab = 0) However, according to Equation (4.31), values of

E can be lower than unity (the cases of slow kinetic regime or some others with

moderate regime), although they have no practical use In Figure 4.3, a plot of E

against Ha1 is shown with the zones of different kinetic regimes Notice that for a

slow kinetic regime (Ha1 < 0.3) Equation (4.26) is used to show the variation of E

with the Hatta number This is because, in the slow kinetic regime, reactions develop

in the bulk liquid and the volume ratio parameter β/aδ L has a great influence on thegas absorption rate

4.2.1.2 Irreversible Second-Order Reactions

These constitute the typical case of most of ozone direct reactions in water Thestoichiometric equation is:

A+zB →k2 P

and the chemical reaction rate referred to the disappearance of A:

(4.33)For this case, both microscopic mass balance equation of A and B have to be simul-taneously solved:

(4.34)

(4.35)Boundary conditions of this mathematical model are:

in Section 4.2.1.1 could be applied with k1 = k2C Br By applying the proposedapproximation,7 the following equation was found for C Br:

(4.37)

FIGURE 4.3 Variation of the reaction factor with the Hatta number for one irreversible

first-or pseudo first-first-order gas liquid reaction.

Then Ha1 [see Equation (4.20)] becomes

(4.38)

Equation (4.38) can be simplified to Equation (4.39):

(4.39)

where Ha2 represents the Hatta number of the irreversible second order reaction

(4.32) and has the same physical meaning as Ha1 in Equation (4.20):

(4.40)

Finally, the absorption rate of A accompanied by an irreversible second order chemical reaction with B is given by Equation (4.25) with Ha1 given by Equation(4.39) The system is solved with Equation (4.31) by a trial-and-error procedure that

allows the values of the reaction factor E to be obtained from the values of Ha2.The solution is usually presented in plots such as Figure 4.5 As deduced fromEquation (4.25) and Equation (4.38), the absorption rate is a function of fourmaximum rates (three of them previously deduced for the first order reaction kinetics)defined as follows:

FIGURE 4.4 Concentration profiles of a gas component A and one liquid component B with

the distance to the gas–liquid interface during their fast chemical reaction while A is diffusing through the liquid film Profiles of B concentration according to the film theory: continuous line; profiles of B concentration according to the simplification of Van Krevelen and Hoftijzer 7 : dotted line (From Van Krevelen, D.W and Hoftijzer, P.J., Kinetics of gas liquid reactions.

Part I General theory, Rec Trav Chim., 67, 563–586, 1948 With permission.)

Gas–liquid interface

=

• R b2 maximum chemical reaction rate in the bulk liquid:

(4.41)

• M1 maximum physical absorption rate at the interface = k L C A*

• R F maximum chemical reaction rate through the film layer:

(4.42)

• M2 maximum physical diffusion of B through the film layer:

(4.43)

For the case of second order reaction there are two new kinetic regimes to consider

in addition to those listed for first order reactions:

• Fast kinetic regime, with C Ab = 0 and Ha2 > 3:

(4.44)

where Ha1 is given by Equation (4.38)

• Instantaneous kinetic regime with C A0 = 0 and Ha2 > 10E i:

(4.45)

where E i is the reaction factor for the instantaneous reaction, defined asfollows:

FIGURE 4.5 Variation of the reaction factor with the Hatta number for one irreversible

second-order gas liquid reaction.

tanh

N A0 =k C E L A* i

where D B is the diffusivity of compound B in the liquid that can be estimated

with the equation of Wilke and Chang8,9 as indicated in Section 5.1.1.Notice that the fast kinetic regime for first order reaction is now called the fastpseudo first-order kinetic regime with the following condition to be fulfilled:

(4.47)The absorption rate law being in this case:

(4.48)

Figure 4.6 to Figure 4.12 show the concentration profiles of A and B through the

liquid that hold for the different kinetic regimes Notice that for fast or instantaneouskinetic regimes, the chemical reaction develops in a zone or plane of the film layer,respectively, while for slow kinetic regimes, the chemical reaction is in the bulkliquid Also notice that for fast or instantaneous kinetic regime there will not be

dissolved gas in the bulk liquid (C Ab = 0) It should be highlighted that the rateequations presented are per unit of interfacial surface area (that is in mols–1m–2) In

a practical situation, the problem is that the interfacial surface area is generallyunknown Nonetheless, this problem is overcome by multiplying both sides of theabsorption rate equation by the specific interfacial area, a, so that the rate will begiven per unit of volume which is known It should also be reminded that depending

on the relative importance of mass transfer and chemical reaction steps, that is,depending on the kinetic regime, the general Equation (4.25), once Equation (4.38)has been accounted for, simplifies in a similar way as shown for the case of firstorder reaction These simplifications will be used in the kinetic study of waterozonation reactions as shown later

4.2.1.3 Series-Parallel Reactions

In most cases, ozone reacts not only with the compound initially present in waterbut also with the intermediate compounds formed in a series-parallel reaction system.Thus, a simplified study of the kinetics (absorption rate law) of these gas–liquid

systems is presented here For so doing, let us assume that a gas component A

transfers into the liquid where it undergoes the following reactions:

(4.49)(4.50)

FIGURE 4.6 Film theory: very slow kinetic regime Concentration profiles for A (gas being

dissolved) and B (liquid component) with the distance to the interface.

FIGURE 4.7 Film theory: slow kinetic regime Concentration profiles for A (gas being

dissolved) and B (liquid component) with the distance to the interface.

FIGURE 4.8 Film theory: diffusional kinetic regime Concentration profiles for A (gas being

dissolved) and B (liquid component) with the distance to the interface.

Gas–liquid interface

Liquid phase Liquid film Bulk liquid

Liquid phase Liquid film Bulk liquid

Liquid phase Liquid film Bulk liquid