Rate equation of isotopic ion exchange reaction with particle diffusion (Pmechanism), film diffusion (Fmechanism), and chemical reaction (Cmechanism) for various mathematical models
Abstract
Generally, unit operation processes that are used in environmental engineering are involved in interfacial reaction where mass transfer is an extremely essential component for system optimization. The purposes of this chapter were intended to provide the information of both theoretical model development and engineering practice for mass transfer of important processes in environmental engineering. Those processes include, but are not limited to, (1) ozonation (gas–liquid process), (2) ion exchange (liquid–solid process), (3) biological activated carbon (liquid–solid process), (4) chlorination (gas–liquid process), and (5) carbonation (gas–liquid–solid process).
Keywords
 ozonation
 ion exchanger
 biological activated carbon
 chlorination
 carbonaion
 rotating packed bed
1. Introduction
1.1. Ozonation process
1.1.1. Model development
1.1.1.1. Process chemistry
Ozone process has been utilized in thousands of water treatment applications such as municipal water (or wastewater) treatment plants, cooling towers, ultrapure water, marine aquaria, beverage industries, industrial process water, swimming pools, bottledwater plants, aquaculture, and food processing. It can oxidize the micropollutants and reduce the amount of disinfection byproduct (DBP) precursors formed in drinking water [1], such as organic peroxides, aliphatic aldehydes, hydrogen peroxides, mixed functional and saturated carboxylic acids, etc. Some of the DBPs such as aldehydes and/or chlorinated aldehydes would be potential health hazards, leading to the increase of bacterial populations in distribution systems. Three oxygen molecules will combine together to form ozone, as shown in Equation (1); this is an endothermic reaction:
The gaseous ozone (O_{3}) will decompose to oxygen gas (O_{2}), as shown in Equation (2). In general, the decomposition rate increases as the temperature increases. The efficiency of ozone production decreases at a higher temperature with a fixed voltage input.
1.1.2. Reaction kinetics and mass transfer
Ozonation is a gas–liquid process under dynamic conditions; the reaction regime, mass transfer characteristics, and reaction kinetics are critical to system design and operation. As O_{3} dissolves in water, it will selfdecompose and generate oxygen and hydroxylfree radicals. In general, the solubility of dissolved O_{3} in ozonated water is merely 1 mg/L; higher dissolved O_{3} concentration will only happen in the environment with better mass transfer (e.g., mixing). The mass transfer rate is influenced by operating variables such as pH, temperature, agitation speed, and gas flow rate. In general, the ozone concentration generated from ozone generator usually is less than 14 % (v/v) that may restrict the mass transfer between gas and liquid phases. Therefore, residual ozone concentration in water treatment ranges from 0.1 to 1 mg/L. A higher dissolved ozone concentration may happen only if under better mass transfer conditions.
It is noted that the specific interfacial area is of great significance to the gas–liquid mass transfer. In ozonation process, the overall liquidphase mass transfer coefficient (K_{L}a) and equilibrium dissolved ozone concentration ([O_{3}]^{*}) can be obtained from the following mass balance equation [2]:
where
where
1.2. Engineering practice
1.2.1. Determination of overall mass transfer coefficients
Figure 1 presents a typical schematic diagram of ozonation process, where the ozone is produced by an ozone generator and able to reach 10 % or higher.
Ouederni et al. [4] proposed the following equation to determine the ozone transfer in stirred reactors:
where K_{L}a is expressed in s^{1}, G is the gas flow rate expressed in 10^{3} m^{3}/s, and n is the number of rotations per second. According to Sheffer and Esterson [5], the volumetric overall liquidphase mass transfer coefficient of ozone in tap water (K_{L}a) was around 1.14 min^{1}. It was also observed that it would reach a higher value at a higher water flow rate (i.e., sufficient agitation). For instance, there was a large decrease in K_{L}a to 0.25 min^{1} without agitation.
In addition, combining the empirical relationship reported by Stankovic [6], the following equation can be derived:
where
1.2.2. Determination of Fractional Ozone Absorption (FOA)
To determine the optimal operation conditions for semibatch ozonation process, the “fractional ozone absorption (FOA),” defined as the ratio of the ozone transferred to water to the ozone applied to the reactor [7], should be applied. The FOA value in semibatch operation can be expressed as [2]
where P_{O3} is the ozone partial pressure (kPa). The value of 1.98 x 10^{5} is the factor which will transfer the term of denominator into the unit of mg/min.
An increase in temperature will increase the ozone diffusivity in water and enhance the liquidphase mass transfer, because the
Figure 2 shows the relationships between the agitation speeds and influent gas flow rates under the specific conditions of K_{L}a and FOA_{0}. The results indicate that the FOA_{0} is mainly controlled by n (not by G). Both the values of
The power consumption (P_{g}) should be taken into consideration for the most efficient design and operating conditions in practice. P_{g} can be calculated as follows [9]:
where
Figure 2 can be used to determine
1.2.3. Integration of mass transfer and reaction kinetics
Two important factors for determining the kinetic regime of ozonation are the enhancement factor (
where
DeCoursey [10] proposed a correlation between
where
Since the prediction model of
Therefore, the marginal level of ozone partial pressure (
Figure 3 gives the relationship between the agitation speed and applied ozone concentration under various retention times. For example, the TOC removal efficiency of 15 % and enhancement factor (E = 1) were held constant under the temperature of 25 °C, a pH value of 7, and an influent gas flow rate of 10 L/min. The higher the ozone partial pressure, the lower the agitation speed needed to achieve the same level of performance. In other words, it suggests that the ozone partial pressure (
2. Ion Exchange (IE)
Deionization is the process of removing the alts of electrolytes from water by ion exchange. “Deionization” is generally considered synonymous with “demineralization,” by various authors, but since some of the ions exchanged are not necessarily of mineral origin, the term “deionization” appears more inclusive.
2.1. Model development
2.1.1. Ratedetermining step
According to the Nernst static diffusion film theory, three mechanisms might be the ratedetermining step under appropriate conditions:
Film diffusion (Fmechanism): transport of ions from the bulk liquid to the liquid solid interface
Particle diffusion (Pmechanism): transport of ions into the resin particle
Chemical reaction (Cmechanism): the chemical process of exchange
Recently, the principle of ion exchange kinetics has been investigated by applying either empirical rate equations or theoretical equations derived from fundamental diffusional mechanisms. In the latter approach, the ratedetermining step could have been film diffusion, surface chemical reaction, or particle diffusion.
According to the kinetics of anion exchange on various resins, diffusion was the ratecontrolling process in all cases. Turse and Rieman III [11] reported that the rate of exchange was controlled by the chemical reaction if chelates were formed by cations and resins. Without formation of chelates, at low concentration, the rate was controlled by “film” diffusion and, on the other hand, at high concentration by “particle” diffusion. In other words, the ratedetermining step of the ion exchange process is always a combination of these two limiting steps of film diffusion and particle diffusion, except for the case of large organic ions (quaternary ammonium ions) which are controlled by particle diffusion.
2.1.2. Crank model
The derivation of the Crank model is given below. For particlediffusion controlled, the rate of exchange could be expressed as Fick’s second law as follows:
With the initial condition:
and the boundary condition:
Equation (14) can be easily solved by the Laplace transform. The
where
In addition,
where
Huang and Tsai [13] have developed a more appropriate and general equation which combined film diffusion, particle diffusion, and chemical surface reaction on the exchange rate for a finite bath. Under a specified case, the rate equation (Table 1) using only isotopic exchange with particle resistance is consistent with the Crank model [12].




Kressman and Kitchener (Limited bath) 



Yagi and Kunii (Shell progressive) 



Boyd (Shallow bed) 



Crank (Finite bath) 



2.2. Engineering practice
2.2.1. Determination of ratedetermining steps
In this chapter, the HPC and BHDAC are demonstrated as the representative quaternary ammonium ions because they are easy to prepare, highly soluble in water, and can be rapidly analyzed by use of UV. In addition, Dow50X8 was utilized as the resin for illustration. By plotting the
2.2.2. Determination of particlediffusion coefficient
The particlediffusion coefficients of LPC, HPC, and BHDAC through the Amb200, AmbDPI, and Dow50X8 resins were determined and summarized in Table 2. As previously cited, the rate of exchange of HPC and BHDAC on various particle sizes on Dow50x8 resins was shown in Figure 4. It was observed that the Dow50X8 resin possessed the lowest particlediffusion coefficient in comparison with the Amb200 and AmbDPI resins. This evidence strongly suggested that the macroporous resin had an advantage over the gelstructure resins for exchange of organic ions.




LPC  3.03  2.35   
HPC  2.43  2.12  0.85 
BHDAC  0.90  1.70  0.57 
It was thus concluded that the Amb200 (strong acid cation exchanger), AmbDPI (weak acid cation exchanger), and XE318 (chelating cation exchanger) resins were the ones to be selected as the representative cation exchangers for the resin selectivity test as described later.
3. Biological Activated Carbon (BAC)
3.1. Model development
3.1.1. Representative models of BAC
The biological activated carbon (BAC) process contains adsorption and biodegradation mechanisms. The BAC process also exhibits lower regeneration cost and prolongs the life of granular activated carbon (GAC) beds. Therefore, it has been widely used in water and wastewater treatments.
A wellvalidated mathematical model can provide valuable information to evaluate and predict the performance of BAC process. Table 3 presents several representative models for BAC process. Chang and Rittmann [14] developed a model that the mass transfer of substrates diffusing through the biofilm, metabolized by microbes, and finally reaching the surface of GAC was illustrated and quantified. However, the limitation of the model is that it cannot be used under unsteady or plug flow conditions. Sakoda et al. (1996) developed a theoretical model for a BAC column considering the mechanisms including dispersion, convection, biodegradation, and adsorption. The assumption included that the substrate concentration on the interface between the biofilm and the GAC is identical to that in the bulk solution. Furthermore, Liang and Chiang [15] developed a nonsteadystate numerical model to differentiate the adsorption and biodegradation quantities of a continuous BAC column including the mechanisms of adsorption, biodegradation, convection, and diffusion.












Complex mixing  A, B  Nonsteady Monod  Monod  Nonsteady  Non equilibrium 
1, 2, 3, 4, 5  Analytical  Chang and Rittmann [14] 
Column  A, B, C, D  Nonsteady, no biodegradation, uniform Monod  n.a.^{c}  Steady  Equilibrium  1  Analytical  Sakoda, Wang and Suzuki [17] 
Column  A, B  Uniform  Monod  Nonsteady  n.a.^{c}  1  Analytical  Walker and Weatherley [18] 
Column  A, B, C  Nonsteady, no biodegradation  Monod  Steady  Non equilibrium 
1, 5  Analytical  Abumaizar, Smith and Kocher [19] 
Column  B, C, D  Nonsteady Monod  Monod  Nonsteady  n.a.^{c}  1, 2, 3  Numerical  Hozalski and Bouwer [20] 
Column  A, B, C, D  Nonsteady, no biodegradation  Monod  Nonsteady  Non equilibrium 
1, 2, 3, 4, 5  Numerical  Badriyha, Ravindran, Den and Pirbazari [21] 
Column  A, B, C, D  Nonsteady, no biodegradation, uniform Monod  Monod  Steady  Non equilibrium 
1, 2, 3, 4, 5  Numerical  Liang and Chiang [15] 
3.1.2. Nonsteadystate models for adsorption and biodegradation of BAC
A numerical model can be developed to simulate both adsorption and biodegradation quantities of a BAC column under a nonsteadystate condition. The governing equation based on mass balance of the substrate concentration in the liquid phase of the BAC column can be expressed as follows [16]:
Table 4 presents the parameter details of Equation (20):



ε    Bed porosity of the BAC column 
S_{b}  M/L^{3}  Substrate concentration in the liquid phase 
D_{b}  L^{2}/T  Dispersion coefficient in the liquid phase 
x  L  Distance along the BAC column 
V_{g}  L^{3}  Volume of a GAC granule 
L_{f}  L  Length of the biofilm 
k_{f}  M/Tcell  Maximum utilization rate in the biofilm 
k_{b}  M/Tcell  Maximum utilization rate in the liquid phase 
X_{f}  cell/L^{3}  Cell density of the biofilm 
X_{b}  cell/L^{3}  Cell density in the liquid phase 
S_{f}  M/L^{3}  Substrate concentration in the biofilm 
K_{f}  M/L^{3}  Monod halfvelocity coefficient in the biofilm 
K_{b}  M/L^{3}  Monod halfvelocity coefficient in the liquid phase 
r_{f}  L  Radius of the biofilm 
r_{g}  L  Radius of the GAC granule 
ρ_{g}  M/L^{3}  GAC granule apparent density 
q_{a}  M/M  Adsorption capacity 
Two boundary conditions (BC) for the equation of dispersion advection reaction can be described as follows:
where Lc is the length of the BAC column (L). If the diffusion and reaction simultaneously occurred under a nonsteadystate biofilm condition, the nonsteadystate form of mass transfer and biodegradation reaction within biofilm, based on Fick’s law and Monod equation, can be expressed as
where D_{f} (L^{2}/T) is the diffusivity within the biofilm. A diffusion layer exists between the bulk solution and the biofilm, and the substrate concentration profile can be solved according to Rittmann and McCarty [22]. Therefore, the boundary condition can be simplified as
where k_{bf} is a factor to estimate the concentration reduction within the diffusion layer. Langmuir isotherm was used for the calculation of the boundary concentration of the biofilm near the GAC side, which was derived from the solidphase concentration of the adsorbates. For singlecomponent adsorbate, the surface concentration (q_{s}, M/M) can be expressed as follows:
where
where m_{g} (M) is the mass of a GAC granule. In a control volume, the average biodegradation rate can be derived by integrating the Monod reaction expression and the amount of biofilm volume:
where ΔV is a control volume unit of the BAC bed (L^{3}) and Ng is the number of BAC granules in a control volume.
3.1.3. Estimation of biofilm thickness
On the other hand, the thickness of biofilm will increase due to growth and decrease by the shear of water and the selfdecay of bacteria. As a result, the biofilm thickness can be described as
where Y is the yield coefficient of biomass (CFU/M) and A_{f} is the surface area of a BAC granule (L^{2}).
In addition, the governing equation for the bacterial density in bulk solution can be simplified as an advection–reaction form:
In this situation, the BCs of Equation (30) can be expressed as follows:
The governing equation is a secondorder partialdifferential equation, which can be approximately solved by the Crank–Nicolson finite differential and Crout factorization methods. Therefore, the developed model can quantify the amounts of adsorption and biodegradation. In addition, the significances of each parameter can be determined throughout the results of sensitivity analysis.
3.2. Engineering practice
3.2.1. Dimensionless analysis
The mass transfer coefficient can be determined by the dimensionless groups of Reynolds (Re), Sherwood (Sh), and Schmidt (Sc) numbers. Since these dimensionless groups are developed for process scaleup, different reactors can be compared to each other through the dimensionless analysis. The Reynolds number represents the ratio between inertial force and viscous force, and the Sherwood number represents the ratio between liquidfilm transfer and biofilm diffusive transfer. The N_{Sh} can be correlated with the dimensionless groups as follows (Wakao and Funazkri, 1978):
Several key operational parameters reasonably dominating the performance of a BAC column include the superficial influent velocity and the particle size of packing. As a result, the effects of particle size on adsorption and biodegradation can be simulated by the dimensional analysis.
3.2.2. Liquidfilm mass transfer
Figure 5 shows the relationship between the Stanton number (N_{St}) and the liquidphase mass transfer coefficient (k_{b}f) reported in the literature. It is observed that the Stanton number (i.e., the ratio of the liquidfilm transfer to the bulk transfer) exhibits an inverse proportion to the particle size (
4. Chlorination process
4.1. Model development
Chlorination (using chlorine as disinfectant) is the common process used in daily water disinfection because of its high efficiency with a relatively low cost. However, as aforementioned in the Ozonation chapter, the disinfection byproducts (DBP), specifically trihalomethane (THM) including chloroform, bromodichloromethane, dibromochloromethane, and bromoform, will also be formed during the chlorination process. Control of the formation potential of THM during chlorination is thus an important target for the supply of drinking water. THMs can be formed in the presence of specific precursors such as humic substances and methyl ketones [23]. Several good engineering practices to prevent THM formation include (1) decrease of chlorine dose, (2) change of the point of chlorination, (3) substitution with an alternative disinfectant, and (4) reduction of precursors in solution by GAC or enhanced coagulation. In addition, several factors including precursor concentration (as represented by total organic carbon concentration), chlorine dose, bromide concentration, reaction temperature, pH of solution, contact time, and reactor configuration (lengthtowidth ratio and mixing conditions) will affect the formation of THMs. To predict the performance of disinfection efficiency and kinetics, and/or THM formation potential, numerous models such as dispersion model have been applied in chlorine process, as illustrated in the following context.
4.1.1. Dispersion model
In general, the chlorination process is applied in either a rectangular basin with baffles (rectangularlong and narrow channel) or a circular basin (circularannular ring surrounding). The hydrology in a rectangular basin with baffles is the most advantageous because of a plugflowlike reactor. It can be examined by the exit age distribution (
where
For a baffled rectangular basin with a total flow length of L and a channel width of W, the dimensionless diffusion coefficient (
where k is a nonideality coefficient of the reactor.
4.1.2. Disinfection kinetics
According to Collins and Selleck [26], the dieoff of microorganisms in a batch reactor due to disinfection can be expressed by Equations (36) and (37):
where
The overall survival fraction of bacteria can be determined by integrating the survival fraction of a series of batch reactors over the
Therefore, the average disinfection efficiency can be calculated by the given chlorine dose (
4.1.3. Formation kinetics of THMs
The reaction chemistry between chlorine and aquatic humic precursors to form THMs can be expressed by the following shorthand way [25]:
Based on the above process chemistry, the kinetics of chlorine decay and the THMs formation can be expressed by Equations (41) and (42), respectively:
When reactions take place in a continuous flow reactor, the overall change of free chlorine and THMs can be obtained by integration of the batch concentration over the entire exit time distribution (E_{θ}) as follows:
4.2. Engineering practice
4.2.1. Determination of coefficients in disinfection kinetic model
In practice, the total coliform and total bacteria count can be measured by the membrane filter method and heterotrophic plate count method, respectively. The effect of chlorine doses and contact time on the survival of total coliform and total bacteria can be determined, as shown in Figures 6 (a) and (b), respectively.
4.2.2. Prediction of THMs Formation
As shown in Figure 7, the effect of the contact time (i.e., 0–70 min) and the chlorine dose (i.e., 0.5–8.0 mg/L) on the THM formation was presented at a TOC of 3 mg/L with an L/W ratio of 20. The THM concentrations increase linearly with the contact time over the range of 0 to 70 min. By knowing the water quality parameters and experimental conditions, it is possible to predict the THM formation in the effluent of the chlorine contact chamber. To keep the THM formation below 100 μg/L at the water treatment plant, it is necessary to maintain the chlorine dose below 2 mg/L for a contact basin with an L/W ratio of 20.
4.2.3. Balancing disinfection efficiency and THM formation
As the aforementioned, balancing the disinfection efficiency (in Section 2.1) and THM formation (in Section 2.2) is an important task in chlorination process. Table 5 provides the engineering practice (exemplified by Case A wastewater treatment plant) to balance the disinfection efficiencies and THM formation, in terms of contact time, L/W ratio, and applied chlorine doses. It can be used to determine the applied chlorine dose needed to meet a specified degree of inactivation (e.g., 99.99 % kill) and level of THM formation. In general, the THM formation increases rapidly with the increases of contact time. Conversely, the disinfection efficiency of chlorine increases gradually as both contact time and chlorine dosage increase.
It is noted that the applied chlorine dose at about 1.0 mg/L exhibits the most economically feasible way to control the THM formation and to maintain the disinfection efficiency at the Case A water treatment plant. Under these conditions, the Case A plant is able to achieve an acceptable level of residual chlorine (i.e., 0.5 mg/L) at the THM formation below 100 μg/L.






Applied chlorine  mg/L  0.5  0.5  0.5  1.0  1.0  1.0  1.25  1.25  1.25  
L/W  –  2  10  20  2  10  20  2  10  20  
Disinfec tion efficiency 
20 min  %  99.40  99.59  99.62  99.96  99.97  99.98  99.98  99.98  99.99 
30 min  %  99.94  99.96  99.96  99.98  99.99  99.99  99.99  99.99  99.99  
40 min  %  99.97  99.98  99.98  99.99  99.99  99.99  99.99  99.99  99.99  
50 min  %  99.98  99.99  99.99  99.99  99.99  99.99  99.99  99.99  99.99  
THM formation  20 min  μg/L  3.54  3.99  4.00  7.08  7.97  7.99  8.85  9.97  9.99 
30 min  μg/L  5.22  5.90  5.91  10.45  11.79  11.83  13.06  14.74  14.79  
40 min  μg/L  6.85  7.75  7.78  13.70  15.50  15.56  17.13  19.37  19.45  
50 min  μg/L  8.42  9.55  9.60  16.85  19.10  19.20  21.06  23.88  23.99 
In addition, the THM concentration and disinfection efficiency associated with a given contact time, L/W ratio, and applied chlorine dose are also presented in Table 5. Prior to chlorine input to the solution, reducing the TOC concentration is definitely the best engineering practice (BEP) for controlling the THM formation, rather than improving the hydraulics through the geometric design of the reactor. On the other hand, the results imply that the geometry of a chlorine contactor can effectively control the disinfection efficiency (not on the formation of THMs), especially for large chlorine doses and a high L/W ratio.
Furthermore, in order to reduce the formation of THM, the ozonation process is sometimes utilized in disinfection for the purpose of eliminating bacteria, viruses, cysts, as well as organic compounds, especially, the precursors of disinfection DBPs [27]. The performance of the ozone disinfection processes is evaluated by the residual ozone and contact time. Therefore, factors such as the efficiency of gas–liquid contact, chemical reactivity of the raw water toward ozone, susceptibility of microorganisms, and hydrodynamic characteristics of the contactor can influence the disinfection performance of an ozonation process. Several researchers have shown that ozonation prior to chlorination can lower the formation potential of THM (trihalomethanes) and HAA (haloacetic acid) [28, 29].
5. Carbonation process
5.1. Model development
5.1.1. Process chemistry
Carbonation is generally classified as a heterogeneous reaction containing gas, liquid, and solid phases. CO_{2} fixation by accelerated carbonation has been regarded as mass transfer limited reaction (i.e., filmdiffusion controlled) according to the findings reported in the literature [3032]. Therefore, different types of approaches such as physical intensification [33] and biological or chemical activation [34, 35] were recently carried out to improve the mass transfer and reaction kinetics. Based on this idea, a rotating packed bed (RPB) reactor has been introduced to improve the mass transfer rate among phases due to its high centrifugal forces and great micromixing ability. RPB, the socalled highgravity (sometimes called the “HIGEE”) process, is able to generate high acceleration via centrifugal force so the mass transfer between gas and liquid and even between liquid and solid can be enhanced. Since RPB can provide a mean acceleration of hundreds, and even thousands, of times greater than the force of gravity, it can effectively lead to the formation of thin liquid films and micro or nanodroplets [3638]. Therefore, the volumetric gas–liquid mass transfer coefficients (K_{G}a) are an order of magnitude higher than those in a conventional packed bed, leading to dramatic reductions in equipment size over that required for equivalent mass transfer in a gravityflow packed bed [36, 38, 39].
According to the findings in the literature [40], the CO_{2} consumption in bulk solution was mainly attributed (over 98 % contribution) to carbonation reaction with calcium ions in solution leached from the steel slag to form calcium carbonate precipitates, where the stoichiometric formula was briefly presented as Equation (45). In this case, the stoichiometric coefficient (
From the gasphase point of view, the CO_{2} removal efficiency (η) for a carbonation process can be calculated as Equation (46):
where the
From the solidphase point of view, the carbonation conversion (X_{B}, %) of the solid can be defined as the ratio of reacted CaO fraction with CO_{2} to the original total available CaO content in solid before carbonation., as expressed in Equation (47):
where Δm_{CO2} is the weight gain due to CO_{2} carbonation per dry weight of the solid sample. MW_{CaO} and MW_{CO2} are the molecular weight of CaO and CO_{2}, respectively. CaO_{total} is the total available CaO content in the solid before carbonation.
5.1.2. Ratedetermining steps
The shrinking core model (SCM) has been utilized to determine the ratelimiting steps in a heterogeneous reaction because of its conceptual and mathematical simplicity [41]. In the SCM, it is assumed that the reaction occurs at the outer skin of solid particle and then proceeds at a narrow front [24]. The narrow front moves into the particle and leaves behind completely reacted product layer and/or reactivespeciesdepleted rims (i.e., ash layer). An unreacted core of material exists at any time, which shrinks in size during the reaction. Based on the above assumption, three factors might affect the reaction (i.e., ratelimiting steps) including (1) fluidfilm diffusion, (2) ashlayer (or product layer) diffusion, and (3) chemical reaction at the unreactedcore surface, as illustrated in Figure 8.
When the reactant transfer through the boundary layer of the liquid–solid interface is the ratelimiting step of a reaction, the relationship between the conversion (X_{B}, %) and reaction time (t, s) can be correlated with Equation (48):
where
When the reactant transfer through the ash layer is the ratelimiting step, the dependence of the reaction time on the conversion and the effective diffusivity of the reactant in the ash layer (D_{e}, cm^{2} s^{1}) can be described by Equation (49):
For simplicity, the particle radius (
If the
When the chemical reaction between reactants is the ratelimiting step, the relationship between the conversion and reaction time can be determined by Equation (51):
where k” (s^{1}) is the firstorder rate constant for the surface reaction.
5.1.3. Mass transfer coefficients
For determining the
where k_{G} is the gasside mass transfer coefficient (m s^{1}), k_{L} is the liquidside mass transfer coefficient (m s^{1}), H is the Henry’s law constant, and I is the enhancement factor.
On the other hand, the driving force between the saturated CO_{2} concentration in the bulk gas and the CO_{2} concentration on the surface of liquid film can be determined by shell mass balance over a thin film of fluid with the RPB as shown in Equations (53) and (54):
where ρ_{co2} is the CO_{2} mass density at the temperature of gas streams, M_{G} is the gas mass flow rate (kg s^{1}), C_{G}^{*} is the saturation concentration of CO_{2} in solution (mg/L), C_{G}^{’} is the concentration of CO_{2} in solution (mg/L), V is the volume of packed bed (m^{3}), h is the height of packing bed (m), and r is the radius of packed bed (m).
In this case, by substitution of Equation (54) into Equation (53), the
Furthermore, Kelleher and Fair (1996) have obtained an overall volumetric mass transfer coefficient in terms of the area of transfer unit (ATU) for the gas side from the literature [44]. The height of transfer unit (HTU) and ATU can be calculated from the experimental data using Equations (56) and (57), respectively [45]:
It is noted that the
5.2. Engineering practice
5.2.1. Determination of diffusion coefficients
According to the findings reported by Changet al. [47], the accelerated carbonation of the basic oxygen furnace slag (BOFS) in an RPB is controlled by the ashlayer diffusion mechanism because the experimental data exhibit a good correlation with Equation (51). Similar findings were observed in the literature [48], which suggests that the carbonation reaction should be ashlayer diffusion controlled. The particles exhibited an ash layer of calcium carbonate (CaCO_{3}) product after carbonation, which could be classified as a shrinking core. As shown in Figure 9, the
In addition, Changet al. [41]) reported an effective diffusivity of steelmaking slag in a slurry reactor of 2.9×10^{7} to 7.3×10^{7} cm^{2} s^{1} according to the results of the SCM. It was found that the rate of CO_{2} molecular diffusion was not significantly enhanced by the RPB because the very similar temperatures were operated in the two reactors. Therefore, the
where
5.2.2. Determination of overall gasphase mass transfer coefficients
Based on the above illustration, the K_{G}a value in an RPB can be estimated by the key operating parameters including gas diffusivity, gas superficial velocity, gas density, gas viscosity, liquid superficial velocity, liquid density, liquid viscosity, centrifugal acceleration, total specific surface area of packings, and effective diameter of packings. Therefore, the mass transfer characteristics of RPB (such as K_{G}a value and HTU) can be obtained by the nonlinear regression of experimental data [49]:
where the ranges of the dimensionless groups in this correlation should be as follows:
Table 6 presents the







1  462.1  0.33  9.33  35.4  10.12  0.672 
2  462.7  0.34  9.33  36.4  10.61  0.650 
3  505.1  0.82  6.50  126.2  12.35  0.715 
4  487.3  0.44  9.17  48.0  11.21  0.680 
5  534.4  0.72  6.33  113.7  12.77  0.632 
6  360.2  0.42  9.00  46.7  10.06  0.699 
On the other hand, several studies have utilized different kinetic models to determine the ratelimiting step of mineral carbonation [31, 41]. In fact, solid–liquid mass transfer is particularly important in mineral carbonation and, in many cases, the rate limiting factor [50, 51] because minerals in solid matrix dissolve partly and passive layers are formed, gradually increasing resistance to mass transfer and eventually leading to incomplete conversion. In general, the solid–liquid mass transfer coefficient is occasionally correlated as itself, where such correlations are specific to the system under consideration and are not generally applicable [50].
In spite of the significant differences between RPB and traditional packed column, penetration theory was still capable of describing the liquidside mass transfer behavior fairly well in RPB [52]. These correlations are most often expressed in terms of dimensionless numbers in the form of a power series. For instance, penetration theory can be applied to RPB to yield as Equation (64):
where the Grashof number which represents the ratio of gravitational to viscous forces can be determined by Equation (65) and the
The Grashof (
and
It is usually difficult to obtain mass transfer coefficients separated from volumetric mass transfer coefficients k_{L}a_{e} and k_{G}a_{e} since the effective interfacial area between the liquid and vapor phase is usually not known [38]. Several correlations have been reported to estimate the wetted surface area (a_{w}), among which Tung and Mah [52] found that the a_{w}/a_{t} predicted value by Perry and Chilton [54] is “reliable” under highgravity RPB process as shown in Equation (69).
The assumption of the above equality is that for packing materials with small static holdup (i.e., large packing size), the wetted area (a_{w}) may equal the interfacial area (a_{e}). In addition, Basic (1992) has correlated the wetted surface area of packing by combining experimental data of liquid holdup and estimation of mean liquid film thickness, as shown in Equation (70):
5.2.3. Balancing mass transfer performance and energy consumption
Balancing the gasphase mass transfer rate for the highgravity carbonation process with a relatively lower energy consumption is also an important task for carbonation process. As shown in Figure 10, the favorable operating conditions can be determined via graphical presentation, where the energy consumptions including rotation, blowers, air compressors, and pumps were measured during operation in terms of kWh per ton CO_{2} capture by the RPB. It suggests that a centrifugal acceleration should be maintained at 475 m/s^{2} for a relatively lower energy consumption (L1
6. Summary
A clear understanding in process chemistry, reaction kinetics, and mass transfer is the essential requirement to achieve the best achievable technology. Many industrial and/or waste treatment processes are involved in diffusioncontrolled mechanism. To increase process efficiency and reaction rate, the mass transfer of those processes should be improved, which can be approached by development of prediction model using theoretical theory such as Fick’s law or penetration theory. For those heterogeneous processes, especially the system containing gas, liquid, and solid phase, assumptions should be appropriately made to simplify the complex governing equation. Lastly, from the viewpoint of engineering practice, the energy consumption of a process is also an important factor to optimize the overall process, thereby reducing the operation costs.
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