Globally Optimal Design and Operation of a Continuous Photocatalytic

Ind. Eng. Chem. Res. 2008, 47, 3591-3600

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Globally Optimal Design and Operation of a Continuous Photocatalytic Advanced Oxidation Process Featuring Moving Bed Adsorption and Draft-Tube Transport David M. Follansbee,† John D. Paccione,§ and Lealon L. Martin*,†,‡ Department of Chemical and Biological Engineering and Department of Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180, and Department of EnVironmental Health Sciences, School of Public Health, State UniVersity of New York at Albany, Albany, New York 12201

In this paper, we present a mathematical model for a continuous, integrated advanced oxidation process consisting of single-component adsorption, UV photocatalytic mineralization, and draft tube transport. The model is developed and employed as part of a strategy to determine optimal values of selected process design parameters and to identify suitable process operating conditions that may lead to an overall robust performance. An optimization-based algorithm is presented and applied to the design of a water purification system utilizing a composite titanium dioxide/activated carbon photocatalyst adsorbent immobilized on a silica substrate for the degradation of reactive red (RR) dye. A sensitivity analysis is performed to identify qualitative trends implicit in the proposed mathematical model and to measure the robustness of the resulting design over a range of process operating conditions. 1. Introduction Increasing demands for potable drinking water, increasingly stringent water quality standards, and the seemingly perpetual discovery of new biological and chemical contaminants all provide incentives to investigate novel water treatment strategies. Over the past few decades, several technologies (e.g., reverse osmosis, activated carbon, and air-stripping processes) have been employed to reduce the concentration of contaminants, particularly organic compounds, in potable water. Although these processes can be effective in certain applications, they typically result in the generation of highly concentrated waste streams that must be processed further. A new and increasingly more attractive method is advanced oxidation via photocatalysis (advanced oxidation mineralizes organic compounds). Several books and reviews have been devoted to applying this fundamental technique to water and air purification.1-4 Recently, advanced oxidation processes employing titanium dioxide-based photocatalysis have received considerable attention as a viable treatment strategy for water contaminated with organic compounds.5 It has been established that UV-irradiated titanium dioxide (TiO2) in the presence of water produces strong oxidizing species that can mineralize organic compounds into carbon dioxide, hydrogen, and (for halo-organics) halogen ions.6 Bench-scale studies have been demonstrated to have degradation half-lives in the 2-4 min range for organic compounds using small concentrations (surface area per unit volume ∼0.12 m2/ L) of fine TiO2 particles (∼1.2 µm).7 It has also been shown that the rate of organic mineralization (oxidation) increases exponentially with an increase in TiO2-water surface concentration.8 These attributes, along with low cost and abundance * To whom correspondence should be addressed. Permanent address: Chemical and Biological Engineering Department, Rensselaer Polytechnic Institute, 110 8th Street, Troy, New York 12180. Phone: (518) 276-3327. Fax: (518) 276-4030. E-mail: [email protected]. † Department of Chemical and Biological Engineering, Rensselaer Polytechnic Institute. ‡ Department of Decision Sciences and Engineering Systems, Rensselaer Polytechnic Institute. § State University of New York at Albany.

of TiO2, have popularized TiO2 photocatalysis in the water treatment, industrial, and academic communities as a potentially high impact advanced oxidation process technology. There are several types of heterogeneous reactor configurations for photocatalytic advanced oxidation processes, including (re-circulating) fluidized beds, hydraulic transport loops,9 fixed beds, and draft tube spout-fluid beds.10 In operating these systems, a ideal photocatalyst should be mechanically robust, readily separated from the reaction stream, rapidly regenerated, and capable of maintaining its activity until the oxidizer can react with the contaminant species. Many of the studies performed on photocatalytic advanced oxidation systems employ fine particles of TiO2 on the order of 1-10 µm. Such systems pose two major problems: (i) expensive downstream particle/ water separation and (ii) turbidity inhibits UV light penetration (even for dilute particle suspensions of TiO2 in water), limiting catalyst regeneration. Fixed bed designs use TiO2 coated surfaces with continuous UV illumination directly onto the exposed surface.11 Although no fluid-particle separation step is required, both the fine particle and fixed bed systems exhibit low surface area to volume ratios. Furthermore, these system designs typically possess inherent mass transfer and kinetics bottlenecks and thus impose difficulties when scaled to meet variable water demands. TiO2 can be immobilized onto larger particles that function as catalyst supports (e.g., silica, zeolites, and other metal oxides) and operated within a fluid or fluidized bed.12 Such reactor configurations can provide large surface areas as well as the benefit of time and energy-efficient solid-phase separation. However, limited light penetration into the bed is a major drawback and such configurations are designed for batch or semi-batch modes. In this paper, we introduce a novel continuous photocatalytic reactor configuration that addresses the difficulties of particle separation, process scalability, and flexibility experienced in many of the aforementioned reactor designs, while providing high surface contact area between the solid particle and aqueous contaminant phases. The key technological differentiation is a novel fluid-particle contacting device called the draft tube spout fluid bed employed for contaminant surface adsorption and particle transport. For clarity, a draft tube spout-moving bed

10.1021/ie071275r CCC: $40.75 © 2008 American Chemical Society Published on Web 04/10/2008

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Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008

Figure 3. Process diagram. Figure 1. DTSMB pilot plant.

Figure 2. Process schematic.

reactor (DTSMB) has the same functionality as a draft tube spout fluid bed reactor. The purpose for referring to this multiphase reactor as a DTSMB in this paper is for better conceptual understanding of a “moving” bed compared to a “fluid” bed. In the next section, we provide a complete description of the continuous adsorptive photocatalytic reactor to be employed for the advanced oxidation of organic contaminants. We then develop a theoretical model that captures the steady-state design and operation of the proposed system. In the following section, we define a problem statement for photocatalytic oxidation applications, formulate this problem statement into a nonlinear mathematical program (NLP) using the developed model, and outline a methodology to identify globally optimal solutions. Finally, we demonstrate the strength of the proposed methodology through an illustrative example involving the treatment of water contaminated with reactive red (RR) using a TiO2-activated carbon (AC) catalyst supported on large silica particles. Using data from the literature, the NLP formulation is then solved to obtain optimal design and process operating conditions at the minimum utility cost. 2. Process Description A prototype of the featured reactor is shown in Figure 1 and a schematic of the process is shown in Figure 2. The system is based on DTSMB technology and heterogeneous photocatalysis

with a photocatalytic-adsorbent composite immobilized on large particles. The reactor design consists of an annular moving packed bed (water purification region) with an internal draft tube for hydraulic transport of the catalyst for regeneration. Contaminated water to be treated enters the annulus through one of several mounted auxiliary flow channels. The contaminantrich stream passes (upward) through the annulus while in counter-current contact with the downward moving photocatalyst-adsorbent catalyst. Treated water exits the column through the outlet of the annulus, whereas spent adsorbent moves to the bottom of the bed for transport (via hydraulic jet flow) through a draft tube to a UV exposure chamber. Particle exposure to UV serves two simultaneous functions (i) activation of the photocatalytic component of composite to promote surface oxidation reactions (to mineralize contaminant molecules) and (ii) regeneration of the adsorbent component. Particles re-enter the water purification region (at the top of the annulus) to again adsorb contaminant from the aqueous phase. The system described is characterized by the following attractive features: • The counter-current configuration in the water purification region allows for high interfacial contact surface area between solid and aqueous phases. • The treatment capacity and quality specifications are governed by the initial catalyst bed height and fluid mass flowrate, which can be manipulated dynamically by the addition or removal of particles. • Immobilized photocatalyst-adsorbent is supported on large, dense particles thus facilitating its seamless in-process separation from the aqueous phase. • There are two mechanisms to vary catalyst UV-light exposure: (a) modification of UV light intensity, and (b) convective manipulation of the phase density within the chamber. • The system allows for an overall high degree of operability, flexibility, and scalability to meet a range of throughput demands, and influent water contaminant levels. Our study is focused on analyzing key design parameters and operating conditions for systems modeling. 3. System Modeling This system is built around the characteristics of a DTSMB reactor, which has two defining components: (a) the draft tube allows for vertical hydraulic transport of the particles in either a lean-phase or dense-phase flow regime, and (b) a moving packed bed of particles surrounding the draft tube moving in a downward motion.13 The decoupling of the auxiliary flow from

Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3593

the particle circulation rate allows for the decomposition of this system into three individual processes: (i) adsorption in the annular bed, (ii) multiphase transport in the draft tube, and (iii) photocatalytic reaction in the UV exposure chamber as shown in Figure 3. These three models are linked through the recirculation of particles within the system. 3.1. Draft Tube Transport Model. The hydrodynamics of the vertical transport of solid particles through smooth pipes has been extensively studied in the literature.9,13-19 The model used to describe multiphase vertical transport of fluid and particles is typically a one-dimensional, two-fluid model with individual fluid and particle momentum balances. The steady state one-dimensional mixture momentum equation with constant fluid properties is given by9,13

where ff is the fanning friction factor and fp is the particle friction factor given by fp ) 7.33 × 10-3 V-2 for 1.94 mm diameter particles. A total dimensionless friction factor that accounts for particle and fluid wall friction can be written based on eqs 6 and 7.

d d Ff (Tu2) + Fp [(1 - T)V2] ) dz dz dp - [(1 - T)(Ff - Fp) + Ff]g - Ff - Fp (1) dz

QfT ) uTAT

(8)

Gp ) V(1 - T)FpAT

(9)

To minimize the complexity of the draft tube model the following assumptions are made:9 ‚ Nonuniformities in fluid/particle flow patterns near the wall are negligible. ‚ Particle acceleration in the draft tube is neglected for the region of interest.13 Because the acceleration region for liquid-solid systems is very small as stated in the last assumption the inertial terms may be neglected so that eq 1 becomes

-

dp - [(1 - T)(Fp - Ff) + Ff]g - Ff - Fp ) 0 dz

(2)

and defining the dynamic pressure as p(z) + Ffgz in eq 2 gives

-

dP - [(1 - T)(Fp - Ff)g] - Ff - Fp ) 0 dz

(3)

Equation 3 can be divided into individual nonaccelerating phase equations as follows Fluid:

( dPdz ) ) - β(u - V) + F

T -

2

(4)

f

Particle

( dPdz ) ) - β(u - V) (1 - )(F - F )g + F

(1 - T) -

2

T

p

f

p

(5)

The most dominant contributions in these conservation equations arise from frictional, gravitational, and inertial effects. These effects are sufficiently captured through relations between the pressure gradient in the draft tube, the voidage in the draft tube, and the interphase drag between the fluid and the particles. The fluid-wall friction (Ff) and particle-wall friction (Fp) can be written in dimensionless pressure loss terms referenced to the static head of the particles in the fluid.9

2ffFfu

2

F/f )

DT(1 - T)(Fp - Ff)g

F/p )

2fpFpV2 DT(Fp - Ff)g

(6)

(7)

F* ) 1 + F/p +

1 - T Ff* T

On the basis of these assumptions, the following equations are used to describe the behavior of particle mass flowrate (Gp), fluid volumetric flowrate (QfT), voidage within the draft tube (T), and the particle and interstitial fluid velocities within the draft tube (u and V).

[ ((

) )]

T - mf β 1 ) (1 - c2) + 1 - λ + c1 βmf λ 1 - mf

[

]

Umf T3(1 - T) βmf Vs ) u - V ) T 3(1 - ) β mf mf

2 1/2

(10)

1/2

xF*

(11)

Equations 8 and 9 describe the volumetric flowrate of liquid and the mass flowrate of particles within the draft tube. Equation 10, developed by Grbavcic et al.,9,13 provides a variational relationship for the drag coefficient as a function of voidage β(). Constants c1 and c2 are variational constants and λ is a Lagrange multiplier. This equation accurately correlates the interphase drag coefficient for the range of voidage between minimum fluidization to lean phase flow. Equation 11 describes the relationship between slip velocity and the other variables in the draft tube that describe the flow. Slip velocity is defined as the relative velocity between the fluid and solid phases. 3.2. Annular Bed Model. Transfer of contaminant from the aqueous medium to the solid phase occurs by a two-step mechanism (i) diffusion through the boundary layer from the bulk liquid to the particle surface, and (ii) adsorption onto the porous surface of the adsorbent. However, it is well-known that physical adsorption occurs instantaneously and diffusion from the bulk to the particle surface is the rate-limiting step.20 Therefore, adsorption of the contaminant onto the surface of the adsorbent depends primarily on the overall mass transfer of the contaminant from the rich (liquid) phase to the lean (solid surface) phase and phase equilibrium. For this reason, the annular bed is modeled as a single-stage, continuous mass exchanger. It has been shown by Manousiouthakis and Martin21 that counter-current mass exchange (analogous to a heat exchange) configurations maximize interfacial contact area between rich and lean phases. However, the simple linear relationship for phase equilibrium employed in their study is not sufficient for this model. Instead, a Langmuir adsorption isotherm is used

x ) xmax

KAy 1 + K Ay

(12)

where KA represents the equilibrium adsorption constant, and xmax represents the maximum adsorptive capacity of the solid phase and corresponds to complete monolayer saturation of contaminant on the surface of the particle. For the case of the Langmuir isotherm, the liquid side concentration (y) is represented by the mass fraction of the contaminant in the rich (liquid)

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phase, resulting in the following model for mass transfer in the annulus

M ) QfAFf∆y ) QfA(yi - yo)

(13)

QfAFf∆y ) Gp∆x

(14)

M AaKla∆ylm

(15)

HA ) ∆ylm )

[ ( )( {[ (

)] [ )] [

(

)] )]}

KAxi xmax - xi KAxo xmax - xo yi - yo xmax xmax xmax xmax KAxo xmax - xo KAxi xmax - xi ln yi / yo xmax xmax xmax xmax

(

(16)

Equation 13 is the mass load of transferable species within the bed. It describes the amount of contaminant that is transferred between phases. Equation 14 represents the conservation of mass for the contaminant transferred between phases. Equation 15 represents the minimum required height of the annular bed (mass exchanger column) based on the overall mass transfer coefficient, mass load, and log mean concentration difference. This equation is analogous to the surface area required in a heat exchange column. Equation 16 is the log mean (geometric) concentration difference, overall driving force of the transferable component from the rich phase into the lean phase throughout the mass exchanger. Note that the Langmuir adsorption (eq 12) is used to describe the equilibrium between the rich and the lean phase. 3.3. UV Chamber Model. The adsorbed contaminant is degraded as the particles come into contact with the ultraviolet light as they fall through the UV chamber. Therefore, the UV chamber model must account for all of the reaction kinetics for organic degradation by the photocatalyst. The following assumptions employed to develop the reactor design model are as follows: • Because the contaminant has been adsorbed onto the particles prior to reaction, there are no mass-transfer limitations. • The UV chamber is assumed to be a plug flow reactor with the particles falling at terminal velocity. • There are no wall effects or effects due to particle-particle interactions within the UV chamber. This last assumption of no particle-particle interactions holds because the sedimenting density of particles throughout the UV chamber is lean through the range of valid particle mass flowrates. Only at higher particle mass flowrates does the particle density transition from lean phase to dense phase. Littman et al. demonstrated that for the sedimentation of glass particles through air the effective drag coefficient varied by 13% from the standard drag coefficient due to wake generation.19 However, with higher density fluids (such as water), this variation will be much lower, allowing for the assumption of no particle-particle interactions to hold true. With these assumptions, the reactor design equation becomes

[ ]

z xi ) xo exp -k Vt

(17)

The reaction rate k, however, is a function of the UV intensity on the particle surface, which decays according to the BeerLambert Law.22

[ ] DUV 2

Io ) I exp R

(18)

where Io is the intensity of the light at the tube wall, I is the intensity of the light at the center of the tube, and R is the absorption coefficient of UV radiation in a given material, thus giving

R ) w(1 - UV) + Ro

(19)

Gp ) Vt(1 - UV)FpAUV

(20)

Equation 19 describes the relationship between UV absorption and voidage within the UV chamber where w is the extinction coefficient specific for a given light source through a designated medium. Equation 20 defines particle mass flowrate as a function of the voidage within the UV chamber. Equations 18-20 demonstrate that as the particle mass flowrate increases the UV chamber becomes more densely populated with particles, therefore causing UV light to rapidly decay as it travels radially through the chamber. It is obvious that in the limit as the voidage in the ultraviolet reactor tends to unity (no particles in the stream), the absorption of the UV light becomes equal to the absorption in the pure medium; and for the case of pure water, the absorption of UV light is very close to zero. Therefore, the power requirement for the UV chamber is modeled on the basis of the intensity of light emitted at the wall and the exposed particle surface area

W˙ UV ) IoπDPz(1 - UV)

(21)

3.4. Pressure Drop and Pumping Requirements. The pressure drop of the fluid through the draft tube is obtained from the Blasius equation for turbulent flow in smooth circular pipes, given the following assumptions: • Newtonian fluid with constant physical properties • No fluid phase change within the system • the pipe is smooth with nonaccelerating fluid • the pressure drop due to particle friction is negligible

Re )

DTFfQfT

(22)

µ f AT

Equation 22 is the Reynolds’ number for flow through circular pipes and is employed here to describe the flow regime in the pipe. The Blasius formula is used to provide an empirical correlation between the fluid friction factor and the Reynolds number for turbulent flow regimes

f) ∆pT )

0.0791 Re1/4

(23)

()

4 LFf f QfT 2 DT AT

2

(24)

Equation 24 is the pressure drop over the length of the draft tube in the direction of the fully developed flow. Static pressure drop is not considered here because the height difference is accounted for in the pump model

hpump )

( )

2 ∆pT 1 QfT +L + Ffg 2g ATFf

(25)


Equation 25 is a simplified steady-flow energy balance where the kinetic-energy correction factor has been assumed to be unity (a common assumption for turbulent flow). Turbine head and friction loss head have been neglected. Finally, the power requirements for pumping is given by

W˙ pump ) FfQfT ghpump

(26)

4. Mathematical Formulation and Solution Strategy Having quantified the mass exchange model, the transport model, and the kinetics model for a photocatalytic water treatment process, we describe the proposed methodology to determine operating conditions based on the following problem statement: GiVen adsorptiVe mass transfer rates, degradation rates for a contaminant species and specific adsorbent and photooxidatiVe materials, a steady-state model quantifying oVerall input-state-output relationships of the system, the influent rate and concentration (QfA, yi), and the target effluent concentration (yo); determine optimal design parameters and operating conditions that satisfy target outlet conditions and minimizes utility cost. 4.1. NLP Formulation for the Optimal Design of DTSMB Advanced Oxidation Processes. The proposed mathematical model employs both empirically and theoretically determined equations. These equations outline the conservation of mass, energy, and momentum throughout the integrated system. However, constraints must be applied to ensure that the mathematical model will be consistent with physically relevant hydrodynamic phenomena, namely

Gp < (1 - min )AaFpVamax

(27)

vc < T < 1

(28)

u g 1.5Vt

(29)

ur )

Umf ; mf

Vamax )

( )

L z 1 + + γ4W˙ δUV + V Vt

2 + γ6Hδ3 (31) γ5W˙ δpump

As written, ζ, can be suitably modified to represent utility cost, capital cost, or total annualized cost. The capacity parameters W˙ UV, W˙ pump, and H represent the power requirement for the UV light reactor, the power requirement for the jet flow pump, and the annular bed height, respectively. The coefficient γ1 represents the cost required to operate the machinery, given in units of $/kWh, whereas the coefficients γ2 and γ3 correspond to the cost required to provide the total amount of desired catalyst (annular bed volume plus particle circulation). γ2 has units of $ per particle diameter per catalyst lifetime and γ3 has units of $ per kg of circulating particles over the catalyst lifetime. The last three terms represent the capital cost and demonstrate how the capacity change of the system effects the capital cost in a nonlinear fashion. However, in this paper, we do not consider how capital cost is affected by changes in operating conditions and design; therefore, the coefficients γ4,5,6 are set equal to zero. Having defined the feasible region based on models developed for draft tube transport, UV degradation of contaminant, and mass exchange in the annulus; and having defined a general cost objective function, the NLP for the DTSMB system is as follows

(P1) min ζ QfT,Gp

(32)

subject to

eqs 8-11 and eqs 13-26 eqs 27-29

Equation 27 places an upper limit on the solid circulation rate to make certain that the annular bed will not become fluidized due to non-settling particles. Equation 28 describes the range of operable voidages within the draft tube. At a voidage lower than vc, the particle phase is too dense and will collapse in the draft tube, causing unsteady transport. Equation 29 describes the lowest acceptable fluid velocity in order to guarantee transport of the particles. This lower limit was reported by Grbavcic et al. to be 1.35Vt for vertical transport of gas-solid flow;23 because we are interested in liquid-solid, this lower limit was increased slightly. To ensure that the true maximum solid circulation rate is being used, we should maximize the particle velocity in the annular bed. When the bed of particles is moving at its maximum speed, it is at the point just before the onset of fluidization corresponding to the minimum fluidization voidage mf. This is shown in eq 30 for any given annular flow rate, where ur is the residual velocity of particles at minimum fluidization in the annular bed.

V a + u a ) u r;

ζ ) γ1(W˙ UV + W˙ pump) + γ2H + γ3Gp

Umf - ua (30) mf

For this problem, we haven chosen an objective, ζ, as a means to quantify system cost.

positivity constraints on all decision variables 4.2. Solution Strategy: Interval Analysis and Direct Optimization. Selected parameters for this problem are listed in Table 1. Given these parameters, this problem has two degrees of freedom, which are chosen to be the jet flowrate (QfT) and the particle mass flowrate (Gp). These quantities are selected because they are easily measured and manipulated in the system. A method for globally optimizing a mathematical model similar in structure to the one being presented here was presented by Chen et al.24 Their solution strategy employed interval analysis to identify acceptable ranges (intervals) for all variables and subsequently optimize within this range. The interval for jet flowrate (QfT) is determined by evaluating its lower limit through eq 8, subject to inequality constraints eqs 28 and 29. The upper limit for jet flow rate is determined by eq 27, solving for the maximum particle mass flowrate with the minimum fluidization voidage, and then using (eqs 8-11) to determine the maximum jet flowrate at a voidage equal to vc. Given the evaluated interval for jet flowrate, the solution proceeds as follows: (i) Based on the interval (value) of jet flow, the interval (value) of interstial fluid velocity is solved for using eq 8 for a given voidage. The value for interstial velocity is a monotonically increasing function of jet flow rate, and thus the lower and upper bound of flowrate correspond to the lower and upper bound for the interstial velocity, respectively. (ii) The interval for the pumping power requirements is found by using the interval of the jet flowrate and (eqs 22-24). The

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Table 1. Design Parameters param Ff µf g DA DT DUV Dp AA AT AUV L z Vt Umf βmf mf vc c1 c2 λ ζ

value

units

1000 1.119 × 10-3 9.81 6 1 2 1.94 πD2a /4 πD2t /4 πD2UV/4 2.5 1.22 0.257 0.0205 1.74 × 106 0.447 0.87 0.9984 -0.06014 -0.9418 9.24

kg/m3 Ns/m2 m/s2 in. in. in. mm m2 m2 m2 m m m/s m/s kg/m4

ref

9, 13 9, 13 9, 13 23 9, 13 9, 13 9, 13

$/kW

pumping requirements are proportional to approximately jet flowrate cubed (W˙ pumping ∝ Gf3D) (iii) Using the lower bound of interstial fluid velocity in conjunction with the lower bound for T (eq 28), the lower bound for particle velocity is found through eqs 10 and 11. This lower bound for particle velocity represents the slowest the particles can be vertically transported up the draft tube without collapsing. (iv) This lower bound for particle velocity translates into the lower bound for particle mass flowrate by eq 9. Now that the interval for particle mass flowrate is completely known, the interval for particle velocity is found again using eq 9 for a given voidage. (v) The interval for the voidage in the UV reactor is determined using the interval for the particle mass flowrate in eq 20. This interval for voidage in UV chamber is used to determine the range of UV intensity at the wall (Io) of the reactor by using eqs 19 and 18. (vi) The interval for energy requirements for the ultraviolet light reactor is found by using the interval of the wall intensity and eq 21. There is an exponential relationship between energy required by the UV reactor and particle mass flowrate (PUV ∝ exp[GP]). (vii) The interval for particle mass flowrate is used to determine the interval for concentration difference of contaminant adsorbed on the catalyst particles (∆x) through eq 13. This interval for ∆x monotonically increases with respect to the interval for Gp. (viii) The interval for both xi and xo are found by using eq 17 and the interval for ∆x. (ix) The interval for the utility cost objective function is determined through the interval of pumping requirements and ultra violet reactor energy requirements using eq 31. (x) Last, the interval for the height objective function is found through the intervals of ∆ylm, xi, and xo based on eq 15. Given the procedure for determining the interval (values) for all of the decision variables, we designed a search algorthim, (for given ranges of QfT and Gp) to find globally optimal values for ζ. A detailed description of the algorithm is shown by the pseudo code in Figure 4. The direct search method employed guarantees that all possible combinations of QfT and Gp from QfT,min to QfT,max, over the entire range of valid Gp, are considered. This systematic procedure ensures that the globally optimal solution is found and demonstrates the trend of ζ with changing input conditions.

Figure 4. Pseudo code for optimization algorithm. Table 2. Material Parameters and Delivery Requirements param Fp yo yi ∆y QfA k w m xmax Kla

value 2507 1 10 yi - yo 0.5 0.00833 300 602430 0.272 0.00615

units

ref

kg/m3 ppm ppm ppm GPM s-1 m-1 units kgcontaminant/kgparticle s-1

25 26 26 appendix

4.3. Degradation of Reactive Red Using a Composite Photocatalytic Adsorbent. To demonstrate the proposed model for a continuous water treatment process, the following system has been chosen: (a) reactive red is the organic compound that will be oxidized, and (b) the catalyst particle is chosen to be TiO2 photocatalyst loaded with activated carbon (AC) on a SiO2 substrate. The organic dye (reactive red) selected has been used extensively in previous photocatalytic and adsorption studies25,26 thus making it an ideal model contaminant based on the amount of data available in literature. TiO2/AC composites have also been extensively studied in literature for both adsorption and oxidation properties, and for synergistic effects.26-32 The use of the SiO2 substrate allows for the control of physical properties of the particles (e.g., density, size, and porosity).33 As catalyst loading increases, the first-order reaction rate constant approaches a value of 0.0013 s-1 ,22 and this reaction rate corresponds to a light intensity of 220 W/m2 by interpola-


Figure 7. Ultraviolet light decay through the UV chamber with Gp equal to 0.04, 0.24, 0.34, and 0.54 kg/s. Figure 5. Utility cost as a function of particle mass flowrate where QfTnorm ) QfT/QfTmax and QfTmax is 1.55 l/s.

Figure 8. Annular bed height over full range of particle mass flowrate.

Figure 6. Effect of jet flowrate (QfT) on utility cost.

tion.25 For this reason, the light intensity at the center of the UV chamber is set to this intensity. The following model parameters affect the overall masstransfer coefficient (Kla), Langmuir equilibrium adsorption constant (KA), maximum capacity for contaminant adsorption (xmax), and the UV extinction coefficient (w). All of the system parameters for this model are either determined theoretically, referenced from literature, or specified as design parameters; they are given in Table 2. The annular flowrate (QfA), influent, and effluent concentrations (yi and yo) are determined on the basis of delivery requirements: 0.5 (GPM), 10, and 1 ppm, respectively. These deliverables are used to test the model. 5. Results and Discussion The algorithm was coded into Matlab and executed on a 2.2 GHz Intel Core 2 Duo processor with a CPU time of 46.0934 s. Optimal solution results suggest an operating range of 0.311.55 L/s for jet flowrate (QfT) and 0.04-0.56 kg/s for particle mass flowrate (Gp). The range of operation and sensitivity of the system to the chosen operating conditions can be seen in Figures 5-8. The utility cost of the system, unlike annular bed height, is sensitive to changes in operating conditions. Figure 5 shows how the utility cost increases as the particle mass flowrate

increases for 3 different jet flowrates. This figure appears to only represent one jet flowrate; however, by focusing on the lower range of mass flow rates and plotting utility cost on a log basis, the distinction between jet flowrates can be established as shown in Figure 6. As expected, increasing the jet flowrate has an impact on the utility cost because of the increased pump work that is needed. This figure also represents the range of operability for lower jet flowrates. When operating the system at a normalized jet flow rate of 0.2, the particle mass flowrate range of operability is 0.04-0.09 kg/s. The reason for this is that for a higher mass flowrate to be achieved, the particle voidage must reach values that are unrealistic for steady flows. The inability to distinguish between flowrates in Figure 5 suggests that the dominant contribution to utility cost is the power requirement for the UV chamber. This high power requirement is a result of the exponential decay of UV light passing through a densely packed chamber. Recalling that the intensity of the middle of the chamber was set constant, an exponential increase in wall intensity with increasing particle mass flowrate, as shown in Figure 7, is expected. Optimal design results for the annular bed height, within operation limits, indicate a range from 52.475 in. to 52.625 in. The curve in Figure 8 represents the minimum bed height required for a given particle mass flowrate. Designs with bed heights and particle circulation rates that fall below this curve cannot satisfy the target concentration specification of 1 ppm imposed on the effluent stream, whereas design and operating conditions that fall above the curve are feasible yet suboptimal. When operating on the design curve in the lower rage of mass

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Figure 9. Change in annular bed height with respect to changes in initial concentration at minimum utility cost operating conditions.

formulation into a NLP mathematical program given our design problem statement, and (iv) its globally optimal solution through a hybrid interval analysis/(polynomial-time) search algorithm, followed by (v) sensitivity analysis. We demonstrated our methodology with a design and operation case study, featuring draft-tube spout moving bed technology, for the cleanup of a water stream contaminated with organic acid (reactive red) using 1.94 mm particles composed of TiO2 photocatalyst integrated with activated carbon immobilized on silica beads (substrate). Our solution strategy identified successive globally optimal particle circulation rates and draft-tube fluid flowrates at minimum utility cost based on transport costs (pumping requirements), photocatalytic costs (UV power requirements), and catalyst utilization. Our sensitivity analysis revealed that UV chamber power requirements, for particle regeneration, represent the dominant contribution to utility cost for all feasible processing conditions considered. This suggests that photocatalytic activity and efficiency may be a key factor in drastically reducing operating costs in (continuous) advanced oxidation processes. On the contrary, our analysis revealed minimal changes in particle bed height (for adsorption) over a range of particle circulation rates within the system. This implicates that (i) DTSMB technology allows for independent regulation of adsorption and oxidation processes through transport; and (ii) capital costs changes are negligible within specific regimes of DTSMB operation. Appendix: Overall Mass Transfer Coefficient

Figure 10. Change in annular bed height with respect to changes in influent flow rate at minimum utility cost operating conditions.

flowrates the annular bed height should be monitored very carefully because slight perturbations in the annular bed height correspond to large changes in mass flowrates. A sensitivity analysis was performed to determine the effects that the influent flowrate and concentration have on the system. Figure 9 shows a logarithmic increase in the annular bed height as the initial concentration of contaminant increases from 10 to 300 ppm. This suggests that changes in annular bed height are large at low concentrations, but small at higher concentrations. Figure 10 shows that the effective annular bed height of particles scales linearly with changes in the influent flow rate, suggesting that by assuming the bed diameter is fixed and the reactor size can accommodate the desired annular bed height dynamic capacity demands can be met by the addition or removal of more catalyst particles. 6. Conclusions We have presented and demonstrated a systematic methodology that identifies globally optimal key design parameter values and associated optimal operating process conditions for a continuous advanced oxidation process. Our methodology involves (i) system decomposition into adsorption, photocatalytic, and transport processes, (ii) process modeling, (iii)

The overall mass transfer coeffiecient is determined through Sherwood’s correlations for an estimated diffusivity coefficient. The use of these correlations slightly overestimates the overall mass transfer coeffiecient for adsorption of organic acid onto activated carbon. In characterizing solid-phase mass-transfer resistance, however, the empirical correlation used provides an accurate approximation even for flow much smaller than those considered in our study.34 The diffusivity coefficient is estimated using the WilkeChang equation and provides the diffusivity for small concentrations of species in aqueous liquid mixture:

xψBMBT

DAB ) 7.4 × 10-8

0.6

(33)

µVA

where ψ is the association parameter and is 2.6 for water and VA is the molar density of the contaminant. The following correlation gives the best prediction for the liquid-film mass transfer coefficient for water treatment processes:

Sh ) (2 + 0.664Re1/2Sc1/3)[1 + 1.5(1 - a)]

(34)

where the following dimensionless numbers (Sh, Re, and Sc) are equal to

dpKL DA

(35)

Ffmfuadp µf

(36)

µf FfD

(37)

Sh ) Re )

Sc )


The mass-transfer area per unit volume of the bed is given by

a ) 6(1 - mf)dp

(38)

With this estimation of the diffusivity coefficient and these correlations the overall mass transfer coefficient is determined to be 0.00615 s-1. Acknowledgment Financial support was provided by Army Research Office W911NF-06-1-0260 and is greatly appreciated. The authors also thank Prof. Howard Littman and Prof. Joel Plawsky of the Department of Chemical and Biological Engineering at Rensselaer Polytechnic Institute for fruitful conversations about multiphase transport. Nomenclature Ff ) fluid density (kg/m3) Fp ) particle density (kg/m3) µf ) fluid viscosity (Ns/m2) g ) gravitational acceleration (m/s2) DT ) inner diameter of transport tube (in) DA ) inner diameter of annular bed (in) DUV ) inner diameter of ultraviolet light chamber (in) Dp ) catalyst particle diameter (mm) AA ) cross-sectional area of annular bed (m2) AT )cross-sectional area of transport tube (m2) AUV ) cross-sectional area of ultraviolet light chamber (m2) Asp ) surface area of particles (mm2) L ) length of transport tube (m) z ) length of ultraviolet light chamber (m) ht ) residence time that particles spend in UV reactor (z/Vt) (s) Vt ) terminal velocity of catalyst particles (m/s) yo ) contaminant concentration in annular effluent (ppm) yi ) contaminant concentration in annular influent (ppm) ∆y ) concentration difference of contaminant in bulk fluid (yi - y o) x0 ) contaminant concentration on catalyst particle at bottom of annular bed (kgcont/kgpart) xi ) contaminant concentration on catalyst particle at top of annular bed (kgcont/kgpart) xz ) contaminant conentration on catalyst particle at end of UV chamber (xz ) xi) (kgcont/kgpart) ∆x ) concentration difference of contaminant on particle from top to bottom of annular bed (x0 - xi) xmax ) contaminant concentration on particle at total coverage of particle (kgcont/kgpart) QfA ) annular influent flowrate (GPM) QfT )fluid flowrate in the draft tube (GPM) Gp ) particle mass flowrate (kg/s) Umf ) superficial fluid velocity at minimum fluidization (m/s) u ) mean interstitial fluid velocity in the transport tube () U/) (m/s) V ) mean particle velocity in the transport tube (m/s) Vamax ) maximum particle velocity in annular bed (m/s) β ) fluid-particle interphase drag coefficient (kg/m4) βmf ) fluid-particle interphase drag coefficient in a particularly fluidized bed at minimum fluidization (kg/m4) mf ) bed voidage at minimum fluidization T ) transport tube voidage uv ) ultraviolet light chamber voidage vc ) vertical collapse voidage c1 ) variational constant in eq 13 c2 ) variational constant in eq 13

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ReceiVed for reView September 21, 2007 ReVised manuscript receiVed March 20, 2008 Accepted March 24, 2008 IE071275R

Globally Optimal Design and Operation of a Continuous Photocatalytic

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