Modeling and control of water disinfection process in annular photoreactors

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1 Modeling and control of water disinfection process in annlar photoreactors K. J. Keesman, D. Vries, S. van Morik and H. Zwart Abstract As an alternative or addition to complex physical modeling, in this paper transfer fnction models of the disinfection process in annlar photoreactors nder different flow conditions are derived. These transfer fnction models allow an analytical evalation of the system dynamics and the control strategies to gain frther insight while preserving the physical process parameters. For diffsive flow conditions a dead-time/padé approximation is proposed to find a low-order linear system description. iven the (approximate transfer fnctions with their physical process parameters, an analytical feed-forward feedback law is frther worked ot. I. INTRODUCTION VER the years chlorination has been the most O preferred disinfection process for water treatment. However, several investigations have proved that chlorine residals are toxic to the aqatic life [1], while at the same time some by-prodcts of chlorination have proved to be mtagenic. Therefore, the se of other disinfection techniqes which are friendlier to the environment and do not arise health concerns is increasing. It is known to scientists for nearly a centry that ltraviolet (UV light is an effective germicidal agent at certain wavelengths. However, the prodction cost of UV light was high. With the development of high intensity, long life lamps, interest in the se of UV as disinfection agent was renewed. Precise modeling of the disinfection process in a UV photoreactor reqires complex analysis of the radiation field []. This analysis needs to be linked to the modeling of the flow dynamics and the reaction kinetics. The models obtained are composed of very complicated differential eqations which reqire demanding nmerical comptations (see e.g. [3]. Conseqently, modeling of the disinfection process in a photoreactor is a qite complicated task. Moreover, phenomena sch as reactivation of disinfected microorganisms make the sitation even less Manscript received October 13, 6. This work was spported by the Dtch organiation for Scientific Research NWO, the Dtch Ministry of Economic Affairs and the technology fondation STW nder the project nmber WWI K. J. Keesman and D. Vries are with the Systems and Control rop, Wageningen University, Bornsesteeg 59, 678 PD Wageningen, The Netherlands (corresponding athor: K. J. Keesman, phone ; fax ; karel.keesman@wr.nl. S. van Morik and H. Zwart are with the Department of Applied Mathematics, University of Twente, P.O. Box 17, 75 AE Enschede, The Netherlands. straightforward. On the other hand, in practice simple models that preferably preserve prior knowledge are needed for fast online calclations. The methods that have been sed so far for the design of water disinfection systems are based on either complex physical models or empirical models. In this stdy or approach is to bild relatively simple mathematical models based on the prior knowledge of the system. After setting p basic eqations for the irradiation field, the effect of the type of flow is examined. Models are obtained for ideal plg flow as well as for diffsive flow. The ltimate goal of this paper is to show how to develop these relatively simple mathematical models that are sitable for dynamical analysis and control. Conseqently, transfer fnctions are derived that connect the otpt of the system (bacteria load after disinfection with the distrbance of system (initial load of bacteria and the control inpts (light intensity and/or flow velocity. In section the UV disinfection process is described in some more detail. The modeling procedre of the disinfection process is presented in section 3. Section 4 presents two model approximation techniqes. The reslting approximate models are sed in section 5 to frther derive an analytical feed-forward feedback control law that explicitly depends on the physical process parameters. II. UV DISINFECTION A UV disinfection system transfers electromagnetic energy from a UV lamp to the genetic material of microorganisms. The absorption of light cases photochemical reactions that alter moleclar components essential to cell fnction. There is scientific evidence to conclde that if sfficient dosages of UV energy reach the organisms, UV can disinfect water to whatever degree is reqired. In [4], the experimental data for UV inactivation of micro organisms have been extensively reviewed and frthermore they tabled the UV dose reqired to achieve the inactivation of bacteria, virses and protooa. Predominantly, there are two types of UV sorces that are sed for water treatment, low pressre (P and medim pressre (MP mercry lamps. The UV dose is the prodct of UV intensity (mw/cm and the average exposre time (s of the water to be disinfected. In theory sing a low intensity lamp for a longer period of time shold give the same microbial inactivation as when a high intensity lamp is

2 sed for a shorter period. However, in [5] it is shown that preferably high intensity lamps shold be sed. Absorption, reflection, refraction and scattering all interfere with the transportation of UV light. Reflection, refraction and scattering only change the direction of the light which is still capable of inactivating microorganisms, while absorbed light is no longer available. The effectiveness of a system is related to the initial load of microorganisms in the water. In general, most of the disinfection models are based on the following expression: c= c exp( Kt (1 where c is the microbial load after disinfection (microorganisms/1 m, c the initial microbial load, K the local inactivation rate constant (1/s and t the time of exposre (s. As can be seen from (1, inactivation of microorganism by UV irradiation is sally expressed in terms of first-order kinetics, which holds at low UV doses for e.g. vegetative bacteria as E. coli (see [5]. III. M ODEIN OF DISINFECTION PROCESS A. Irradiation Field in Annlar Reactor Analysis of light energy distribtion in the annls is important in order to determine the local inactivation rate constant. It becomes apparent that, since the amont of energy varies with space in the photoreactor, the same holds for the inactivation rate for the microorganisms. For the development of a light distribtion model for the annls it is obvios that working in three dimensions leads to nnecessarily complex mathematical expressions. Therefore, it is assmed that both variations in light intensity in the longitdinal direction of the photoreactor and end effects of the lamp do not play a role. The developed model is also based pon the following assmptions: The UV lamp emits rays radially from the entire srface The attenation of light depends on the concentration of solids in the medim and the length of the light path Solids are homogenosly sspended in the medim, ths all the properties of the medim are assmed constant throghot the reactor There is monochromatic UV-light at 53.7nm at which the DNA of all microorganisms is altered casing the inactivation of virses and bacteria (see [6] The irradiation of the field is not time varying, it is only a fnction of the space coordinates of each point The effects of reflection and/or refraction are negligible There is only one species of microorganisms which follow first-order kinetics in the process of inactivation (see [7] Water has been pre-filtered, ths the concentration of sspended solids is small and irradiation field is only affected by the attenation in water. Under these assmptions and sing ambert s law, the light intensity at any point in the reactor is related to the srface flx (see [8]: 1 dri ( r dr = EI ( where r is the radial distance in the reactor, I is the light intensity of the irradiation field at a distance r from the lamp (mw/cm and E is the monochromatic absorbance of water (cm -1. Integration of (, sing the bondary condition I = I when r =, gives: Er ( I = I e (3 r where I is the light intensity of the irradiation field on the srface of the UV-lamp (mw/cm and r is the oter radis of the UV lamp (cm. The reaction constant (K is the prodct of the available energy from the field mltiplied by the ssceptibility factor of the microorganism. Under the assmption that disinfection of a specific microorganism follows first-order kinetics we obtain: Er ( Kr ( = ε I e (4 r where K( r is the spatially dependent reaction constant (s -1 and ε is the ssceptibility factor of the microorganism (cm / mw s. The average light intensity related reaction constant across a cross-section of the tbe will then be: R e Er ( ε dr r R K = I = I β Hence, the reaction constant K depends on the light intensity on the srface of the lamp mltiplied by the parameter β. It is also possible to take into accont the effects of reflection and refraction, see e.g. [6], [9] and [1], bt then the reflection and refraction coefficient has to be identified in sit for each specific reactor. B. Flow in Annlar Reactor In addition to the assmptions made in the previos section for the irradiation field, in order to develop the model for the case of ideal plg flow, the following assmptions have been made: The liqid is ideally mixed in the radial direction Every volme of the liqid has exactly the same retention time in the reactor (5

3 Every volme is receiving the same amont of radiation The only mechanism of mass transfer is convection (as yet, diffsion is neglected The eqation that describes the disinfection process nder the above assmptions is: ct (, ct (, + + Kc(, t = t After applying aplace transformation with bondary condition at = C(, s = C ( s the soltion of the partial differential eqation at = is given by: ( K+ s Cs (, = C ( se (7 Conseqently, the concentration at the end of the reactor is the otpt of the system, whereas the concentration at the entrance is the inpt. Therefore the above eqation can be written in inpt-otpt form with transfer fnction (,s: ( K+ s Y( s s (, = = e (8 U( s Frthermore, we define: t R = / (residence time. If, however, to be more realistic, we also assme diffsion in the -direction with diffsion coefficient D the partial differential eqation that describes this phenomenon is: (6 ct (, ct (, ct (, + D + Kc(, t = (9 t The soltion of this differential eqation, in terms of the aplace variable s, is: K+ s + 4 D D D Cs (, = Ce 1 (1 If we introdce the Peclet nmber, i.e. Pe =, and D sbstitte the following bondary conditions: C (, =, C(, s = C ( s and lim Cs (, =, the transfer fnction (s is given by: IV. MODE APPROXIMATIONS A. Padé approximation In the previos section transfer fnctions have been derived from partial differential eqations. However, as it can be seen from (8 and (11 these transfer fnctions contain an exponential term in s. For frther dynamical analysis or for controller design, preferably rational transfer fnctions (polynomial qotients in s are reqired. Pre τ s dead-time terms (of the form e are also allowed, becase nowadays a vast amont of literatre on so-called dead-time systems is available (see e.g. [11], [1] and the references therein. Conseqently, (8 is a pre dead-time system with τ Kt = t R and with a constant gain e R. However, for (11 with its sqare root of s there is a need for a model approximation step. Instead of the commonly sed Padé approximations, as in e.g. [13], we now derive a dead-time/padé[,1] approximation of (11, that is s b s τ ( = e s + a (1 (See Appendix A for details of this approximation. For Pe = 1 and K = 1 Bode plots of the original and the approximate system with a =.43, b = 8.36 and τ =.95 are obtained (see Fig. 1, where the approximation is appropriate for a freqency smaller than rad/s. Amplitde Phase ω (rad/s Figre 1. Comparison between original convectiondiffsion-reaction transfer fnction (ble line and its deadtime/padé[,1] approximation (dashed red line. Pe Pe + 4 Pe ( K + s s (, = e with dimensionless reaction constant K = Kt. R (11 B. ineariation Notice that so far the transfer fnction between the distrbance inpt C (s and the concentration at the end of the reactor C(,s for constant flow velocity and light intensity has been considered. For control of the disinfection process, in addition to possible shaping of the distrbance inpt by bffering, light intensity (I, or in what follows K (= βi with β constant, and flow velocity ( can be

4 considered as control inpts or maniplated variables. For simplicity of the expressions only, in what follows we will focs on the ideal plg flow case; extension to diffsive flow in the -direction is more or less straightforward. From (6 it follows that both control inpts appear in a bilinear form together with the concentration c(,t. For small pertrbations from the steady state (denoted by an overbar, e.g. c a linearied system description of the disinfection process can be obtained (see Appendix B for details. After some algebraic maniplations the following inpt-otpt relationship, relating the pertrbed distrbance inpt C, the pertrbed reaction constant K = β I and the pertrbed flow velocity to the pertrbed system otpt C(,s, can be fond: Cs (, = 1( s, C( s + ( s, Ks ( + 3( s, ( s (13 Herein: C 1 1(, e s s = e τ with C1 = K = Iβ and τ = (14a K C C τ s (, s = + e with C = c e (14b s s K C3 C3 τ s ck 3(, s = + e with C3 = e (14c s s Conseqently, the MISO system has three inpts and ths three transfer fnctions. Notice that 1 (,s is a pre deadtime system with gain e and both (,s and K/ 3 (,s are parallel integrators with some time shift. In case of diffsive flow terms like in (13 will appear. For c =.75 kg/m 3, K =.4 s -1, =5 m and =1 m/s the following step responses for the distrbance inpt and the light intensity related control inpt nder ideal plg flow are presented in Fig.. From Fig. a it is immediately clear that after a dead time of 5 s, the nit change in bacteria concentration at the entrance of the reactor (dotted line is redced to 3% of its initial vale. Fig. b shows that an increase of the light intensity initially redces the bacteria concentration linearly with time and after 5 s a constant redction is obtained. For se in a feed-forward controller design procedre (described in the next section, a step response of an approximate system of the form of (1, with n τ =, a / = (n = and K b = c e, is also shown in Fig. b (thin line. Amplitde Amplitde Time (sec Figre. (a Unit step ( c = 1 kg/m 3 response and (b step ( K =.1 s -1 response (solid thin line: approximate system. In the next section some sggestions for control of the disinfection process in the annlar photoreactor are given and frther analysed. V. PROCESS CONTRO DESIN A. Feed-forward control When the distrbance of a system is known or measred on-line, the se of a feed-forward controller can prove to be beneficial. The design of a feed-forward controller is rather simple bt reqires good models. C 1 1 F Figre 3. Feed-forward controller scheme for constant velocity. In Fig. 3 a feed-forward controller scheme for light intensity as control inpt is shown. iven the objective that the otpt shold be close to ero, the design of the feedforward transfer fnction F simply follows from the algebraic eqation: Cs (, = 1( sc, + F ( s ( sc, = (15 + C so that 1(, s F ( s = (, s, where and follow from 1 e.g. (14. et s evalate the scheme in Fig. 3 for the ideal plg flow case with light intensity as control inpt in some more detail.

5 In this case, in (15 1 is fond from (14a and from (14b. Conseqently, s / 1 (, s se F ( s = = (16 / (, s s c (1 e which is a non-rational transfer fnction. et s therefore se the approximation of presented in Fig. b. Then, s / 1 (, s e ( s+ n / F ( s = = (, s c ( s+ m / (17 where the factor ( s + m / with m large is added to make this controller physically realiable. A similar filter term is sally added in the D-action of a PID controller. The overall transfer fnction H(s from C to C is then given by ( K / ( s+ n / / / ( s 1 1 s H s = e e e ss ( + m / (18 Fig. 4 presents the simlation reslts of the linearied system with the physically realiable feed-forward controller for m =1 and as before for c =.75 kg/m 3, K =.4 s -1, =5 m and =1 m/s. Since s / ( 1 e lims = (19 s K / lim ( n s H s = e 1. ( m Conseqently, for the parameter vales given above and as a reslt from the approximations, the deviation in the otpt at = is eqal to.3 kg/m 3 for c =.1 kg/m 3 and t. C C( K Time (s Figre 4. Simlation reslts for feed-forward control strategy. B. Combined feed-forward and feedback control Althogh feed-forward can theoretically reslt in perfect control of a process and perfect attenation of known distrbances, in practice it is not always the case. That is mainly becase it reqires, as qoted before, very precise models. In practice, there are always deviations between the model and the real process and frthermore the measred or predicted distrbance inpt contains errors. Therefore the combination of feed-forward and feedback control can reslt in more precise control of the process (see e.g. [14]. For or application with constant flow velocity, Fig. 5 shows an appropriate feed-forward feedback controller scheme. C Figre 5. Feed-forward - feedback controller scheme. The closed loop transfer fnction for the changes of the distrbance is: 1+ (, Cs = F C ( s 1 B (1 so that again 1(, s F ( s = (, s or for implementation in practice 1(, s. From (1, however, it can be seen that (, s the stability of the system and ths the denominator of the fraction, only depends on the process and the feedback controller B. This gives the opportnity to tne both controllers separately and deal with the stability of the system. The feedback controller B can, for instance, take a PI-controller strctre with controller parameters that directly depend on the dead-time first-order properties of. Conseqently, 1 B( s = Kp 1+ Ts r 1 F + B ( From the Cohen-Coon reaction crve method and after sbstitting from (1, for the ideal plg flow case, the time n constant and gain, that is a / =, K b= c e and after choosing τ = τ, we obtain + C

6 K / Te 1 K p = (3 c T and (3T + 3 Tr = τ (4 (9T + where T =. Hence, the controller parameters are nτ flly expressed in terms of the process parameters. Again, from (1 the overall transfer fnction can be fond. Fig. 6 presents the simlation reslts of the feed-forward feedback scheme (Fig. 5 for n =, m =1, c =.75 kg/m 3, K =.4 s -1, =5 m and =1 m/s..1 A. Padé approximation APPENDIX A dead-time-padé[,1] approximation of (s in s = is of the form s b s τ ( = e s + a (A.1 where the coefficients a, b and τ are determined by setting ( = d( d ( d ( d ( (, = and =. ds ds ds ds iven the convection-diffsion transfer fnction (s as in (11, we obtain (rather complicated expressions for a, b and τ in terms of Pe, K and s. However, the following relationships hold: C Pe Pe + 4 PeK b 1 Pe = e and τ + = a a Pe + 4PeK (A. C( K Time (s Figre 6. Simlation reslts for feed-forward feedback control strategy. Hence, these analyses show how prior non-rational process knowledge, which freqently appears in processes with flow components, can be directly implemented in a controller design procedre that conserve the knowledge of physical process parameters. Clearly, given the dynamical models of the previos sections, even a mltivariable (optimal controller cold have been designed (see e.g. [11], [1]. VI. CONCUSIONS For dynamical analysis and model-based controller design of a water disinfection process in annlar reactors, described by convection-diffsion-reaction type of differential eqations, a transfer fnction modelling approach, sing analytical expressions in terms of the aplace variable s and the original physical process parameters, is possible and provides frther insight into the process (see also [13]. where b/a is the steady-state gain of the system. This procedre can be repeated for different orders m and n, bt then most often we mst rely on nmerical schemes for the estimation of the coefficients. In general, an appropriate choice of the orders n and m in a Padé[n,m] approximation is made by observation of the Bode plot of the original transfer fnction. B. ineariation et s as an example of the lineariation procedre write (6 in terms of the steady states (denoted by an overbar and small pertrbations, indicated by : ( c + c ( c + c + ( + + ( K + K( c + c = t (B.1 c + c + c c c c t t + Kc + K c+ Kc + K c = (B. Sbtracting from (B. the steady state terms that obey (6 c and neglecting the second-order terms and K c the following eqation in the so-called deviation variables is obtained: c c c K c+ Kc = t (B.3 For both and K the constant steady state vales can be sbstitted, bt for the concentration c the steady state soltion mst be fond from:

7 dc + Kc = d (B.4 [14]. C. oodwin, S. F. raebe and M. E. Salgado, Control System Design. Prentice Hall, New Jersey, 1. / which is given by = K c c e, so that in (B.3 c Kc = K / e. Hence, after sbstittion of the steady state soltions and after defining 1 := K and :=, (B.3 becomes c c Kc K / / + K + + e K c ce 1 = t (B.5 After aplace transformation and re-ordering the eqation we obtain: C ( s+ K Kc K / c / + C e U K + e U 1 = (B.6 which is a linear first-order eqation with the initial condition: C(= C, the distrbance inpt of the reactor system and U 1 = K(s, U = (s the control inpts. After solving (B.6 for =, the transfer fnctions 1 (,s to 3 (,s in (14 are obtained. REFERENCES [1] R. W. Ward and. M. Derave, Residal toxicity of several disinfectants in domestic wastewater. Jornal of Water Polltion Control Federation, 5(1: 46 6, [] A. E. Cassano, C. A. Martin, R. J. Brandi and O. M. Alfano, Photoreactor analysis and design: Fndamentals and applications. Indstrial and Engineering Chemistry Research, 34: 155 1, [3].. Pma and P.. Ye, Modeling and design of thin-film slrry photocatalytic reactors for water prification. Chemical Engineering Science, 58(11: 69-81, 3. [4] H. B. Wright and. Sakamoto, UV dose reqired to achieve incremental log inactivation of bacteria, virses, and protooa. Trojan Tech Inc, revision of Sept 1, 1. [5] I. Woitenko, M. K. Stinson and R. Field, Challenges of Combined Sewer Overflow Disinfection by Ultraviolet ight Irradiation. Critical Reviews in Environmental Science and Technology, 31(3: 3 39,. [6] J. R. Bolton, Calclation of ltraviolet flence rate distribtions in an annlar reactor: Significance of refraction and reflection. Water Research, 34: ,. [7] B. F. Severin, M. T. Sidan, B. E. Rittmann and R. S. Engelbrecht, Inactivation kinetics in a flowthrogh UV reactor. Jomal of the Water Polltion Control Federation 56: , [8] M. T. Sidan and B. F. Severin, ight intensity models for annlar UV disinfection reactors. AIChE Jornal, 3(11: ,1986. [9] E. R. Blatchley, III, Nmerical modeling of UV intensity: Application to collimated-beam reactors and continos-flow systems. Water Research, 31(9: 5-18, [1] K. V. Pareek and A. A. Adesina, ight intensity distribtion in a photocatalytic reactor sing finite volme. AIChE Jornal, 5(6: , 4. [11]. Meinsma and H. Zwart, On H control for dead-time systems, IEEE Trans. At. Control, 45(: 7-85,. [1] A. A. Moelja, H Control of Systems with I/O Delays. PhD thesis University of Twente, 5. [13] S. van Morik, H. Zwart and K. J. Keesman, Analytic control law for a food storage room. Sbmitted, 6.