Design and validation of an innovative soft-sensor for pharmaceuticals freeze-drying monitoring

Transcription

1 This is an electronic version (author's version) of the aer: Bosca S., Fissore D. (0). Design and validation of an innovative soft-sensor for harmaceuticals freeze-drying monitoring. Chemical Engineering Science (Elsevier), 66(), DOI: 0.06/j.ces Design and validation of an innovative soft-sensor for harmaceuticals freeze-drying monitoring Serena Bosca, Davide Fissore Diartimento di Scienza dei Materiali e Ingegneria Chimica, Politecnico di Torino, corso Duca degli bruzzi 4, 09 Torino (Italy) Corresonding author davide.fissore@olito.it Tel: +39-(0) Fax: +39-(0)

2 bstract The aer is focused on the develoment of a Kalman filter tye observer for the monitoring of the rimary drying hase of the freeze-drying of harmaceutical roducts in vials. The roosed soft-sensor is able to estimate the evolution of temerature of the roduct, the duration of the rimary drying hase, the value of the overall heat transfer coefficient between the heating shelf and the roduct, as well as the mass transfer resistance of the dried cake to vaor flow. ccurate results are obtained for various tyes of roducts, characterized by a different deendence of the mass transfer resistance on the dried cake thickness. Theoretical results are confirmed by exerimental tests carried out in a ilot-scale freeze-dryer. Finally, the strength and the weakness of the roosed observer are ointed out. Key words Drying; Product rocessing; Simulation; Biorocessing; Freeze-drying; Soft-sensor; Monitoring; Extended Kalman Filter.

3 . Introduction Lyohilization is a drying rocess based on the sublimation of the solvent contained in the roduct to be treated. It is characterized by low oerating temeratures and it is articularly suitable for those roducts, such as harmaceuticals, which could be damaged by high temerature drying. Furthermore, the rocess allows obtaining a roduct with a orous structure that can be easily rehydrated and that can comletely recover its roerties after rehydration (see, among the others, Nail and Gatlin, 99; Oetjen and Haseley, 004; Franks, 007). The lyohilization rocess is carried out in three successive stes: freezing, rimary drying, secondary drying. The most imortant hase is rimary drying, when the vaor flows through the dried layer into the sublimation chamber, and then is continuously removed by a condenser. During this hase, rocess monitoring is crucial in order to obtain a roduct having the desired quality. In fact, it is required to monitor roduct temerature because it has to be maintained below a limit value. In the case of crystalline solutes, this limit temerature corresonds to the eutectic oint, in order to avoid the formation of a liquid hase; if the solutes are amorhous, the limit value is the glass transition temerature in order to avoid the collase of the dried cake. nother imortant variable that is useful to monitor is the osition of the sublimation interface, whose evolution is reresentative of the rogress of rimary drying. This variable allows to know when it is ossible to increase the set-oint of the temerature of the heating fluid in order to allow the desortion of the water bounded to the roduct (secondary drying). If the secondary drying is started too early, the temerature of the roduct is increased when sublimation is still occurring, thus causing the melting of the ice and damaging the roduct. If secondary drying is delayed, the length and cost of the rocess are increased. The use of an effective in-line monitoring system allows closing a control loo, thus maniulating the oerating conditions, namely shelf temerature and chamber ressure, in order to reserve (and also to build) roduct quality during rocessing (Fissore et al., 008). This is one of the targets indicated by the Guidance for Industry PT (Process nalytical Technology), issued by US-FD in 004: quality has no longer to be tested into roducts, but it has to be built-in or it should be by design In order to monitor the freeze-drying rocess it is ossible to use a soft-sensor (observer), i.e. a device that uses a mathematical model of the rocess and the exerimental measurements of one or more hysical variables to rovide an in-line estimation of the

4 roduct state and of other variables of interest. Various techniques were roosed in the literature to design an observer, e.g. the Extended Kalman Filter and the High Gain Technique. Velardi et al. (009) roosed to use the Extended Kalman Filter to monitor the rimary drying hase of a freeze-drying rocess: they used the measurement of the temerature of the roduct at the bottom of the vial, obtained inserting a thermocoule in the vial, and a simle mono-dimensional model of the rocess (Velardi and Barresi, 008) to estimate in-line the temerature of the roduct, the osition of the sublimating interface, the mass transfer resistance to vaor flow and the heat transfer coefficient. Velardi et al. (00) designed also a High Gain observer to monitor the rimary drying of a lyohilization rocess, evidencing that the accuracy of the estimations is similar to that obtained using the Extended Kalman Filter, but the time required by the calculations is significantly reduced, even if it is required to maniulate the system of differential equations modeling the dynamics of the system in order to get a structure suitable for the design of the High Gain observer. Both the Extended Kalman filter and the High Gain observer were develoed for a articular tye of roduct, characterized by a linear deendence of the mass transfer resistance ( ) on the dried layer thickness (H); thus, these tool are only suitable for that tye of roduct and they can not be used to monitor those roducts characterized by a non-linear deendence of on H. ctually, the majority of roducts dried by means of a lyohilization rocess is characterized by a non linear deendence of on H: this motivates the need for a new observer that is able to monitor all tyes of roducts. The Extended Kalman Filter aroach will be chosen to built the desired observer, as the rocess is highly non-linear. The aer is structured as follows: at first, the structure of the roosed observer is introduced and, then, it will be validated by means of numerical and exerimental investigations. The numerical validation is carried out to verify that the estimations rovided by the observer converges to the true values (as in this case the state of the roduct is fully known from mathematical simulation), while the exerimental validation is carried out to rove the adequacy of the soft-sensor to monitor real lyohilization cycles.. Non-linear observer design The Kalman filter was firstly roosed by Rudolh E. Kalman in 960 as a solution for the linear discrete data filtering roblem (Kalman, 960): it is a set of equations imlementing a redictor-corrector estimator that minimizes the covariance of the estimation error. The filter 3

5 was develoed for a discrete-time system, but it has also been alied to continuous-time systems, as well as to non-linear rocesses (Becerra et al., 00; Judd, 003). Let us consider the dynamics of a system described by the following set of differential equations: x f x,u () where x is the array of state variables, u is the maniulated inut and f is a non-linear function of x and u. Let y be the measured variable: y h x, u () The Kalman filter tye observer for the system described by equations ()-() is based on the following equations: xˆ f xˆ K ˆ (3),u ty y yˆ h x ˆ, u (4) h t t - K S x xˆ T (5) T T f f h h S t S t S t S t S t (6) x xˆ x xˆ x xˆ x xˆ detailed derivation of equations (3)-(6) can be found in various aers, e.g. in the endix of the aer by Velardi et al. (009). It has to be remarked that these equations have been obtained after a linearization of model equations ()-() and, thus, the estimations of the state variables rovided by the observer ( ˆx ) are guaranteed to converge to the real values (x) only in case the aroximation introduced by the linearization is small. This makes the first guess of the estimated variables a critical ste. The equations of the observer are based on the mathematical model of the rocess. For a vial freeze-drying rocess various models have been roosed to describe the evolution of the temerature of the roduct and of the amount of ice vs. time as a function of the oerating conditions (a brief review can be found in Velardi and Barresi, 008). In this aer we use the simlified mono-dimensional model of Velardi and Barresi (008): according to this model, the heat flux to the roduct and the solvent sublimation flux are calculated using the following equations: q v fluid B J K T T (7) J (8) w w, i w, c R 4

6 K v is the overall heat transfer coefficient between the heating fluid and the roduct at the bottom of the vial and is the resistance of the dried roduct to vaor flow. The coefficient K v is a function of chamber ressure, for a given vial-freeze-dryer system: K v bk v c ak v c Kv c while is a function of the thickness of the dried layer (H), for a given roduct. The following equation is generally roosed to describe the deendence of on H: R R H = R + + B H, 0 R (9) (0) where R, 0,, B deend on the roduct under investigation. In case R 0 0 and, B 0, the deendence of on H is linear: R = H () as in the model used by Velardi et al. (009, 00) to design a soft-sensor for the vial freezedrying rocess, where: R æ RT ö i = H çm k è ø t the moving interface there is no heat accumulation, and the heat flux is used for ice sublimation: () L H Kv kii R Tfluid Ti HS w, i w, c The following equation gives roduct temerature at the vial bottom: (3) L H TB Tfluid Tfluid Ti (4) Kv Kv kii Finally, the evolution of frozen roduct thickness is calculated by solving the following equation: dh dt R II Ie wi, wc, This model has been extensively validated by means of exeriments (see, among the others, Velardi and Barresi, 008, and Giordano et al., 0). coefficients Various methods have been roosed to calculate model arameters (K v and ). The a K v, b K v and ck v (5) can be calculated through various theoretical exressions 5

7 rovided in the literature (see, among the others, Pikal et al., 984); in any case exerimental investigation is necessary to determine these values reliably. To this urose a gravimetric method can be used. It requires to carry out a lyohilization cycle under well defined oerating conditions (temerature and ressure) and filling the vials of the batch with water (or a solution). The ice temerature at the bottom of the vial (T B ) and the weight loss of water (m) in each vial have to be measured. The value of K v can be calculated using the following equation: K s v t Tshelf TB v mh s an alternative, the value of the sublimation flux ( m t v ) required to calculate K v can be determined using the Tunable Diode Laser bsortion Sectroscoy (Kessler et al., 006; Gieseler et al., 007; Kuu et al., 009). nother method to obtain the value of K v is based on the use of the algorithms roosed for the monitoring of the rimary drying using the ressure rise test (Milton et al., 997; Chouvenc et al., 004; Velardi et al., 008; Fissore et al., 0). In any case, at least three measurements at three different values of c, are required to determine the values of a K v, b K v and The arameter R P can be determined from the measurement of the solvent flux, obtained using the Tunable Diode Laser bsortion Sectroscoy sensor, as shown in the following equation: R c K v. wi, c (7) J w or one of the algorithms used to interret the ressure rise test. Furthermore, Fissore et al. (00) roosed to use a secial device laced in the lyohilization chamber to calculate using of the measurement of the weight loss due to the sublimation of ice vs. time (i.e. of J w ) and of roduct temerature in the weighed vials (used to calculate w,i ). For the rocess under investigation the state vector that the soft sensor has to estimate is the following: T x x x3 Ti R K v T x (8) (6) It can be remarked that the arameter has been selected to account for the value of (the values of R, 0 and B have been ket constant and equal to their first estimations) thus reducing the number of variables that the soft-sensor has to estimate. The measured variable of the rocess (y) is the roduct temerature at the bottom of the vial. The maniulated 6

8 variable (u) is the temerature of the heating fluid. Model equation can then be written as in equations () and (): x f x, x, x, u 3 x fx, u x 0 x 3 0 y h x u h x x x3 u,,,, where the first derivative of x and x 3 are set equal to zero because the arameters (9) and K v do not change during the freeze-drying rocess. The following equation is used to calculate the first derivative of x = T i with resect to time: dx dh H du H dt dt u dt x (0) Equation (0) requires the artial derivatives of H with resect to x, u and t. To this urose, it is necessary to determine the deendence of H on the state variables. By means of simle maniulations of equation (4) we get: H = where: a-b b g - a B R, 0 H x x x () s, 3 w,i w,c x3 x, u R u x,0 (3) x R B x (4),0 Using equations ()-(4) we calculate: () x3 B dh R,0 dt R B,0 R R (5) B H R x x x B R R,0 R,0 where (7) H S II Ie (6) 7

9 du t first we assume that 0 dt = and, thus, the calculation of H is not necessary: this u assumtion is justified by the fact that in most freeze-drying rocesses the shelf temerature is maintained constant throughout rimary drying. Furthermore, this assumtion allows to simlify the equations of the observer, and it can be useful also for validation uroses du because the degrees of freedom of the system are reduced. In case 0 dt ¹ subsequent du equations are modified and they are shown in endix. When = 0, equation (0) dt becomes: dx dh H (8) dt dt x By combining equations (5) and (6) into equation (8), and exressing the measured variable as a function of the state variables, we get: x3 BR R P,0 x x x 0 0 y u hx, u R,0 R, u,0br f x In order to calculate the gain of the observer ( t) the vector h x: (9) K we need the Jacobian matrix f x and f f f f x x x x xˆ h h h h x x x x xˆ xˆ 3 xˆ (30) (3) The comonents of f xand of h xare obtained from the equations (9): 8

10 x3 BR x 3 BR B R f x R,0 R,0 R,0 x x x R,0BR x x B R,0 x3 R x x R,0BR x x BR R,0 BR B R f x R 3,0 R,0 R,0 x x R,0BR x x (3) (33) B x B B x3 R,0BR x x R 3 R R f R,0 R,0 x3 B RP,0 3 3 x3 R x x xx R,0BR x x h x R x,0 (34) (35) h x h 0 (36) x 3 Finally, it is necessary to calculate the derivative of the functions x, x, x u, that can be determined from equations ()-(4): 3,, x H d s wi, x x dx 3, x x3 dx S 3 x3 H S d wi H wi, x c (37) (38) (39) 9

11 H d s wi x x x dx x x, x 0 x x,0 (40) (4) (4) (43) 3. Case studies The validation of the observer is carried out in two stes. In the first hase, rocess simulation using the model of Velardi and Barresi (008) has been used to calculate the evolution of a roduct characterized by a non-linear deendence of the resistance on dry cake thickness, as described by equation (0), and, thus, the temerature of the roduct at the bottom of the vial required by the observer. By this way it is ossible to comare the estimations of interface roduct temerature, frozen layer thickness, and model arameters with the true values, thus validating the tool. Two different case studies have been considered to this urose: in the first case the roduct is characterized by a high value of resistance to vaor flow (i.e. of ) and a high value of K v ; in the second case we considered lower values of and K v. By this way we are able to account for the large variability of and K v when rocessing different roducts and, consequently, we are able to test the effectiveness of the soft sensor in case the rocess is heat-transfer controlled, as well as when freeze-drying is mass-transfer controlled. fterwards, the erformance of the observer used to monitor the dynamics of a roduct characterized by a linear deendence of resistance on dry cake thickness, as shown in equation (), has been investigated, in order to demonstrate the ossibility of alying the roosed observer to a wide variety of roducts. lso in this case the values of the temerature at the bottom of the vial obtained by mathematical simulation are rovided to the observer, and the convergence of the roosed tool is verified. Finally, exerimental investigation has been used to verify the adequacy of the tool to monitor the rimary drying hase of a real freeze-drying cycle. In this case a thermocoule 0

12 laced at the bottom of the vial is used to rovide the roduct temerature measurement. The lyohilization cycle is carried out in a LyoBeta 5 freeze-dryer (Telstar, Sain), using ISO 836-0R tubing vials filled with.5 ml of a 0% w/w aqueous solution of an active harmaceutical ingredient, corresonding to 7.6 mm of roduct thickness. In this case the temerature of the heating shelf was not constant, while chamber ressure was ket constant and equal to 5 Pa. Product temerature was measured using a T-tye miniature thermocoule (by Tersid S..., Milano, Italy). It has to be remarked that the evolution of the roduct in vials laced in various ositions over the shelf can be different according to various heat transfer mechanisms affecting the thermal balance of the roduct (Barresi et al., 00a, 00b). With this resect, vials of the first rows receive additional heat due to radiation from chamber walls. Thus, for this test we have monitored the dynamics of roduct temerature in a vial laced in a central osition over the shelf as it is reresentative of the largest art of the batch 4. Results and discussion Numerical validation of the observer is carried out by using the mathematical model to simulate the rimary drying. Figure shows the results obtained for two different case studies, characterized by different values of K v and : in the first K v = 30 J s - m - K - and = 0 9 s -, while in the second K v = 0 J s - m - K - and = 0 8 s -. Both shelf temerature and chamber ressure have been considered constant throughout rimary drying. In order to test the erformance of the observer, the initial values of K ˆ v and ˆ R have been set with an error of 5% comared to the reorted true values. n excellent agreement is obtained between the temerature estimated by the observer and the true value; a good agreement is obtained with resect to the osition of the moving interface vs. time, with an error on the estimation of the duration of rimary drying of -.6 % in the first case, and of -3.9 % in the second case. s far as model arameters are concerned, the error on the estimation of remains very small (lower than 5%), while the estimated value of K v moves from the wrong initial guess to the true value in both cases. This is of outmost imortance as the observer is not only able to rovide a reliable estimation of roduct temerature at the interface of sublimation and of the sublimation flux (i.e. of the osition of the sublimating interface, and of the duration of the rimary drying), but it rovides also reliable values of

13 model arameters and, thus, it can be used in a model-based control loo (Fissore et al., 008). In both cases we tested also the erformance of the observer roosed by Velardi et al. (009), evidencing that it does not converge to the correct values: as an examle, the error on the duration of the rimary drying is about 50%. These results confirm that the observer roosed by Velardi (009) is able to monitor only the dynamics of those roducts that exhibit a linear deendence of the mass transfer resistance on the dried layer thickness. s it has been ointed out in the Introduction section, the convergence of the Kalman filter is guaranteed only in case the initial values of the estimated variables are sufficiently close to the real values. For this urose it is worthwhile to analyze the effect of the initial error on the state variables on the erformance of the observer. For this study we have considered a roduct whose resistance R P is calculated using the following arameters: R, 0 = 300 m s -, = s - and B = 90 m - ; the value of the heat transfer coefficient is K v = 7. J s - m - K -. The values of K v and considered in this study corresonds to those obtained exerimentally for the 0% w/w aqueous solution of the drug considered at the end of this section as case study for the exerimental validation of the observer. Chamber ressure is considered constant at 8 Pa, and the heating fluid temerature is equal to -0 C. Wrong values of the arameters K v and are rovided to the observer at t = 0 in order to verify the ability of the observer to converge to the correct values, starting from a bad estimation; the ˆ 0 ˆ t 0 are summarized in Table. Figure shows the observer values of K t and v redictions in all the cases. In Figure (a), the evolution of roduct temerature is shown, thus evidencing that the observer is able to rovide good estimates regardless of the initial error on the value of arameters. The same results are obtained when considering the evolution of the moving front, shown in Figure (b), and this roves that the evaluation of the duration of the rimary drying obtained using the observer is reliable in all the cases (the error ranges from -0.7 % in case 4 to -5 % in case ). Finally, Figures (c) and (d) show that also the arameters K v and are correctly estimated, desite their final values are not erfectly coincident with the true ones, used by the mathematical model. This fact can be exlained by considering that the observer, during its calculations, minimizes the difference between the model estimated temerature and the calculated one, and it is ossible to find various coules of values of the arameters resulting in the same value of temerature. This leads to the conclusion that the final values of K v and are the results of a comensation between the values of the same arameters. Furthermore, by observing the Figure (c) and (d) it is

14 ossible to underline that if initial guesses of K v and are both higher (or lower) than the true values, the tool is able to reduce the estimation error, otherwise the error can be increased. With resect to the convergence of the observer calculations, it is interesting to investigate the maximum error allowed for the initial estimation of the arameters K v and to obtain the convergence of the observer. In this study we have considered that the initial value of roduct temerature is erfectly known, as it is generally the case when using a thermocoule to monitor roduct drying. The results of this study are resented in Figure 3, ˆ 0 ˆ t 0 that guarantee the convergence of the where the set of errors on K t and on v observer, is evidenced. More recisely we observe that the estimation error on the initial estimation of can roughly vary between -5% and +0% comared to the true value, while the error on the initial estimation of K v can vary between -5% and +5% comared to the true value. In those ranges of arameters values the observer rovide correct estimations, i.e. an error on the duration of the rimary drying lower than 5%, and a maximum error on roduct temerature lower than 0.5 C. The occurrence of a relevant error on the initial estimation of roduct temerature has also been investigated, evidencing that the convergence of the observer is not significantly affected by the initial error on roduct temerature. Results of Figure 3 are of outmost imortance if we consider that, using the methods reviously described (e.g. the gravimetric method for K v or the ressure rise test for ) to determine the arameters K v and, their values are affected by uncertainty (generally about 5-0%). Thus, the observer is able to correct the initial estimation of model arameters in order to rovide reliable estimations of the temerature at the interface of sublimation and of the sublimation flux (i.e. of the drying time), using the measurement of roduct temerature at the bottom of the container. Figure 4 shows the results obtained when using the roosed observer to monitor the dynamics of a roduct whose deendence of the mass transfer resistance on the dry cake thickness is linear, as for the roduct considered by Velardi et al. (009). In this case we considered the following arameters to calculate :,0 = m s -, = s - and B = 0 m - ; the arameter K v has been considered equal to 6. J s - m - K -. These arameter corresond to the values exerimentally determined for a 5% w/w aqueous solution of lactose, rocessed in vials characterized by an internal diameter of 6. mm and a mean thickness of the vial bottom of 0.84 mm. s in the revious cases, chamber ressure and 3

15 heating fluid temerature have been considered constant and equal to 8 Pa and -0 C resectively. lso in this case the agreement between the correct values of roduct temerature, interface osition, and model arameters with the values estimated by the softsensor is very good. It can be ointed out that the deendence of the estimated values of vs. L dried is not erfectly linear: this is due to the fact that the observer continuously estimates during rimary drying and, in this case, small variations in the estimated value of this arameter are resonsible for the small variation in the sloe of the curve of R ˆ vs. L dried. Thus, we can conclude that the roosed observer can be alied to various tye of roducts, and in all cases it rovides a good estimation of the variables of interest. Once the observer has been validated by means of numerical simulation, it is ossible to rove its adequacy to monitor a real lyohilization cycle. The exerimental validation is carried out with data obtained in freeze-drying cycles erformed in laboratory and, as indicated in the revious section, the roduct that is freeze-dried is a 0% w/w aqueous solution of an active harmaceutical ingredient. Preliminary investigation was required to determine the values of K v and for this system: the reviously described gravimetric method was used to determine the overall heat transfer coefficient to the roduct, while the DPE algorithm (Velardi et al., 008), roosed to interret the ressure rise test, was used to determine. Both values were adoted as the initial estimations of model arameters required by the observer. In order to test the robustness of the soft-sensor we considered also ˆ 0 ˆ t 0, as shown in Table. The initial value of different values of K t and v roduct temerature, in all the cases, has been considered equal to the first measure rovided by the thermocoule at the bottom of the vial laced in a central osition over the shelf. It has to be remarked that in this test the temerature of the heating shelf was not constant, and this is resonsible of the evolution of roduct temerature that shows a maximum at about.5 h from the beginning of the rimary drying. lthough du dt 0, we used the observer designed assuming du dt 0 and a good agreement between the true and the estimated values of roduct temerature was obtained, as shown in Figure 5. The error in the estimation of the duration of rimary drying ranges from -3% to 5%. This result is due to the fact that the convergence of the observer is very fast, and the rate of change of fluid temerature is quite slow, as in most freeze-drying rocesses. This allows to use the observer ()-(43). Obviously, in case the rate of change of T fluid is high, the observer given in endix has to be used. lso in this case we have tested the robustness of the observer, looking for the 4

16 maximum errors on Kˆ t 0 and ˆ 0 v t that guarantee the convergence of the observer. Results are shown in Figure 6, evidencing that the set of values of errors on K v and on is similar to that shown in Figure 3 for the simulated exeriment. 5. Conclusions. new observer for monitoring the rimary drying hase of a lyohilization rocess has been designed in this aer. It is able to estimate reliably roduct temerature and the duration of rimary drying, as well as model arameters describing heat transfer to the roduct and mass transfer from the interface of sublimation. The soft-sensor has been validated by means of numerical simulations and exerimental tests, thus ointing out the strength of this tool, that is able to reliably monitor the dynamics of the temerature of the roduct and of the osition of the interface of sublimation, and also to estimate the arameters of a simle model of the rocess, thus allowing the use of model-based control systems, as roosed by Fissore et al. (008). Moreover, this soft-sensor aears to be quite robust with resect to the initial estimations of model arameters and of roduct temerature, and this is an imortant issue as the values of these arameters can be determined mainly with an exerimental investigation and, thus, their values are affected by uncertainty. Moreover, using the roosed soft-sensor it is ossible to monitor various vials laced in different ositions in the freeze-dryer, thus allowing taking into account the heterogeneity of the batch due to the different heat transfer mechanisms to the roduct (e.g. radiation from chamber walls, or non-uniform temerature of the heating fluid). Finally, the soft-sensor has rovided reliable estimations with various tyes of roducts, characterized by a different deendence of on dried cake thickness, even in case the temerature of the heating fluid is not constant. The main weakness related to the use of this sensor is due to the use of a thermocoule to get the temerature measurement. The use of wireless sensors recently roosed (Vallan et al., 005; Corbellini et al., 00) may allow the use of thermocoules with the automatic loading/unloading systems used in industrial freeze-dryers. Finally, the insertion of a thermocoule in the vial could not be allowed because of sterility reasons, or because the resence of the thermocoule can affect the dynamics of the roduct. In this case it could be ossible to lace the thermocoule on the external wall of the vial (Grassini et al., 009), thus avoiding any interference with the roduct. n examle of observer for a roduct exhibiting a 5

17 linear deendence of on dried cake thickness and using the measurement of the temerature of the external wall was atented by Barresi et al. (007). cknowledgements Valuable suggestions of rof.. Barresi (Politecnico di Torino) are gratefully acknowledged. 6

18 List of Symbols a K v arameter used to calculate K v, J m - s - K - arameter used to calculate, s - v cross sectional area of the vial, m b K v arameter used to calculate K v, J m - s - K - Pa - B arameter used to calculate, m - c K v arameter used to calculate K v, Pa - e K v error on the initial estimation of K v, e R error on the initial estimation of, K ˆ 0 v t K K v ˆ 0 R t ˆ f vectorial function giving the derivatives of the state h state sace equation of the measured variable H moving front osition (thickness of the dried layer), m ΔH s enthaly of sublimation, J kg - J q heat flux, J m - s - J w sublimation flux of water, kg m - s - k effective diffusivity coefficient, m s - k II thermal conductivity of the frozen layer, J m - s - K - K observer gain K v overall heat transfer coefficient, W m - K - L total roduct thickness, m Δm weight loss of water, kg M water molecular weight, kg kmol - c chamber ressure, Pa w,c water artial ressure in the drying chamber, Pa w,i water ressure at the interface of sublimation, Pa R ideal gas constant, J mol - K - mass transfer resistance, m s -,0 arameter used to calculate, m s - S matrix used to calculate the observer gain R v R 7

19 T i T B T fluid t u x y roduct temerature at the sublimation interface, K temerature of the roduct at the bottom of the vial, K heating fluid temerature, K time, s system inut (maniulated variable) array of the state variables system outut (measured variable) Greeks variables defined by equations ()-(4) variable defined by equation (7) Λ matrix of tuning arameters of the Kalman observer ρ Ie aarent density of the dried roduct, kg m -3 ρ II density of the frozen roduct, kg m -3 Suerscrits ^ observer estimate first time derivative 8

20 References Barresi,.., Baldi, G., Parvis, M., Vallan,., Velardi, S.., Hammouri H., 007. Otimization and control of the freeze-drying rocess of harmaceutical roducts. U.S. Patent lication No. /96,80 (0/04/007). Barresi.., Pisano, R., Fissore, D., Rasetto, V., Velardi, S..; Vallan,., Parvis, M., Galan, M., 009. Monitoring of the rimary drying of a lyohilization rocess in vials. Chemical Engineering and Processing 48, Barresi,.., Fissore, D., Marchisio, D.L., 00a. Process nalytical Technology in industrial freeze-drying. in: Rey, L., May, J.C. (Eds.), Freeze-drying/lyohilization of harmaceuticals and biological roducts, 3rd revised Ed., Informa Healthcare, New York, Barresi,.., Pisano, R., Rasetto, V., Fissore, D, Marchisio, D.L., 00b. Model-based monitoring and control of industrial freeze-drying rocesses: effect of batch nonuniformity. Drying Technology 8, Becerra, V.M., Roberts, P.D., Griffiths, G.W., 00. lying the extended Kalman filter to systems described by nonlinear differential-algebraic equations. Control Engineering Practice 9, Chouvenc, P., Vessot, S., ndrieu, J., Vacus, P., 004. Otimization of the freeze-drying cycle: a new model for ressure rise analysis. Drying Technology, Corbellini, S., Parvis, M., Vallan,., 00. In-rocess temerature maing system for industrial freeze-dryers. IEEE Transactions on Instrumentation and Measurement, 59, Fissore, D., Velardi, S.., Barresi,.., 008. In-line control of a freeze-drying rocess in vial. Drying Technology 6, Fissore, D., Pisano, R., Barresi,.., 00. model-based framework to get quality-bydesign in freeze-drying of harmaceuticals. Proceedings of Freeze Drying of Pharmaceuticals and Biological Conference, Garmisch-Partenkirchen, Setember 8 - October, Fissore, D., Pisano, R., Barresi,.., 0. On the methods based on the Pressure Rise Test for monitoring a freeze-drying rocess. Drying Technology 9, Franks, F., 007. Freeze-drying of harmaceuticals and bioharmaceuticals. Royal Society of Chemistry, Cambridge. Gieseler, H., Kessler, W.J., Finson, M., Davis, S.J., Mulhall, P.., Bons, V., Debo, D.J., 9

21 Pikal, M.J., 007. Evaluation of Tunable Diode Laser bsortion Sectroscoy for inrocess water vaor mass flux measurement during freeze drying. Journal of Pharmaceutical Sciences 96, Giordano,., Barresi,.., Fissore, D., 0. On the use of mathematical models to build the design sace for the rimary drying hase of a harmaceutical lyohilization rocess. Journal of Pharmaceutical Sciences 00, Grassini, S., Mombello, D., Neri,., Parvis, M., Vallan., 009. Plasma deosited thermocoule for non-invasive temerature measurement. Proceedings of International Instrumentation and Measurement Technology Conference - IMTC 09 (IEEE), Singaore, Reublic of Singaore, 5-7 May 009, [ISBN: , DOI: 0.09/IMTC ] Judd, K., 003. Nonlinear state estimation, indistinguishable states, and the extended Kalman filter. Physica D 83, Kalman, R.E., 960. new aroach to linear filtering and rediction. Transaction of the SME Journal of Basic Engineering 8, Kessler, W.J., Davis, S.J., Mulhall, P.., Finson, M.L., 006. System for monitoring a drying rocess. United States Patent No Kuu, W.Y., Nail, S.L., Sacha, G., 009. Raid determination of vial heat transfer arameters using tunable diode laser absortion sectroscoy (TDLS) in resonse to ste-changes in ressure set-oint during freeze-drying. Journal of Pharmaceutical Sciences 98, Milton, N., Pikal, M.J., Roy, M.L., Nail, S.L., 997. Evaluation of manometric temerature measurement as a method of monitoring roduct temerature during lyohilisation. PD. Journal of Pharmaceutical Sciences and Technologies 5, 7-6. Nail, S.L., Gatlin, G.., 99. Freeze drying: rinciles and ractice, in: vis, K.E., Lieberman, H.., Lachman, L. (Eds.), Pharmaceutical dosage forms: Parenteral medications, vol., Marcell Dekker Inc., New York, Oetjen, G.W., Haseley, P., 004. Freeze-Drying, nd edition. Wiely-VHC, Weinheim Pikal, M.J., Roy, M.L., Shah, S., 984. Mass and heat transfer in vial freeze-drying of harmaceuticals: role of the vial. Journal of Pharmaceutical Sciences 73, Velardi, S.., Barresi,.., 008. Develoment of simlified models for the freeze-drying rocess and investigation of the otimal oerating conditions. Chemical Engineering Research and Design 87, 9-. Velardi, S.., Rasetto, V., Barresi,.., 008. Dynamic Parameters Estimation Method: 0

22 advanced Manometric Temerature Measurement aroach for freeze-drying monitoring of harmaceutical. Industrial and Engineering Chemistry Research 47, Velardi, S.., Hammouri, H., Barresi,.., 009. In line monitoring of the rimary drying hase of the freeze-drying rocess in vial by means of a Kalman filter based observer. Chemical Engineering Research and Design 87, Velardi, S.., Hammouri, H., Barresi,.., 00. Develoment of a high gain observer for in-line monitoring of sublimation in vial freeze-drying. Drying Technology 8,

23 endix. Observer equations in case of non-constant heating fluid temerature du In case ¹ 0 it is necessary to calculate the artial derivative of H resect to variable u, dt thus obtaining: R u u B R,0 H,0 b dx with =, 0. In this case, equation (0) is used to calculate u dt dx dt x3 BR R,0 du R,0 u dt R,0B R B x x R,0 x x, thus obtaining: (.) (.) Equation (.) is then used to calculate the Jacobian matrix f x, whose comonents are the followings: x3 BR x 3 BR B R f x R,0 R,0 R,0 x x x R,0BR x x B R,0 x3 R x x R,0BR x x R du,0 x R,0 x x u dt BR B R R,0 x x R,0 x x (.3)

24 BR R,0 BR B R f R x3,0 R,0 R,0 x x R,0BR x x (.4) du R,0 R,0 udtx x B B R R,0 R R R,0 x x,0 x x B x B B R R x x3 R,0BR x x R 3 R R f,0,0 3 BR,0 3 3 x3 R x x xx R,0BR x x du x 3 R,0 R,0 x x3 x x3 udt BR B R R,0 x x R,0 x x (.5) 3

25 List of Tables Table. Values of Kˆ t 0 and ˆ 0 data. v t used for observer validation with simulated Table. Values of Kˆ t 0 and ˆ 0 exerimental data. v t used to validate the observer with 4

26 List of Figures Figure. Comarison between estimated (lines) and true values (symbols) of roduct temerature (to grahs; : T B, : T i, : T i ), moving front osition (middle grahs) and model arameters (bottom grahs; and :, and _ : K v ) for two case studies (case : K v = 30.0 J s - m - K -, = s - ; case : K v = 0.0 J s - m - K -, = s - ; R,0 =. 0 4 m s -, B = m -, L = 7.6 mm, T fluid = -0 C, c = 8 Pa). Figure. Comarison between true ( : T B, _ : T i ) and estimated values (symbols: T i ; : case, : case, Δ: case 3, : case 4) of roduct temerature (grah a), moving front osition (grah b), heat transfer coefficient (grah c), and roduct resistance to vaor flow (grah d). Figure 3. Range of values of errors on initial estimations of K v and that ensures the convergence of the observer: symbols ( : values of e K v and e R for which the convergence of the observer is obtained; : values of convergence of the observer is not obtained). e K v and e R for which the Figure 4. Comarison between estimated (lines) and true values (symbols) of roduct temerature at the interface of sublimation (grah a), moving front osition (grah b), heat transfer coefficient (grah c), and roduct resistance to vaor flow (grah d) (K v = 6. J s - m - K -,,0 = m s -, L = 3.4 mm, T fluid = -0 C, c = 8 Pa). = s - and B = 0 m -, Figure 5. Comarison between exerimentally measured values (symbols) of roduct temerature and the values estimated by the observer using the initial estimations of model arameters given in Table (lines; : T i, _ : T B ). Figure 6. Range of values of errors on initial estimations of K v and that ensures the convergence of the observer ( : values of ek v and e R for which the convergence 5

27 of the observer is obtained; : values of of the observer is not obtained). ek v and e R for which the convergence 6

28 Table ( ) Kˆ t = 0, J s - m - K - Kv v e ˆ ( 0) t =, s - e R case % % case % % case % % case % %

29 Table ( ) Kˆ t = 0, J s - m - K - Kv v e ˆ ( 0) t =, s - e R case % % case % % case % % case % %

30 Figure Temerature, K case Temerature, K case Front Position, H/L x 0-5, m s case case Kv, J s- m - K - Front Position, H/L x 0-5, m s case case Kv, J s- m - K - Time, h Time, h

31 Figure 45.0 Temerature, K (a) Front Position, H(t)/L (b) K v, J s - m - K Time, h (c) x 0-5, m s (d) Time, h

32 Figure 3 e, % e Kv, %

33 Figure Temerature, K (a) Front Position, H(t)/L (b) K v, J s - m - K Time, h (c) x 0-5, m s (d) Time, h

34 Figure 5 4 Temerature, K () () Temerature, K (3) Time, h (4) Time, h

35 Figure 6 e, % e Kv, %