Most batch processes, including many fermentation processes, pass through several phases based on complex physiological phenomena during the progress of the batch (Figure 6.13). In this book, we used the term "stage" to refer to different process operations such as fermentation and separation, and the term "phase" to refer to distinct episodes in time during the progress of the batch where qualitatively different activities take place. Since batch fermentation time varies from batch to batch due to complex physiological behavior and operational changes, the data sets for different batches will have different lengths and shifted phase changing points or process landmarks. These shifts can affect monitoring activities and generate false alarms. Consequently, alignment of landmarks is necessary for comparing similar events. Multivariate analysis requires the data to be stacked in a matrix (or in a three-way array) prior to empirical modelling. Several techniques have been suggested for batch data length synchronization and equalization [270, 296, 418, 522, 641], Cutting batch data lengths to length of the variable with the shortest data sequence is not recommended because of significant information loss generated by discarding data. When the time between the shortest batch and the longest batch is large, or the process in question is very sensitive to small changes in operational or environmental conditions, robust and generic methods are needed to synchronize and equalize data lengths.
Two problems will be addressed in this section: equalization of batch data lengths, and detection and alignment of phase change landmarks. Three methods are discussed for equalizing batch data lengths:
• Indicator Variable Technique
• Dynamic Time Warping (DTW)
• Curve registration
The simple popular technique based on an indicator variable is discussed in Section 6.3.1. Then, the dynamic time warping method is presented in Section 6.3.2. Finally, time warping by functional data analysis (also called curve registration) is discussed in Section 6.3.3 and comparative examples are provided.
This technique is based on selecting a process variable to indicate the progress of the batch instead of time. Each new observation is taken relative to the progress of this variable. The indicator variable should be smooth, continuous, monotonic and spanning the range of all other process variables within the batch data set. Linear interpolation techniques are used to transform batch-time dimension into indicator variable dimension. This variable should be chosen appropriately such that it also shows the maturity or percent completion of each batch. This variable can be for example percent conversion or percent of a component fed to the fermenter. For monitoring new batches, data are collected from all process variables o a.
at specified time intervals and then adjusted with respect to the indicator variable. In this technique, a measure of the maturity or percent completion of any batch is provided by the percentage of its final value that has been attained by the indicator variable at the current time. Several successful applications of this approach can be found in the literature, mostly for batch/semi-batch polymerization processes, reaction extent or percent of component fed being the indicator variables [296, 418]. An application for fermentation processes has been also given in the literature ,
Choosing an indicator variable in batch fermentation processes depends on the process operation and characteristics. If the process is a batch fermentation, the choice of this variable is simpler than processes with batch and fed-batch phases. For batch fermentations, there may be several variables, which can serve as indicator variables such as substrate concentration, product concentration or product yield. In the fed-batch case, in addition to the aforementioned variables, percent substrate fed is also an indicator variable. This percentage is calculated by fixing the total amount of substrate added into the fermenter based on some performance criteria. This end point (total amount of substrate fed), which is eventually reached in all batches, defines a maturity point. For more complex operations such as batch operation followed by fed-batch operation, which is very common for non-growth associated products such as antibiotics, different approaches to choosing indicator variables can be considered. Batch and fed-batch phases of the operation can be treated separately so that appropriate indicator variables can be determined for individual phases. Implementation of this two-phase operation is illustrated in the following example.
Example. Assume that data are available from 5 runs of a batch followed by fed-batch penicillin fermentation. Potential process variables are shown in Figure 6.14 for all batches before data pretreatment. Based on simulation studies, data were collected using 0.2 h of sampling interval on each variable for each batch resulting in total batch lengths varying between 403.8 h (2019 observations) and 433.6 h (2168 observations). When these variables are assessed for use as an indicator variable, none of them seem appropriate. Most of these variables contain discontinuities because of the two operating regions (batch and fed-batch) and some of them are not smooth or monotonically increasing/decreasing. Since none of the variables can be chosen as an indicator variable that spans the whole duration of fermentation, a different approach is suggested. The solution is to look for different indicator variables for each operating region. In order to achieve this mixed approach fermentation data are analyzed. For the first operating (batch operation) region, substrate concentration in the fermenter can be
considered as a good candidate since it can be started from the same initial value and terminated at the same final value for each batch. The initial and final substrate concentrations are fixed to 15 g/L and 0.4 g/L, respectively to implement this idea. Instead of reporting data as a function of time for these batches, data are reported on each variable for each batch at every decrease of 0.5 g/L in substrate concentration using linear interpolation.
Choosing substrate concentration decrease (substrate consumption) as an indicator variable for the batch operation provides another advantage, it defines the end of batch operation or in other words the switching point to fed-batch operation. Since the operating conditions are slightly different and there are some random changes in microbial phenomena, the switching point is reached at different times for each batch resulting in different number of observations (Figure 6.15). While the number of observations is varying between 210 and 218 before equalization, after implementing the indicator variable technique there are only 147 observations taken from each batch.
Substrate concentration cannot be used for the fed-batch operation region since substrate is added continuously to promote penicillin production and biomass maintenance such that substrate concentration approximately constant until the end of the run. In the second region (fed-batch), amount of substrate fed to the fermenter can be considered as an indicator variable since it somehow defines the end point of the fermentation. Figure 6.16 is used to decide the approximate amount of substrate needed to reach the desired final penicillin concentrations under normal operating conditions. Analysis shows approximately 16 L of substrate would be necessary for each batch. To calculate this amount, fermentations were carried out fairly long enough (approx. 600 h) to see a decline in penicillin concentration. A mean trajectory is then calculated and its maximum is used to determine the total amount of substrate to be added. Batch/fed-batch switching times, maximum and final penicillin concentrations for each batch including mean trajectory values are given in Table 6.3. The following calculations then become straightforward
Pfinai = 1.3525 g/L , iswitch = 42 h, ifinai = 420 h,
i where Pfinai denotes final penicillin concentration, ¿switch beginning of the fed-batch period, tfinai end of fermentation for the mean trajectory, Fz, instantaneous value of the substrate feed rate and Aij, instantaneous sampling interval (which is 0.2 h in the example). When the maximum of penicillin concentration of the mean trajectory is used by assuming that the values Pmax = 1.3664 g/L, iswjtch = 42.0 h, tfjna] = 446.2 h and Fi — 0.0394 L/h are predetermined by analyzing the mean trajectory for penicillin concentration, the approximate total amount of substrate is calculated as Fl x (tfina, - iswitch) = 0.0394 x (446.2 - 42.0) = 15.9055 L. This amount can be calculated more accurately by using Eq. 6.47 resulting a closer value to 16 L. Although 16 L is not the exact outcome of the calculations, it was chosen to round off the number and introduce a little safety margin (using a little more substrate than the required minimum). The resulting final penicillin concentrations do not deviate substantially from their maximum values (Table 6.3) verifying this choice. The result of equalization is shown in Figure 6.17 for several variables based on the calculations above.
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