(373g) GPU Acceleration As Low Cost Enabling Technology for Optimization Based Synthesis of Novel Integrated Crystallizer-Wet Mill Systems
There are numerous approaches to control the CS during the crystallization. To understand the basic principle of the state-of-the art crystallization control techniques as well as their fundamental limitations, it is important to understand the basics of crystallization. For simplicity, in this work we consider cooling crystallization process, which relies on the fact that the solubility is temperature dependent. The solubility is an increasing line in the concentration-temperature (c-T) phase diagram: under the solubility concentration the crystals dissolve, whereas over the solubility spontaneous growth and nucleation may happen. There is another important line in the supersaturated region: the metastable limit (ML). Between the ML and solubility, spontaneous nucleation does not happen, but the existing crystals grow. The solubility is a thermodynamic property, but the ML is an uncertain kinetic curve. Hence, the factors impacting the notoriously uncertain nucleation rate, including the solid volume fraction, cooling rate, certain impurities even in ppm amounts, also impacts the ML.
These information gives the foundation of a simple and yet powerful control strategy: the supersaturation control (SSC). The essence of SSC is to seed at high temperature solution within the metastable zone, and then keep the concentration under the ML limit. Hence, SSC is a concentration based feedback control, by manipulating the crystallizer temperature, and it is to reject nucleation and promote the growth of seeds which will then become a population of large crystals with narrow distribution. Although, due to the uncertain nature of the ML, secondary nucleation often happens during SSC. Since SSC control the CS indirectly, it is not able to detect nucleation, which is its major limitation. There are numerous in-situ techniques that provides the relative crystal number (RCN), including the focused beam reflectance measurement (FBRM) and in-line imaging tools. Direct nucleation control (DNC) relies on controlling the RCN and uses the simple principle that the RCN is inversely proportional to the mean CS. DNC is generally started from high temperature, highly concentrated but not necessarily supersaturated solution with cooling to nucleation, which will likely increase the RCN over the setpoint. Then, automatic heat-up stage is triggered for selective dissolution. When the RCN sinks below the setpoint, the system switches to cooling mode again. The cycles of heating and cooling are repeated until the desired temperature is reached. There are two important observations: (i) DNC automatically employs temperature cycles, which is known to improve the crystal quality; (ii) no preliminary system information is required, in contrast, the DNC automatically determines the solubility as well as primary and secondary nucleation ML. On the downside, DNC is not always converging to the desired temperature (e.g. generates sustained oscillations), and it often leads to excessively long batch times. Various model based control (MBC) techniques were successfully employed to overcome the limitations of the DNC and SSC. The major limitation of MBCs remains the high expertise required to their implementation, and its high complexity level. In addition, MBC faces the same limitations by the achievable CS range as DNC and SSC. Hence, new operation policies that combine the advantages of the existing techniques (temperature cycles for improved quality, operates at low supersaturation for nucleation rejection and involve the prediction power of the process models) are largely required by the chemical industries and academic research communities.
Most recently, wet milling was applied during crystallization. These pioneering studies revealed that wet-milling increases tremendously the overall system flexibility as well as the achievable product property space. Being a promising system that addresses multiple issues for various industries, the interest of generating high level guidelines for quick yet efficient operation design cannot be overestimated. Although the pure experimental investigation of novel systems is always a straightforward path to follow for fine-tuning the operation for a particular process, this is time and resource demanding and can hardly provide answers for general design questions. This brings to the conclusion that process models are to be employed to gain valuable system understanding that permits to generalize the optimum operating profiles, subject to various constraints and objectives. This requires highly detailed and accurate process models with sufficiently low computational burden.
Population balance models (PBMs) are applied for decades to describe the population density function dynamics during solution crystallization. PBMs generally takes into account the nucleation, growth and dissolution of crystals, but secondary mechanisms such as agglomeration and breakage are also often considered. Hence, PBMs are well suited to describe the crystallization in integrated crystallizer wet-milling systems. One of the greatest challenges, if one wants to fit the process model to experimental data, is to find the most suitable thermodynamic and kinetic rate equations. However, in our work the main goal to accomplish is the deep understanding of the integrated process dynamics and behaviour, therefore we apply the standard rate equations for the nucleation, dissolution, crystal growth (which are considered along the length and width axes as independent rate equations) and breakage. A similarly important challenge is the application of the right model solution technique. For extended high dimensional PBMs the preferred techniques are the moment based methods (MBMs), due to their great accuracy and low computational expense. There are two problems associated with the MBMs: they cannot handle dissolution, and cannot provide the CSD, only its mean values. Any of these is disqualifying reason for this application. Finite volume based methods (FVM) were successfully applied for the solution of PBMs with acceptable accuracy. However, especially for high dimensional extended PBMs, the computational burden becomes too large for optimization based process synthesis applications.
Accelerating the FVM solution seems to be the enabling technology for optimization based crystallizer-wet mill design. There are several ways to accelerate the numerical solution, from parallel/cloud computing to numerical improvements. For an engineering application, beyond off-line optimization based system design, the feasible way forward is to use locally available computational power that also enables the on-site implementation of model based control approaches. This excludes the application of supercomputers or cloud computing services. In contrast, we propose the application of graphical processing units (GPUs) as low-cost high performance accelerators. GPUs, due to their massively parallel hardware architecture are well suited to execute large number of parallel tasks significantly faster than the central processing unit (CPU). We developed a hybrid CPU-GPU implementation of the FVM model-equations of the integrated system, which mutually benefits from the high single core CPU performance for serial operation (mass and energy balances, kinetic rate equations etc.) and the parallel architecture of the GPU for the FVM and integral calculations. Naturally, the code is optimized to account for the GPU specific issues, such as minimized data transfer between the main and GPU memory as well as optimized parallel computing by careful synchronization of the thousands of parallel GPU threads. The hybrid CPU-GPU solver written in CUDA C brings up to two order of magnitude speedup compared to the pure serial C implementation, which is up to two orders of magnitude faster than the standard Matlab program.
GPU acceleration enabled to speed the numerical solution up to the level required for process optimization. The solution of large number of highly diverse optimization problems involving both 1D and 2D PBMs, various breakage descriptions, initial conditions and optimization objectives revealed clear similarities: the optimum operating pattern is system/material independent and it relies on crystallization mechanisms scheduling. At the beginning of the process, there is a simultaneous cooling and milling for nucleation, attrition and breakage, which is followed by a heating stage. During the heat-up the mill is switched off, and the excess of small crystals is selectively dissolved, and the damaged crystal layers (by the wet mill) are removed by dissolution. The in-situ seed generation is followed by a traditional cooling crystallization. This general pattern permits to create a low dimensional design of experiment (DOE) framework which is to determine the system dependent milling as well as heating/cooling rates.
B.Szilagyi, Z.K.Nagy,Comp.Chem.Eng, Under revision
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