Modelling and optimization of seeded batch crystallizers

doi:10.1016/j.compchemeng.2004.09.011

Computers & Chemical Engineering

Volume 29, Issue 4, 15 March 2005, Pages 911-918

https://doi.org/10.1016/j.compchemeng.2004.09.011 Get rights and content

Abstract

The existing model identification methods for estimating nucleation and crystal growth kinetic parameters are generally based upon simplified population balance models, such as moment equations, which contain insufficient information on the crystal size distribution (CSD). This paper deals with model identification and optimization of batch-seeded crystallizers. The final product CSD, temperature profile and concentration profile are used for identification. The reliability and precision of estimates are analyzed. Optimal temperature profile is determined such that an objective function pertinent to the final product qualities is minimized. The optimization algorithm finds the minimum of an objective function with respect to a parameter vector temperature input, subject to the mass balance and energy balance dynamics as well as the population balance equation (PBE). Simulation results for different objective functions are given to show the effectiveness of the proposed method. Compared with linear cooling profile, the optimal cooling profile is able to reduce the volumes of fines by 53.2% to improve product quality.

Introduction

The product quality is usually determined by the crystal size distribution (CSD). In the case of batch crystallization, temperature profile affects supersaturation profile which has strong influence on the CSD. Thus, optimization of temperature profiles is of great importance. Designing the optimal temperature profiles relies on the accuracy of the population balance model.

Significant previous work has focused on optimization-based methods for kinetic parameter estimation for batch crystallizers. Palwe, Chivate, and Tavare (1985) used three different methods including polynomial fitting, initial derivatives and optimization procedure to determine the growth rate kinetics of ammonium nitrate. They noted that the least squares error optimization procedure was potentially the most accurate and precise. Witkowski, Miller, and Rawlings (1990) implemented optimization-based estimation of crystallization kinetics by minimizing a weighted least squares objective function which represented the error between the estimated and experimental data. They used on-line concentration and transmittance data from a batch naphthalene–toluene system, in which the error weights were chosen arbitrarily. Qiu and Rasmuson (1991) used concentration and final product CSD data to determine kinetic parameters for an aqueous succinic acid system. They simplified the population balance equation (PBE) to a first-order quasilinear equation by assuming that the crystal growth rate can be expressed as a product of a function of supersaturation and a function of crystal size. Dash and Rohani (1993) estimated crystallization kinetics of potassium chloride. They included crystal mass distribution from a sieve analysis in the objective function. Monnier, Fevotte, Hoff, and Klein (1997) used calorimetry and image analysis to identify nucleation and growth kinetic parameters of adipic acid in water.

Mullin and Nyvlt (1971) studied the cooling control of aqueous potassium and ammonium sulfate system to maximize the final size of the seed crystals. Ajinkya and Ray (1974) developed an optimal operating policy using the ammonium sulfate model to maximize the average crystal size with respect to the temperature profile. Jones and Mullin (1974) applied optimal control theory to obtain an optimal cooling profile using the method of moments for a size-independent growth population balance model. They verified that the optimal cooling trajectory increases average crystal size as compared with linear and natural cooling strategies. Miller and Rawlings (1994) developed an optimal control scheme with constraints. Using a method similar to Miller and Rawlings (1994) and Matthews (1997) designed an optimal temperature schedule to improve the filtration of final-time slurries of a photochemical. The profile minimizes the mass of nucleated crystal relative to seed crystal mass, subject to supersaturation and yield constraints. Similarly, Chung, Ma, and Braatz (1999) investigated the seed size effect on the final-time product. Zhang and Rohani (2003) developed an on-line optimal control method for a seeded batch cooling crystallizer to improve the product quality. Ma, Tafti, and Braatz (2002) simulated an optimal control of a batch crystallization process. Most of these optimal profiles are resulted from applying the method of moments to the population balance equation (PBE), and therefore are limited to processes with size-independent growth rate model only.

In this work, we solve the PBE directly instead of reducing it to a set of ordinary differential equations (ODEs) or changing the PBE to simple forms under some assumptions, so it can be applied to cases with size-dependent growth rate. This paper presents a new optimization-based methodology to identify kinetic parameters of batch crystallization. An optimal temperature profile is determined for a chosen performance measure. The proposed strategy is evaluated through simulations for a seeded batch crystallizer.

Section snippets

Modelling and dynamics of a seeded batch crystallizer

A mathematical framework suited to modelling crystallization processes is the population balance, which describes the state of the CSD. The process model of a seeded batch crystallizer has the following form (Rawlings, Miller, & Witkowski, 1993; Shi, El-Farra, Li, Mhaskar, & Christofides, 2004): $\frac{\partial n (r, t)}{\partial t} + \frac{\partial (G (r, t) n (r, t))}{\partial r} = 0$ $\frac{d C}{d t} = - 3 ρ k_{v} G (t) μ_{2} (t)$ $\frac{d T}{d t} = - \frac{U A}{M c_{p}} (T - T_{j}) - \frac{Δ H}{c_{p}} 3 ρ k_{v} G (t) μ_{2} (t)$ where $ρ$ is the density of crystals, $k_{v}$ is the volumetric shape factor, U is the overall heat-transfer coefficient, A is

Model solution

In the absence of aggregation and breakage, a representation of population balance can be shown in Fig. 1, where population balance distributions at time $t = t_{1}$ and $t = t_{1} + Δ t$ are demonstrated. The particles grow into the size range $r_{2} + Δ r_{2}$ from size range $r_{1} + Δ r_{1}$ over the time interval $Δ t$ . $n (r_{1}, t_{1})$ and $n (r_{2}, t_{1} + Δ t)$ represent the population density at time $t_{1}$ and $t_{1} + Δ t$ , respectively. The population balance implies (Hu, Rohani, Wang, & Jutan, 2004): $n (r_{1}, t_{1}) Δ r_{1} = n (r_{2}, t_{1} + Δ t) Δ r_{2}$ Based on the definition of

Parameter estimation

If we define the error $e_{i j}$ as the difference between the measured and predicted values of a variable, $e_{i j} (θ) = y_{i j} (θ) - {\tilde{y}}_{i j} (θ)$ where $\tilde{y}$ denotes the predicted value, and a variable subscripted by ij denotes the j th value of the i th variable. The least squares procedure consists of finding the values of $θ$ which minimize the function $Φ (θ) = \sum_{i = 1}^{N_{m}} \sum_{j = 1}^{N_{i}} e_{i j}^{2} (θ)$ i.e., we minimize the sum of squares of the residuals. $N_{m}$ is the number of variables and $N_{i}$ is the number of measurements of each variable.

The

Generation of optimal control profiles

Once the model of a batch crystallization process is identified through nucleation and growth kinetics parameter estimation, the optimal open-loop control profile can be calculated by solving a nonlinear program. A general statement of the optimal control problem is: $\min_{T (t)} Ψ (T (t), n (r, t_{f}))$ $\begin{matrix} subject to: & Crystallizer model \\ h_{1} (T (t), n (r, t)) = 0 \\ h_{2} (T (t), n (r, t)) \geq 0 \end{matrix}$ where $T (t)$ is the vector of the piecewise continuous input temperature profile, and $h_{1}$ and $h_{2}$ are path constraints which could be imposed on

Conclusions

An approach to estimate the kinetic parameters of crystal nucleation and growth is presented. The kinetic parameters are estimated using the final product CSD, the solute concentration and reactor temperature profiles. The reliability and precision of the estimated are investigated by the 95% confidence interval. The determination of optimal temperature profiles for batch crystallizers is discussed. Objective function minimizing the third moment between the nucleated crystal and grown seed

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Citation Excerpt :
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Modelling and optimization of seeded batch crystallizers

Abstract

Introduction

Section snippets

Modelling and dynamics of a seeded batch crystallizer

Model solution

Parameter estimation

Generation of optimal control profiles

Conclusions

Chemical Engineering Science

Computers and Chemical Engineering

Chemical Engineering Science

Chemical Engineering Science

Chemical Engineering Science

On the optimal operation of crystallization process

Chemical Engineering Communications

Nonlinear control of particular processes

AIChE Journal

Optimal seeding in batch crystallization

The Canadian Journal of Chemical Engineering