SOLUTION OF POPULATION BALANCE EQUATIONS BY MWR
SOLUTION OF POPULATION BALANCE EQUATIONS BY MWR
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MWR 求解人口平衡方程
DOI：
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发表时间：
2001
期刊：
影响因子：
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通讯作者：
D. RAMKRIsHNAt
中科院分区：
文献类型：
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作者：
N. P.;D. RAMKRIsHNAt
用加权残值法求解了考虑任意晶体断裂影响的连续混合悬浮法和混合产物脱除结晶器中的结晶粒数平衡方程。所使用的试函数是由Gram-Schmidt正交化过程生成的特定于问题的多项式，在内积的定义中具有适当的权函数。权函数来自原始人口平衡方程的多个版本的解析解。所使用的方法还包括Vorobyev矩方法，该方法本质上是加权残值法的变体。只有四个特定问题的多项式可以得到精确的解，因此这些试函数的评级高于更标准(和传统)的选择，如拉盖尔多项式，即使使用大量的函数，也不能获得满意的解。范围-粒子数平衡[L]对描述分散相体系的行为很重要，但由于由此产生的积分方程组或积分-微分方程解的困难，在化学工程中的应用有限。过去的尝试集中在通过解矩方程来估计出现在方程中的数密度函数的矩[L，2，9]。这种方法有很大的局限性[3]。本文提出了求解结晶过程中选定的布居平衡方程的有效方法。这些例子包括一个在过去曾违抗解决方案的例子。本文提出的求解方法是一般加权残值法的应用，其中前面提到的矩方程法是特例(见[3])，试函数是专门构造的[6]。这些特定的试函数是通过使用适当加权的内积对集合(~‘7)进行Gram-Schmidt正交化来获得的。内积中的权函数的性质在[6]之前已经讨论过了。
The population balance equation for crystallization in a continuous mixed suspension and mixed product removal crystallizer accounting for the effects of arbitrary crystal breakage (an outstanding problem) has been solved by the method of weighted residuals. The trial functions used were problem-specific polynomials generated by the Gram-Schmidt orthogonalization process with a suitable weight function in the definition of the inner product. The weight function emerged from the analytical solution of multilated versions of the original population balance equation. The methods used also included a Vorobyev’s method of moments which is essentially a variation of the method of weighted residuals. Accurate solutions were obtained with only four problem-specific polynomials thus rating these trial functions above the more standard (and traditional) choices like Laguerre polynomials with which no satisfactory solutions could be obtained even when a large number of functions were employed. Scope-Population balances[l], which are important to the description of the behavior of dispersed phase systems, have found limited applications in chemical engineering because of the difficulty in obtaining solutions to the integral or integro-differential equations which arise therefrom. Attempts in the past have concentrated in estimating the moments of the number density function appearing in the equation through the solution of moment equations[l, 2,9]. This approach has powerful limitations[3]. In this paper, efficient methods of solution have been presented to solve selected population balance equations arising in cystallization. The examples include one that has defied solution in the past. The method of solution presented in this paper is an application of the general method of weighted residuals, of which the method of moment equations referred to earlier is a special case (see, e.g. [3]), with trial functions that were constructed specifically[6]. These specific trial functions are obtained by Gram-Schmidt orthogonalization of the set (~‘7 using suitably weighted inner products. The nature of the weight functions in the inner product has been discussed before [6].