Analysis of breakage kernels for population balance modelling

Patruno, L.E.; Dorao, Carlos Alberto; Svendsen, Hallvard F.; Jakobsen, Hugo A.

doi:10.1016/j.ces.2008.09.029

Cited by 41 publications

(18 citation statements)

References 16 publications

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“…An efficient algorithm is required for this task. 38,1], and test various polynomial orders in [0, 0.38]. For example, the element solution for = 100 in Figure 7(e) is improved in Figure 8(a).…”

Section: The Element Frameworkmentioning

confidence: 99%

“…[1,[16][17][18]28] Compared to the orthogonal collocation method and the low order finite difference and finite volume methods, the Galerkin and tau methods are rarely observed in the solution algorithm of chemical reactor model equations. [32][33][34][35] The least-squares method has been adopted for the solution of population balance problems, [36][37][38][39][40][41][42] fixed packed bed reactors, [43,44] catalyst pellet problems [45,46] and Knuii-Levenspiel type of model for fluidized bed reactors. [32][33][34][35] The least-squares method has been adopted for the solution of population balance problems, [36][37][38][39][40][41][42] fixed packed bed reactors, [43,44] catalyst pellet problems [45,46] and Knuii-Levenspiel type of model for fluidized bed reactors.…”

Section: Introductionmentioning

confidence: 99%

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Evaluation of spectral, spectral‐element and finite‐element methods for the solution of the pellet equation

Solsvik

Jakobsen

2014

Can J Chem Eng

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Several numerical methods (orthogonal collocation, Galerkin, tau, least-squares and least-squares with a direct minimization algorithm) are applied to solve a linear diffusion-reaction problem. The spectral, finite-element and spectral-element frameworks are employed to investigate the methods. Overall, the Galerkin and tau methods are considered the most universal methods. Spectral framework: With sufficient diffusion limitations, the least-squares method suffers in general from significantly lower accuracy than the Galerkin, tau and orthogonal collocation methods. On the other hand, the least-squares method with a direct minimization algorithm provides favourable lower system matrix condition numbers than the conventional least-squares approach. Hence, for higher diffusion limitations, the least-squares direct minimization formulation provides higher numerical accuracy than the conventional least-squares method, but is still not as accurate as the Galerkin, tau and orthogonal collocation techniques. The accuracy of the least-squares solution can compete with the other methods only in cases with low gradients in the solution. Element framework: For a highly diffusion limited problem, the element framework is considered favourable as compared to the spectral framework. On the other hand, the element approach is not as efficient as the spectral solution for small Thiele modulus solutions.

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Section: The Element Frameworkmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Evaluation of spectral, spectral‐element and finite‐element methods for the solution of the pellet equation

Solsvik

Jakobsen

2014

Can J Chem Eng

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“…(Mahoney et al, 2002; Muralidar and Ramkrishna, 1986; Muralidar and Ramkrishna, 1989; Patruno et al, 2008; Patruno et al, 2009; Sathyagal et al, 1995; Wright and Ramkrishna, 1992). While these studies can provide invaluable guidance, their methods cannot be directly applied to address the challenges of inverse problems resulting from cell population balance models.…”

Section: Solving the Inverse Problemmentioning

confidence: 99%

Single-cell behavior and population heterogeneity: Solving an inverse problem to compute the intrinsic physiological state functions

Spetsieris

Zygourakis

2012

Journal of Biotechnology

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The dynamics of isogenic cell populations can be described by cell population balance models that account for phenotypic heterogeneity. To utilize the predictive power of these models, however, we must know the rates of single-cell reaction and division and the bivariate partition probability density function. These three intrinsic physiological state (IPS) functions can be obtained by solving an inverse problem that requires knowledge of the phenotypic distributions for the overall cell population, the dividing cell subpopulation and the newborn cell subpopulation. We present here a robust computational procedure that can accurately estimate the IPS functions for heterogeneous cell populations. A detailed parametric analysis shows how the accuracy of the inverse solution is affected by discretization parameters, the type of non-parametric estimators used, the qualitative characteristics of phenotypic distributions and the unknown partitioning probability density function. The effect of finite sampling and measurement errors on the accuracy of the recovered IPS functions is also assessed. Finally, we apply the procedure to estimate the IPS functions of an E. coli population carrying an IPTG-inducible genetic toggle network. This study completes the development of an integrated experimental and computational framework that can become a powerful tool for quantifying single-cell behavior using measurements from heterogeneous cell populations.

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“…Para um reator de batelada perfeitamente agitado, a equação de balanço populacional para esta situação pode ser escrita na forma (RAIKAR et al, 2006;RAMKRISHNA, 2000;PATRUNO et al,2009;COULALOGLOU;TAVLARIDES, 1977, NARSIMHAM et al, 1979NAMBIAR et al, 1992): …”

Section: Estruturação Da Equação De Balanço Populacionalunclassified

“…Para uma fragmentação binária, dois fragmentos por partículas, a restrição acima é simplificada. Sendo equivalente a uma taxaUma expressão para a taxa, em que as partículas de tamanho v são formadas devido à quebra, pode ser deduzida para um caso simples de quebra binária: A taxa de geração de partículas de tamanho v é igual à soma de todos os tamanhos maiores que v, visto que, partículas menores são formadas pela quebra, de partículas maiores: Com as equações (1.1.2) e (1.1.3), podemos escrever um balanço populacional de quebra em (1.1.5) ou, de outro modo, uma formulação para o caso da quebra dominante, como descrito por PATRUNO et al (2009). Este balanço só será apresentado mais adiante, após outras considerações.…”

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Estudo do problema inverso em balanço populacional aplicado a degradação de polímeros.

Uliana¹

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Uliana, MuriloEstudo do problema inverso em balanço populacional aplicado a degradação de polímeros / M. Uliana. SATHYAGAL et al. (1995) e a obtida no presente trabalho usando o algoritmo do problema inverso, para o casoestudo 1; (b) Distribuição de partículas filhas obtida no presente trabalho para o casoestudo 1. σ -desvio padrão φ -Função para minimização na equação (3.4.2.28) dentro da função objetivo. Palavras-chave: taxa de quebra, balanço populacional, problema inverso, polímeros, modelagem. xiv ABSTRACT.Computational algorithms are used to obtain quantitative information from experimental observations. The aim of the present work was the application of the methodology of inverse problem in population balance used to describe the evolution of the chain length distribution in different polymer degradation processes. The time evolution of the chain length distribution during the polymer breakage can be mathematically described by a population balance equation.In the so-called inverse problem, experimentally measured distributions are used to estimate the parameters of the population balance, such as the distribution of breakage rate along the chain and as function of the chain length. The inverse problem is known to be an ill-conditioned numerical problem. An algorithm previously developed in the literature for liquid droplet breakage in liquid emulsions, based on the concept of self-similarity of the distributions, was adapted and applied in the present work for the problem of polymer scission during polymer degradation. Nestes estudos têm se observado que, para um tempos longos (t→∞), a distribuição atinge o estado estacionário sem ocorrer mais mudanças. Para uma fragmentação binária, dois fragmentos por partículas, a restrição acima é simplificada. Sendo equivalente a uma taxaUma expressão para a taxa, em que as partículas de tamanho v são formadas devido à quebra, pode ser deduzida para um caso simples de quebra binária: A taxa de geração de partículas de tamanho v é igual à soma de todos os tamanhos maiores que v, visto que, partículas menores são formadas pela quebra, de partículas maiores: Com as equações (1.1.2) e (1.1.3), podemos escrever um balanço populacional de quebra em (1.1.5) ou, de outro modo, uma formulação para o caso da quebra dominante, como descrito por PATRUNO et al. (2009). Este balanço só será apresentado mais adiante, após outras considerações.Os métodos de diferença finita no domínio das partículas são usados com muita freqüência na literatura para a discretização, como nos ensina RIBEIRO et al. (1995).Recentemente para a solução do BP foi desenvolvido no trabalho de HAGESAETHER; JAKOBSEN; SVENDSEN (2002) Dessa forma, a função de quebra também será discretizada, e resultando em β(v i ,v j ). O sucesso no uso das equações fundamentais de balanço populacional para predizer quantitativamente a distribuição de partículas em um sistema em que a quebra é dominante requer duas funções chave: a taxa de quebra que corresponde a um valor para cada volume 5 de partícula, e a distribu...

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Analysis of breakage kernels for population balance modelling

Cited by 41 publications

References 16 publications

Evaluation of spectral, spectral‐element and finite‐element methods for the solution of the pellet equation

Evaluation of spectral, spectral‐element and finite‐element methods for the solution of the pellet equation

Single-cell behavior and population heterogeneity: Solving an inverse problem to compute the intrinsic physiological state functions

Estudo do problema inverso em balanço populacional aplicado a degradação de polímeros.

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