First-order differential equations in chemistry

Scholz, Gudrun; Scholz, Fritz

doi:10.1007/s40828-014-0001-x

Cited by 66 publications

(18 citation statements)

References 7 publications

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“…Another important criterion for the mixing characterization is the characteristic mixing time. For that, we consider that the segregation intensity can be simply modeled by a dynamic first order system model without (∝ e −t/τ ) or with time delay (∝ e −(t−t d )/τ ) [29]. In this case, the time constant of the mixing can be deduced by the fitted curve of the first order system, indicated in Figure 8.…”

Section: Methodsmentioning

confidence: 99%

Mixing intensification under turbulent conditions in a high pressure microreactor

Zhang

Marre

Erriguible

2020

Chemical Engineering Journal

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The turbulent mixing of two miscible fluids is investigated in a high pressure (HP) coflow microreactor operated at 100 bar. Ethanol and CO 2 are selected as model solvents to mimic the final targeted application, i.e.: a supercritical antisolvent process at microscale (µSAS). We first demonstrate experimentally that turbulent mixing can be reached in a microchannel using HP microfluidics. A computational fluid dynamic (CFD) model, performed using direct numerical simulation (DNS) down to the Kolmogorov scale has been applied for the turbulent mixing simulations. The effects of the main operating parameters on the final mixing efficiency has been studied, namely: the temperature, the fluid flowrates, the microchannel dimensions and the capillary inner and outer diameters. According to a predefined intensity of segregation, the characteristic mixing times are determined and used for determining mixing efficiency. The ratio of the total mixing time to the diffusion time depends on the ratio of the kinetic energies (the outer fluid to the inner one). The obtained micromixing times have been related to the turbulent energy dissipation rate ϵ, calculated directly from the velocity fluctuations. The mixing intensification is obtained with much lower characteristic mixing times in the microreactor (one order of magnitude) than previously reported. This fundamental study is an indispensable guidance for several processes, including the µSAS applications.

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Section: Methodsmentioning

confidence: 99%

Mixing intensification under turbulent conditions in a high pressure microreactor

Zhang

Marre

Erriguible

2020

Chemical Engineering Journal

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“…According to the chemical reaction kinetics and the above equations or chemical reaction schemes [27], the following differential‐equation‐based model structure can be obtained to represent the major photosynthesis kinetics for CO 2 fixation and diffusion in C 3 plants. These state equations were developed from the four subprocesses of photosynthesis

\frac{d x_{1}}{d t} = 2 x_{1}^{2} x_{2}^{3} (- k_{1}) + x_{6} k_{4} (s_{1} - x_{1})

\frac{d x_{2}}{d t} = 3 x_{1}^{2} x_{2}^{3} (- k_{1}) + x_{5} k_{3} (s_{2} - x_{2}) + 3 x_{7}^{5} k_{5} (s_{2} - x_{2})^{3}

\frac{d x_{3}}{d t} = x_{3} x_{4} m (- k_{2}) + 3 x_{7}^{5} k_{5} (s_{2} - x_{2})^{3}

\frac{d x_{4}}{d t} = x_{3} x_{4} (- k_{2}) + (x_{9} - x_{4}) k_{6}

…”

Section: Model Structure Developmentmentioning

confidence: 99%

Modelling and simulation of photosynthetic activities in C ₃ plants as affected by CO ₂

Wang

Tang

Xia

et al. 2019

IET Systems Biology

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CO 2 concentration ([CO 2 ]) in a greenhouse may be a limiting factor for plant growth. Current greenhouse CO 2 control strategy usually depends on expert experience, which may control [CO 2 ] in a moderate range but cannot make it optimal due to lack of considering plant photochemistry reactions. A state-space kinetic model structure covering major photosynthetic reactions as affected by CO 2 is useful for [CO 2 ] control strategy development in a greenhouse because modern control theories are usually based on state-space models. In this work, a state-space kinetic model structure for photosynthesis was built, which describes the major reaction cascades of photophosphorylation, Calvin cycle, and biophysical processes such as CO 2 transport through the stomata under moderate [CO 2 ] range without considering photorespiration. Simulations were performed with a large range of model parameters to demonstrate the effect of [CO 2 ] on stable sugar production and the flexibilities of the developed model structure. The results clearly show whether increasing of CO 2 will lead to more production of sugar or not in different scenarios. The model structure may be extended to cover other photosynthetic influence factors such as temperature by using the well-known Arrhenius equation.

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“…Understanding the relationships among these parameters and their dynamic processes is important to optimize the system for electricity generation and wastewater treatment [19,66,67]. Specifically, DEs are the most frequently applied techniques [52][53][54][55]. Based on the common mathematical methods used in BES, in the following, DEs are divided into ordinary differential equations (ODEs) and partial differential equations (PDEs) to discuss their motivations, advantages and limitations.…”

Section: Overviewmentioning

confidence: 99%

“…The objective of ODE is to find the relationships between input x and output y by describing the differential changes of output under change of input [52,68].…”

Section: Introduction Of Ordinary Differential Equationsmentioning

confidence: 99%

A Review of Modeling Bioelectrochemical Systems: Engineering and Statistical Aspects

et al. 2016

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Bioelectrochemical systems (BES) are promising technologies to convert organic compounds in wastewater to electrical energy through a series of complex physical-chemical, biological and electrochemical processes. Representative BES such as microbial fuel cells (MFCs) have been studied and advanced for energy recovery. Substantial experimental and modeling efforts have been made for investigating the processes involved in electricity generation toward the improvement of the BES performance for practical applications. However, there are many parameters that will potentially affect these processes, thereby making the optimization of system performance hard to be achieved. Mathematical models, including engineering models and statistical models, are powerful tools to help understand the interactions among the parameters in BES and perform optimization of BES configuration/operation. This review paper aims to introduce and discuss the recent developments of BES modeling from engineering and statistical aspects, including analysis on the model structure, description of application cases and sensitivity analysis of various parameters. It is expected to serves as a compass for integrating the engineering and statistical modeling strategies to improve model accuracy for BES development.

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First-order differential equations in chemistry

Cited by 66 publications

References 7 publications

Mixing intensification under turbulent conditions in a high pressure microreactor

Mixing intensification under turbulent conditions in a high pressure microreactor

Modelling and simulation of photosynthetic activities in C ₃ plants as affected by CO ₂

A Review of Modeling Bioelectrochemical Systems: Engineering and Statistical Aspects

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First-order differential equations in chemistry

Cited by 66 publications

References 7 publications

Mixing intensification under turbulent conditions in a high pressure microreactor

Mixing intensification under turbulent conditions in a high pressure microreactor

Modelling and simulation of photosynthetic activities in C 3 plants as affected by CO 2

A Review of Modeling Bioelectrochemical Systems: Engineering and Statistical Aspects

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Modelling and simulation of photosynthetic activities in C ₃ plants as affected by CO ₂