Mathematical Modeling in Chemical Engineering: A Tool to Analyse Complex Systems

Mathematical modeling is an attempt to describe a slice of reality in mathematical terms. In Chemical Engineering, mathematical modeling is used for simulation, control and optimization of a process and it is also a tool to design the industrial devices. Mathematical modeling is a technique commonly in place also in both theoretical and experimental studies of chemical processes. In the present chapter mathematical modelling applications to complex systems as a consequence of structure heterogeneity and involved various physical-chemical phenomena are presented. Particular attention will be focused on improving the quantitative understanding of the basic phenomena of a process that can come from the use of mathematical models. Specific task is also demonstrating how, through the use of information coming from experimental investigations and simulations, it is possible checking on the validity of the assumptions made and fine tuning the predictive mathematical model capability. The possibility of analyzing and quantifying the role played by each single step of the process is examined in order to define the relevant mathematical expressions. The latter allows getting useful indications about the impact of different operating conditions on the role of each single step and at the very end it gives indication about the efficiency of the process itself. Next step focuses on the estimation of the significant parameters of the process. In complex systems the determination \u201ca priori\u201d of some parameters is not always feasible and they are therefore determined as a comparison of experimental and simulation data. The final result is therefore the availability of a tool, the verified and validated (V&V) mathematical model, that can be used for simulation, process analysis, process control, optimization, design. Specific reference will be made to the use of the proposed methodology on a system whose behaviour, on varying the agitation level, was quantified and validated against the results of an experimental investigation in a pilot plant. A second application will allow to analyse the effect of transport phenomena in multi-phase heterogeneous systems in order to detect the conditions at which production plant efficiency is improved

Thermal modeling of industrial-scale vanadium redox flow batteries in high-current operations

A cell-resolved model that simulates the dynamic thermal behavior of a Vanadium Redox Flow Battery during charge and discharge is presented. It takes into account, at a cell level, the reversible entropic heat of the electrochemical reactions, irreversible heat due to overpotentials, self-discharge reactions due to ion crossover, and shunt current losses. The model accounts for the heat transfer between cells and toward the environment, the pump hydraulic losses and the heat transfer of piping and tanks. It provides the electrolyte temperature in each cell, at the stack inlet and outlet, along the piping and in the tanks. Validation has been carried out against the charge/discharge measurements from a 9kW/27kWh VRFB test facility. The model has been applied to study a VRFB with the same stack but a much larger capacity, operating at \uf0b1400 A for 8 h, in order to identify critical thermal conditions which may occur in next-generation industrial VRFB stacks capable to operating at high current density. The most critical condition has been found at the end a long discharge, when temperatures above 50\ub0C appeared, possibly resulting in \u3016VO\u3017_2^+ precipitation and battery faults. These results call for heat exchangers tailored to assist high-power VRFB systems

Multiphysics Finite\u2013Element Modelling of an All\u2013Vanadium Redox Flow Battery for Stationary Energy Storage

All-Vanadium Redox Flow Batteries (VRFBs) are emerging as a novel technology for stationary energy storage. Numerical models are useful for exploring the potential performance of such devices, optimizing the structure and operating condition of cell stacks, and studying its interfacing to the electrical grid. A one-dimensional steady-state multiphysics model of a single VRFB, including mass, charge and momentum transport and conservation, and coupled to a kinetic model for electrochemical reactions, is first presented. This model is then extended, including reservoir equations, in order to simulate the VRFB charge and discharge dynamics. These multiphysics models are discretized by the finite element method in a commercial software package (COMSOL). Numerical results of both static and dynamic 1D models are compared to those from 2D models, with the same parameters, showing good agreement. This motivates the use of reduced models for a more efficient system simulation

Engineering a 3D in vitro model of human skeletal muscle at the single fiber scale

The reproduction of reliable in vitro models of human skeletal muscle is made harder by the intrinsic 3D structural complexity of this tissue. Here we coupled engineered hydrogel with 3D structural cues and specific mechanical properties to derive human 3D muscle constructs ("myobundles") at the scale of single fibers, by using primary myoblasts or myoblasts derived from embryonic stem cells. To this aim, cell culture was performed in confined, laminin-coated micrometric channels obtained inside a 3D hydrogel characterized by the optimal stiffness for skeletal muscle myogenesis. Primary myoblasts cultured in our 3D culture system were able to undergo myotube differentiation and maturation, as demonstrated by the proper expression and localization of key components of the sarcomere and sarcolemma. Such approach allowed the generation of human myobundles of ~10 mm in length and ~120 \u3bcm in diameter, showing spontaneous contraction 7 days after cell seeding. Transcriptome analyses showed higher similarity between 3D myobundles and skeletal signature, compared to that found between 2D myotubes and skeletal muscle, mainly resulting from expression in 3D myobundles of categories of genes involved in skeletal muscle maturation, including extracellular matrix organization. Moreover, imaging analyses confirmed that structured 3D culture system was conducive to differentiation/maturation also when using myoblasts derived from embryonic stem cells. In conclusion, our structured 3D model is a promising tool for modelling human skeletal muscle in healthy and diseases conditions

Hydrogel-in-hydrogel live bioprinting for guidance and control of organoids and organotypic cultures

Three-dimensional hydrogel-based organ-like cultures can be applied to study development, regeneration, and disease in vitro. However, the control of engineered hydrogel composition, mechanical properties and geometrical constraints tends to be restricted to the initial time of fabrication. Modulation of hydrogel characteristics over time and according to culture evolution is often not possible. Here, we overcome these limitations by developing a hydrogel-in-hydrogel live bioprinting approach that enables the dynamic fabrication of instructive hydrogel elements within pre-existing hydrogel-based organ-like cultures. This can be achieved by crosslinking photosensitive hydrogels via two-photon absorption at any time during culture. We show that instructive hydrogels guide neural axon directionality in growing organotypic spinal cords, and that hydrogel geometry and mechanical properties control differential cell migration in developing cancer organoids. Finally, we show that hydrogel constraints promote cell polarity in liver organoids, guide small intestinal organoid morphogenesis and control lung tip bifurcation according to the hydrogel composition and shape

Electrochemical sensing with carbon nanotubes

In this chapter we aim at reviewing the most relevant contributions in the development of electrochemical sensors and biosensors based on carbon nanotubes (CNTs). Particular attention will be focused on the different strategies to modified the electrodes for sensing application. First, we will discuss about the electrochemical properties of nanotubes, their peculiar physical-chemical features that make them very useful for sensing. For instance, CNTs have been incorporated in electrochemical sensors to decrease overpotential and to improve sensitivity. In addition, concerning biosensing, the use of nanotubes enhancing the electrochemical reactivity of important biomolecules by promoting the electron-transfer reactions of biomolecules with catalytic activity. Different ways of integrating nanotubes and electrodes will be presented: from a bulk modification to a more sensible surface one. After a survey of bulk modification strategies, we will show an in-depth study of the surface modification technique developing the random and oriented choice to modify the surface, especially focusing on the vertical alignment. A structured methodological point of view will be given in this latter section. We will review the state of the art of sensing and biosensing with CNTs. In the first part of this section we will present how CNTs modified electrodes can act as sensors without any further modification of their structure. In the latter we will first focalize the attention to the functionalization techniques required in order to make CNTs able to detect biological substrates. Therefore we will review how CNTs modified electrodes will detect biological species such as neurotransmitters, proteins, enzymes and proteins enhancing the detection limits obtained using other techniques. Finally a future perspectives section is provided in which we will analyze the possibility to integrate the CNTs electrochemical devices in microfluidic platforms in order to diminish the average dimensions of the substrates, to enhance the selectivity and other appreciable advantages

Biosensing with electroconductive biomimetic soft materials

The development of smart biomaterials able to quantitatively analyse the dynamics of biological systems with high temporal resolution in biomimetic environments is of paramount importance in biophysics, biology and medicine. In this context, we develop a biosensing water-based soft biomaterial with tunable mechanical properties through the generation of an electroconductive nano-element network. As a proof of concept, in order to detect glucose concentration, we fabricate an electroconductive polyacrylamide glucose oxidase (GOx) loaded hydrogel (HY) modified with a small amount of single-walled carbon nanotubes (SWNTs) (up to 0.85 wt%). MicroRaman maps and optical analysis show the nanotube distribution in the samples at different mass fractions. Electrochemical impedance spectra and their fitting with equivalent circuit models reveal electron conduction in the charged hydrogels in addition to ionic conductivity. The effective resulting resistance of the nanostructured network is comparable to that of a gold electrode. These findings were also confirmed by cyclic voltammetry. Interestingly, heterogeneous clustering of SWNTs shows double electric mechanisms and efficiencies. GOx-SWNT doped hydrogels show a linear glucose concentration response in the range between 0.1 mM and 1.6 mM; taken together these results show high detection limits for glucose (down to 15 \u3bcM) and a sensitivity of 0.63 \u3bcA mM-1. In the perspective of monitoring cell dynamics, hydrogel functionalization allows cell adhesion and long-term cell culture, and atomic force microscopy is used for mapping the doped hydrogel stiffness. Myoblasts, cells sensitive to mechanical substrate properties, show proper differentiation of phenotype in the SWNT-HYs with nominal physiological stiffness