Mathematical Model Of Mass Transport Research Articles

A comprehensive approach to the mathematical modeling of mass transport in biological systems: Fundamental concepts and models

Multiscale mathematical modeling of transport phenomena across different levels of biological systems, such as cells, capillaries, tissues, and organs, has been increasingly helpful in describing how interactions among these systems lead to their function and dysfunction. The development of models across these scales is based on knowledge from various fields, such as engineering, physiology, and biophysics, and it requires significant collaboration among scientists from the relevant areas. Therefore, a unified framework to describe the fundamental principles and unite the established models could support this growing research community. In this regard, the present work deals with the essential terminology required to understand and model biological transport mechanisms, as well as to compile currently available models. An inclusive mathematical framework for models of mass transport mechanisms in different tissue compartments, including cells, capillaries, and gland ducts, is developed, with a primary focus on the mechanisms of membrane-mediated transporters such as channels, uniporters, symporters, pumps, and antiporters. The main objective of this study is to provide a comprehensive tool to facilitate the analysis of biological mass transport mechanisms and substantially decrease the time taken to find the appropriate model for a study.

Kinetics and Thermodynamics of Gum Extraction from Wild Sage Seed

Abstract Wild sage seed (Salvia macrosiphon) has practical amounts of gum with good functional properties. In this work, extraction of gum from wild sage seed was studied. Effect of pH, temperature and water/seed ratio on the kinetic and thermodynamic parameters and entropy, enthalpy and free energy of extraction were investigated. The maximum gum yield was 12.5% at 55°C for pH=7 and water/seed ratio 40:1. In this study, the experimental data were fitted to a mathematical model of mass transport and constants were obtained. The kinetic of wild sage seed gum extraction was found to be a first-order mass transfer model. Statistical results indicated that the model used in this study will be able to predict the gum extraction from wild sage seed. It is also found that ΔH and ΔS were positive and ΔG was negative indicating that the extraction process was spontaneous, irreversible and endothermic. The ΔH, ΔS and ΔG values were 0.52–14.99 kJ/mol, 6.3–52.2 J/mol K and 0.14–2.44 kJ/mol, respectively.

Preparation of a Three-Dimensional Ordered Macroporous Carbon Nanotube/Polypyrrole Composite for Supercapacitors and Diffusion Modeling

Three-dimensional ordered macroporous (3DOM) carbon nanotube (CNT)/polypyrrole (PPy) composite electrodes for supercapacitor application were prepared through cyclic voltammetric copolymerization from a solution containing both acid-treated CNTs and pyrrole monomers. A self-assembled SiO2 colloidal crystal was used as the sacrificial template. After electrochemical copolymerization, the template was removed, and a 3DOM CNT/PPy composite electrode was obtained. The specific capacitance of the composite reached 427 F g–1 at the scanning rate of 5 mV s–1, and it is calculated that ion diffusion contributed approximately 30% to the specific capacitance of the composite. A mathematical model of mass transport was proposed to evaluate the ion diffusion capability on the surfaces of 3DOM, nanoporous, and planar films. The calculation results showed that the flux (i.e., ion flux per unit length) of 3DOM film was larger than that of planar film, while the flux of nanoporous film was close to that of planar film. The model indicates that 3DOM film is favorable for ion transportation, while nanoporous film does the opposite. The model partially explains the reason why the specific capacitance of the prepared 3DOM CNT/PPy composite is far above the specific capacitance values of other reported CNT/PPy composites, even the nanoporous CNT/PPy composite.

Effect of electrons involved in charge transfer reactions in ECE processes under linear sweep voltametry: A theoretical model

Electrochemical methods have been found to be more efficient but of all the methods, which are vogue for treatment of effluent generated from various industries containing toxic substances like heavy metals, cyanide, organic compounds etc. Mathematical models play an important role in investigating the whole environmental management problems through statistical empirical modeling to more sophisticated statistical computer simulations. The mathematical model of mass transport for LSV under hydrodynamic conditions at tubular electrodes for ECE processes in which an irreversible chemical reaction is coupled between two reversible charge transfer reactions is considered. The resulting boundary value problem is converted into system of two integral equations, which is solved numerically. The effect of electrons (n2/n1) involved in two charge transfer reactions on cv-voltammograms are investigated and shown graphically.

Non-Fickian mass transport in fractured porous media

The paper provides an introduction to fundamental concepts of mathematical modeling of mass transport in fractured porous heterogeneous rocks. Keeping aside many important factors that can affect mass transport in subsurface, our main concern is the multi-scale character of the rock formation, which is constituted by porous domains dissected by the network of fractures. Taking into account the well-documented fact that porous rocks can be considered as a fractal medium and assuming that sizes of pores vary significantly (i.e. have different characteristic scales), the fractional-order differential equations that model the anomalous diffusive mass transport in such type of domains are derived and justified analytically. Analytical solutions of some particular problems of anomalous diffusion in the fractal media of various geometries are obtained. Extending this approach to more complex situation when diffusion is accompanied by advection, solute transport in a fractured porous medium is modeled by the advection–dispersion equation with fractional time derivative. In the case of confined fractured porous aquifer, accounting for anomalous non-Fickian diffusion in the surrounding rock mass, the adopted approach leads to introduction of an additional fractional time derivative in the equation for solute transport. The closed-form solutions for concentrations in the aquifer and surrounding rocks are obtained for the arbitrary time-dependent source of contamination located in the inlet of the aquifer. Based on these solutions, different regimes of contamination of the aquifers with different physical properties can be readily modeled and analyzed.

Chemical Engineering Challenges and Investment Opportunities in Sustainable Energy

The chemical and energy industries are transforming as they adjust to the new era of high-priced petroleum and severe global warming. As a result of the transformation, engineering challenges and investment opportunities abound. Rapid evolution and fast growth are expected in cathode and anode materials as well as polymeric electrolytes for vehicular batteries and in high-performance polymer-ceramic composites for wind turbines, fuel-efficient aircraft, and lighter and safer cars. Unique process-engineering opportunities exist in sand-oil, coal, and possibly also shale liquefaction to produce transportation fuel; and also in genetic engineering of photosynthesizing plants and other organisms for their processing into high-performance biodegradable polymers and high-value-added environmentally friendly chemicals. Also, research on the feasibility of mitigation of global warming through enhancement of CO(2) uptake by the southern oceans by fertilization with trace amounts of iron is progressing. Because chemical engineers are uniquely well trained in mathematical modeling of mass transport, flow, and mixing, and also in cost analysis, they are likely to join the oceanographers and marine biologists in this important endeavor.

Impact of the Fåhraeus effect on NO and O2 biotransport: a computer model.

Nitric oxide (NO) and oxygen (O2) transport in the microcirculation are coupled in a complex manner, since enzymatic production of NO depends on O2 availability, NO modulates vascular tone and O2 delivery, and tissue O2 consumption is reversibly inhibited by NO. The authors investigated whether NO bioavailability is influenced by the well-known Fåhraeus effect, which has been observed for over 70 years. This phenomenon occurs in small-diameter blood vessels, where the tube hematocrit is reduced below systemic hematocrit as a plasma boundary layer forms near the vascular wall when flowing red blood cells (rbcs) migrate toward the center of the bloodstream. Since hemoglobin in the bloodstream is thought to be the primary scavenger of NO in vivo, this might have a significant impact on NO transport. To investigate this possibility, the authors developed a multilayered mathematical model for mass transport in arterioles using finite element numerical methods to simulate coupled NO and O2 transport in the blood vessel lumen, plasma layer, endothelium, vascular wall, and surrounding tissue. The Fåhraeus effect was modeled by varying plasma layer thickness while increasing core hematocrit based on conservation of mass. Key findings from this study are that (1) despite an increase in the NO scavenging rate in the core with higher hematocrit, the model predicts enhanced vascular wall and tissue NO bioavailability due to the relatively greater resistance for NO diffusion through the plasma layer; (2) increasing the plasma layer thickness also increases the resistance for O2 diffusion, causing a larger P(O2) gradient near the vascular wall and decreasing tissue O2 availability, although this can be partially offset with inhibition of O2 consumption by higher tissue NO levels; (3) the Fåhraeus effect can become very significant in smaller blood vessels (diameters <30 microm); and (4) models that ignore the Fåhraeus effect may underestimate NO concentrations in blood and tissue.

Transperitoneal transport of glucose in vitro.

The effect of fluid mixing intensification, damage of mesothelial cells, gentamicin, and icodextrin on the diffusive glucose transport across the peritoneal membrane were evaluated in in vitro studies. A mathematical model of mass transport was used to calculate the diffusive permeability, expressed as a diffusive permeability coefficient (P). In the control conditions, the rate of glucose transfer from the interstitial to the mesothelial side of membrane (I-->M) and in the opposite direction (M-->I) remained constant, and the P value at mean was 2,731 +/- 1,493 x 10-4 (cm x s-1). The change of the stirring rate from 5.5 to 11 ml/min increased P values by about 74% for transport direction I-->M and 58% for M-->I, and the change from 11 to 22 ml/min enhanced P at mean by about 42% for both directions. The damage of the mesothelial layer, using sodium deoxycholate (2.5 mmol/L; 103.6 mg%), increased the glucose transfer from the interstitial to the mesothelial side of the peritoneum by 41% and to the opposite direction by 70%. Addition of icodextrin to the glucose solution increased glucose bidirectional transport at mean by about 14% for I-->M and 24% for M-->I. Furthermore, gentamicin did not change the I-->M transfer, but diminished M-->I transport by about 12%. In conclusion, the reduction of unstirred fluid layers at the mesothelium and the interstitium-fluid interfaces, removal of mesothelium, and addition of icodextrin increased the diffusive glucose transport in vitro; unstirred fluid layers restricted glucose transfer (I-->M) more than the mesothelium; and peritoneal glucose transport, directed from the mesothelial to the interstitial side of the peritoneum, decreased slightly after the addition of gentamicin.

Mass transfer to tubular electrodes: ECE process

The mathematical model of mass transport for linear sweep voltammetry under hydro-dynamic conditions at tubular electrodes has been studied for ECE processes in which an irreversible chemical reaction is coupled between two reversible charge transfer reactions. The resulting boundary value problem is converted into system of two integral equations, which is solved numerically. The effects of axial flow rate, scan rate, potential difference, variation of chemical reaction rate and the effect of the ratio of number of electrons (n2/n1) involved in two charge transfer reactions on CV-voltammograms are investigated and shown graphically.

Mathematical model of mass transport through dispersed‐phase polymer networks

AbstractA mathematical model of mass transport through dispersed‐phase networks to be used for the sustained release of drugs or other solutes at steady rates is presented. The drug is assumed to be encapsulated within the dispersed microdomains and transported to the bulk by diffusion across the interface. A drug is released to the surroundings by diffusion through the bulk. The results show that the desired steady flux of a drug to the surroundings may be obtained given appropriate values of structural properties of the network. These properties may be manipulated easily in the fabrication of dispersed‐phase networks reported previously.

Development of a bioartificial liver: Properties and function of a hollow-fiber module inoculated with liver cells

We have developed a bioartificial liver support system utilizing hollow-fiber bioreactor, plasmapheresis and microcarrier cell culture technologies. Liver cells were obtained through portal vein perfusion with ethylenediaminetetraacetate or ethylenediaminetetraacetate/collagenase. A mathematical model of mass transport in a hollow-fiber module, at various plasma flow velocities and system configurations, was developed. The bioartificial liver's ability to carry out specific differentiated metabolic liver functions was tested in vitro and in vivo. A reproducible large-animal model of acute ischemic liver failure was developed. Most major first-generation cyclosporine and 19-norterstosterone metabolites were isolated after substrate addition to the bioartificial liver in vitro. After bioartificial liver treatment for 6 hr (with dog or pig liver cells), dogs with acute liver failure had significantly lower serum ammonia and lactate levels and significantly higher serum glucose levels than did control animals treated with a bioartificial liver system inoculated with microcarriers alone. In addition, bioartificial liver–treated animals had significantly higher mean systolic blood pressures than did controls. Liver cell viability at the end of the 6-hr in vivo experiment was greater than 90%. (HEPATOLOGY 1993;17:258–265.)

Development of a bioartificial liver: Properties and function of a hollow-fiber module inoculated with liver cells

We have developed a bioartificial liver support system utilizing hollow-fiber bioreactor, plasmapheresis and microcarrier cell culture technologies. Liver cells were obtained through portal vein perfusion with ethylenediaminetetraacetate or ethylenediaminetetraacetate/collagenase. A mathematical model of mass transport in a hollow-fiber module, at various plasma flow velocities and system configurations, was developed. The bioartificial liver's ability to carry out specific differentiated metabolic liver functions was tested in vitro and in vivo. A reproducible large-animal model of acute ischemic liver failure was developed. Most major first-generation cyclosporine and 19-norterstosterone metabolites were isolated after substrate addition to the bioartificial liver in vitro. After bioartificial liver treatment for 6 hr (with dog or pig liver cells), dogs with acute liver failure had significantly lower serum ammonia and lactate levels and significantly higher serum glucose levels than did control animals treated with a bioartificial liver system inoculated with microcarriers alone. In addition, bioartificial liver-treated animals had significantly higher mean systolic blood pressures than did controls. Liver cell viability at the end of the 6-hr in vivo experiment was greater than 90%.

Mathematical modeling of mass transport and chemical reaction in crevice and pitting corrosion

A theoretical model is developed to evaluate transport processes and chemical reaction in crevices and pits. The model is general in form, allowing application to a variety of systems. The model is applied to recent experimental data on crevice corrosion in iron. Two different electrolytes are analyzed: acetate and sulfuric acid. These experimental systems are of interest because active crevice corrosion is obtained in the absence of localized acidification and aggressive anions. Excellent agreement is found between model predictions and experimental results

Mathematical model of mass transport throughout the kidney: Effects of nephron heterogeneity and tubular-vascular organization

A model of transport processes throughout the mammalian kidney is presented. Structural features of the model include labyrinth and medullary ray subsystems in the cortex, distinct vasa recta and capillary plexus subsystems in the outer and inner stripes, and an inner zone subsystem. Distinctions are also made between short and long nephrons, juxtamedullary and nonjuxtamedullary nephrons, vasculature, and interstitial spaces. Salt, urea, plasma proteins, and miscellaneous “nonreabsorbable” solutes in tubular and interstitial fluids are chosen as the essential solutes for describing renal mass transport. Geometric and number density data of vessels are taken from anatomical studies. Membrane parameters are evaluated principally from isolated rabbit tubule perfusion and rat micropuncture experiments. By numerical solution of the mass transport equations, the mild antidiuretic, steady state of a rat is simulated. To assess the importance of special structual features of the model, simulations were performed with different values of the anatomical parameters. By distributing more arterial blood toward the inner cortex and medulla, the osmotic gradient in the inner stripe is reduced while the urinary water and solute excretion rates are increased. When the reabsorption of urea from the papillary collecting duct is enhanced, the interstitial osmotic gradient monotonically increases throughout the inner medulla.

A mathematical model of mass transport in a long permeable tube with radial convection

Laplace transforms and regular double asymptotic expansions are used to solve the problem of ordinary chemical mass transport in a permeable tube, where there is small radial convection through the membrane wall and where the length-to-diameter ratio is large. The system is taken to be dilute and Newtonian and the solution is found to higher order in two small parameters. Results indicate that the exit concentration decreases markedly as the diameter, membrane permeability and tube length increase, and that changes in mass transport owing to variations in radial convection are much more significant than those due to the same order of magnitude changes in the resistance of the chemical solute to passage through the membrane (transmittance). In addition, the maximum effects of changes in the radial convection and transmittance are not at the membrane itself (r = 1), but rather roughly at radial values of 0·6 and 0, respectively.

A mathematical model of mass transport in the lung

A mathematical model of mass transport in the lung is developed. This model is concerned with gaseous diffusion in the region distal to the terminal bronchioles, following the suggestion that in this region the airflow is small and gaseous diffusion accounts for practically all of the mass transport. Differential equations for diffusion in a ternary mixture are written in the form of state equations. The resulting two equations are solved by state variable techniques utilizing a computer program which calculates the partial pressures of oxygen and carbon dioxide in ventilated blood as a function of ventilation rate. The program includes the effects of oxygen and carbon dioxide dissociation curves as well as conducting the calculations for each lobe of the lung independently. The model shows good correlation with published data.