Extension of Murray's law using a non-Newtonian model of blood flow

<p>Abstract</p> <p>Background</p> <p>So far, none of the existing methods on Murray's law deal with the non-Newtonian behavior of blood flow although the non-Newtonian approach for blood flow modelling looks more accurate.</p> <p>Modeling</p> <p>In the present paper, Murray's law which is applicable to an arterial bifurcation, is generalized to a non-Newtonian blood flow model (power-law model). When the vessel size reaches the capillary limitation, blood can be modeled using a non-Newtonian constitutive equation. It is assumed two different constraints in addition to the pumping power: the volume constraint or the surface constraint (related to the internal surface of the vessel). For a seek of generality, the relationships are given for an arbitrary number of daughter vessels. It is shown that for a cost function including the volume constraint, classical Murray's law remains valid (i.e. Σ<it>R</it>^{<it>c </it>}= <it>cste </it>with <it>c </it>= 3 is verified and is independent of <it>n</it>, the dimensionless index in the viscosity equation; <it>R </it>being the radius of the vessel). On the contrary, for a cost function including the surface constraint, different values of <it>c </it>may be calculated depending on the value of <it>n</it>.</p> <p>Results</p> <p>We find that <it>c </it>varies for blood from 2.42 to 3 depending on the constraint and the fluid properties. For the Newtonian model, the surface constraint leads to <it>c </it>= 2.5. The cost function (based on the surface constraint) can be related to entropy generation, by dividing it by the temperature.</p> <p>Conclusion</p> <p>It is demonstrated that the entropy generated in all the daughter vessels is greater than the entropy generated in the parent vessel. Furthermore, it is shown that the difference of entropy generation between the parent and daughter vessels is smaller for a non-Newtonian fluid than for a Newtonian fluid.</p

A planar anode-supported Solid Oxide Fuel Cell model with internal reforming of natural gas

Solid Oxide Fuel Cells (SOFCs) are of great interest due to their high energy efficiency, low emission level, and multiple fuel utilization. SOFC can operate with various kinds of fuels such as natural gas, carbon monoxide, methanol, ethanol, and hydrocarbon compounds, and they are becoming one of the main competitors among environmentally friendly energy sources for the future. In this study, a mathematical model of a co-flow planar anode-supported solid oxide fuel cell with internal reforming of natural gas has been developed. The model simultaneously solves mass, energy transport equations, and chemical as well as electrochemical reactions. The model can effectively predict the compound species distributions as well as the cell performance under specific operating conditions. The main result is a rather small temperature gradient obtained at 800 °C with S/C = 1 in classical operating conditions. The cell performance is reported for several operating temperatures and pressures. The cell performance is specified in terms of cell voltage and power density at any specific current density. The influence of electrode microstructure on cell performance was investigated. The simulation results show that the steady state performance is almost insensitive to microstructure of cells such as porosity and tortuosity unlike the operating pressure and temperature. However, for SOFC power output enhancement, the power output could be maximized by adjusting the pore size to an optimal value, similarly to porosity and tortuosity. At standard operating pressure (1 atm) and 800 °C with 48% fuel utilization, when an output cell voltage was 0.73 V, a current density of 0.38 A cm-2 with a power density of 0.28 W cm-2 was predicted. The accuracy of the model was validated by comparing with existing experimental results from the available literature

The robustness of the permeability of constructal tree-shaped fissures

WOS:000365361400027International audienceHere we develop analytically the formulas for effective permeability in several configurations using the closed-form description of tree networks designed to provide flow access. The objective was to find the relation between the permeability and porosity of tree-shaped fissures. We found the effect of the fracture size on the permeability for fixed number of bifurcation and the results showed that the permeability of the fracture network increased rapidly with the size of the fracture. Next, we found a relation between the Reservoir Quality Index (RQI) and the porosity of the fracture. The results in this paper have been validated by comparison with experimental and numerical results. We show that the permeability formulas do not vary much from one tree design to the next, suggesting that similar formulas may apply to naturally fissured porous media with unknown precise details, which occur in natural reservoirs. (C) 2015 Elsevier Ltd. All rights reserved