Influence of root resistivity on plant water uptake mechanism,Part II: analytical solutions for low/moderate soil-root conductivity ratio

A one-dimensional approximate analytical model, which preserves the main features of soil-crop-atmospheric hydrodynamics, has been suggested for plant roots of low soil-root conductivity ratio (SRCR). The proposed approach involves physically based concepts, such as mass balance equation, Darcy’s law, and related water uptake and plant transpiration functions. Two main assumptions have been made to derive the analytical solution: (1) gravitational flow is adopted and (2) the uniform soil moisture distribution within the root water activity zone is supposed. The mass balance equation in its integral form is solved by the method of characteristics. This leads to the two functional equations for soil pressure head and root potential, which can be solved simultaneously by using common software. The model has been further verified against the numerical one. The model represents a reasonable compromise between the complicated mechanism of unsaturated water flow with root water uptake (RWU) and still insufficient knowledge of the soil-plant-atmospheric continuum. It is able to account for temporal fluctuations in root activity zone and provides a relatively simple algorithm for investigation of RWU-mechanism. Besides the theoretical and applicative importance, this flow model yields water and velocity distributions within soil profile, and, thereby, constitutes a preliminary step toward solution of contaminant transport problems in vadose zone. Deceased     

Minimizing hydraulic resistance of a plant root by shape optimization

Optimum plant root structure that minimizes a single root’s hydraulic resistance to water-uptake is studied in this paper with the constraint of constant root volume. Hydraulic resistances under the slender body approximation and without such a limitation are considered. It is found that for large stele-to-cortex permeability ratio, there exists an optimum root length-to-base-radius ratio that minimizes the hydraulic resistance. A remarkable feature of the optimum root structure is that the optimum dimensionless stele conductivity depends only on a single geometrical parameter, the stele-to-root base-radius ratio. Once the stele-to-root base-radius ratio and the stele-to-cortex permeability ratio are given, the optimum root length-to-radius ratio can be found. While these findings remain to be verified by experiments for real plant roots, they offer theoretical guidance for the design of bio-inspired structures that minimize the hydraulic resistance for fluid production from porous media.     

Numerical perturbation method for approximate solution of poisson's equation on a moderately deforming grid

In problems such as the computation of incompressible flows with moving boundaries, it may be necessary to solve Poisson's equation on a large sequence of related grids. In this paper the LU decomposition of the matrix A ₀ representing Poisson's equation discretized on one grid is used to efficiently obtain an approximate solution on a perturbation of that grid. Instead of doing an LU decomposition of the new matrix A , the RHS is perturbed by a Taylor expansion of A ^?1 about A ₀. Each term in the resulting series requires one ‘backsolve’ using the original LU . Tests using Laplace's equation on a square/rectangle deformation look promising; three and seven correction terms for deformations of 20% and 40% respectively yielded better than 1% accuracy. As another test, Poisson's equation was solved in an ellipse (fully developed flow in a duct) of aspect ratio 2/3 by perturbing about a circle; one correction term yielded better than 1% accuracy. Envisioned applications other than the computation of unsteady incompressible flow include: three-dimensional parabolic problems in tubes of varying cross-section, use of ‘elimination’ techniques other than LU decomposition, and the solution of PDEs other than Poisson's equation.     

Numerical investigation on cold flow dynamics of supercritical water fluidized bed reactor with inclined distributor: Design and scale up

Supercritical water fluidized bed reactor (SCWFBR) is a novel concept for the gasification of coal and biomass to produce hydrogen. In this work, to enhance the mixing in the axial direction, an inclined distributor is introduced to optimize the flow dynamics in SCWFBR with partitioned fluid supply. Through numerical simulations based on the two fluid model (TFM), the effects of the inclined distributor structure and operating parameters on the solid distribution and the residence time are evaluated with the optimal values determined. Numerical results show that, area ratio = 2:1, SCW velocity ratio = 3:1, flow ratio = 3.36:1 and inclination angle = 20° are the optimal design in this paper. A predictive correlation of the minimum fluidization velocity for the improved SCWFBR is also proposed based on the numerical data. The average error between the correlation and numerical simulation results is approximately 1.4% which strongly demonstrates its capability. Finally, based on the optimal design, the lab-scale reactor is further scaled up and the studies about two scale-up rules are carried out. Only the cold flow is simulated in this study without considering chemical reaction which would be involved in future work.