Critical Biot Number of a Periodic Array of Rectangular Fins

We consider the heat transfer problem associated with a periodic array of rectangular fins subjected to convection heat transfer with a uniform heat transfer coefficient. Our analysis differs from the classical approach as (i) we consider two-dimensional (2D) heat conduction and (ii) the wall, to which the fins are attached, is included in the analysis

Optimal Design of Helical Springs of Power Law Materials

In this paper the geometric dimensions of a compressive helical spring made of power law materials are optimized to reduce the amount of material. The mechanical constraints are derived to form the geometric programming problem. Both the prime and the dual problem are examined and solved semi-analytically for a range of spring index. A numerical example is provided to validate the solution

Heat transfer enhancement of a periodic array of isothermal pipes

We address the problem of two-dimensional heat conduction in a solid slab whose upper and lower surfaces are subjected to uniform convection. In the midsection of the slab there is a periodic array of isothermal pipes of general cross section. The main objective of this work is to find the optimum shapes of the pipes that maximize the Shape Factor (heat transport rate). The Shape Factor is obtained by transforming the periodic array of pipes into a periodic array of strips, using the generalized Schwarz- Christoffel transformation, and applying the collocation boundary element method on the transformed domain. Subsequently we pose the inverse problem, i.e. finding the shape that maximizes the Shape factor given the perimeter of the pipes. For large Biot number the optimum shapes are in agreement with the isothermal case, i.e. circular for sufficiently small perimeters/heat transfer, and elongated towards the surfaces of the slab for larger perimeters/heat transfer. Furthermore, for the isothermal case, we were able to discover a new family of optimum shapes for large thickness of the slab and large perimeters, which do not have their maximum width on the horizontal axis of symmetry. For small Biot number the optimum pipes are flatter than the isothermal ones for a given perimeter. The flatness becomes more apparent for larger perimeters. Most important, for large perimeters there exists a critical thickness which is characterized by maximum heat transfer rate. This is further investigated using the finite element method to obtain the critical thickness of a slab and the critical depth of the periodic array of circular pipe

Optimal Design of Helical Springs of Power Law Materials

In this paper the geometric dimensions of a compressive helical spring made of power law materials are optimized to reduce the amount of material. The mechanical constraints are derived to form the geometric programming problem. Both the prime and the dual problem are examined and solved semi-analytically for a range of spring index. A numerical example is provided to validate the solution

Experimental study of zeolitic diffusion by use of a concentration-dependent surface diffusion model

https://www.sciencedirect.com/science/article/pii/S2405844019358037Surface diffusivity in adsorption and ion exchange processes is probably the most important property studied expensively in the literature but some aspects, especially its dependence on solid phase concentration, is still an open subject to discussion. In this study a new concentration-dependent surface diffusion model, equipped with a flexible double selectivity equilibrium relationship is applied on the removal of Pb2+, Cr3+, Fe3+ and Cu2+ from aqueous solutions using a natural zeolite. The model incorporates the Chen-Yang surface diffusivity correlation able to deal with positive and negative dependence with surface coverage. The double selectivity equilibrium relationship successfully represents the experimental equilibrium data, which follow Langmurian isotherm type for Pb2+, sigmoidal for Cr3+ and Fe3+ and linear for Cu2+. The concentration-dependent surface diffusion model was compared with the constant diffusivity surface diffusion model and found to be moderately more accurate but considerably more useful as it provides more insights into the diffusion mechanism. The application of the model resulted in an average deviation of 8.56 ± 6.74% from the experimental data and an average solid phase diffusion coefficients between 10−9 and 10−10 cm2/s. The results showed that the diffusion of metal ions in the zeolite structure is unhindered following the surface diffusion mass transfer mechanism

Phosphate removal by Ca(OH)2-treated natural minerals : experimental and modeling studies

Adsorption of phosphate phosphorus (PO4-P) from wastewater onto eco-friendly geosorbents has gained great attention aiming at recovering an essential nutrient for crop production. Notably, the literature on PO4-P adsorption kinetics is limited to the application of either empirical reaction-based models lacking a physical significance or over-simplified diffusion-based models frequently used outside their applicability area. In this study, equilibrium and kinetic experiments are presented under a wide range of phosphate concentrations (50-500 mg P/L) using sustainable and low-cost modified adsorbents. The kinetics of PO4-P adsorption from aqueous solutions onto Ca(OH)2-treated zeolite (CaT-Z) and bentonite (CaT-B) was analyzed by a dimensionless two-phase homogeneous surface diffusion model (TP-HSDM) assuming constant diffusivity and coupled with the double selectivity isotherm model (DSM). The TP-HSDM fit to the data at four initial P concentrations (50, 100, 200 and 300 mg/L) resulted in an average relative error of 14.6% and 17.4% from the experimental data for CaT-Z and CaT-B, respectively. The average surface diffusion coefficient (Ds) ranged from 2.5 × 10-10 to 8.7 × 10-10 cm2/s for CaT-Z and from 1.6 × 10-10 to 4.78 × 10-9 cm2/s for CaT-B. The external mass transfer coefficient (kf) ranged from 2.72 × 10-4 to 8.38 × 10-4 cm/s for CaT-Z and from 5.63 × 10-4 to 2.24 × 10-3 cm/s for CaT-B. The dimensionless Biot (Bi) number exhibited values in the order of magnitude of 105 indicating that intraparticle diffusion is the controlling mass transfer mechanism for both materials

Optimum interfaces that maximize the heat transfer rate between two conforming conductive media

Abstract We consider conjugate heat transfer between two conductive and conforming media, with isothermal boundary conditions on the exposed surfaces, and continuity of the temperature and the heat flux along their interface. We address the inverse problem of finding the shape of the interface such that the heat transfer rate is maximized. We formulate three isoperimetric, shape optimization problems associated with three different applications: i) the optimal shape of corrugations (surface “roughness”), ii) the optimal shape of high conductivity inserts (inverted fins) and iii) the optimal shape of high conductivity fins. As expected, the optimal geometries have the shape of an extension of the high conductivity material into the low conductivity material. For the case of corrugations and inserts, the optimum shapes are triangular for small perimeters; for large perimeters and thick slabs they are elliptical and tend to cover the whole width/period of the domain. Optimum fins are characterized by long, shallow valleys and deep, narrow protrusions of the high conductivity material. For the parameters considered in this study, the width of the protrusion is approximately one quarter of the period