J. Prieur du Plessis

Analytical quantification of coefficients in the Ergun equation for fluid friction in a packed bed

A new analytical derivation for momentum transport during laminar flow through granular porous media is discussed and some of its implied results described. In the very low Reynolds number regime fully developed laminar flow is assumed and in the higher laminar Reynolds number regime the Forchheimer (non-Darcy) effect is modelled through reference to form drag induced by the solid constituents of the porous medium. The results are compared to the Ergun equation, which is empirically based on experimental measurements, and the correspondence is shown to be remarkably close.

An attempt to quantify fibre bed permeability utilizing the phase average Navier-Stokes equation

The mathematical development relating the micro-scale surface integral term of the phase average Navier-Stokes equation and the Darcy permeability is reviewed, as well as a proposed closed-form solution for the orthotropic permeability of unidirectional fibre beds. A simplistic extension of the solution is proposed to account for pinch-off effects during crossflow through the fibre bed. A solution for the in-plane permeability of plain weave fabrics is then constructed, following a similar methodology to that employed for unidirectional fibre beds. Encouraging agreement with experimental permeability measurements is listed throughout the paper in an attempt to demonstrate that sufficient potential exists to pursue this approach further.

Quantification of Unidirectional Fiber Bed Permeability

A new approach to the closed form quantification of the permeability of a bed of unidirectional fibers is proposed. The methodology is founded on the phase average form of the Navier-Stokes equation and is applied here to low Reynolds number laminar flow of an isotropic non-homogeneous Newtonian fluid through a bed of randomly packed unidirectional fibers. Permeability in both the longitudinal and transverse directions is shown to be a function of fiber volume fraction and fiber diameter only. The results are combined in a flow resistance tensor through which the phase-average Navier-Stokes equation is transformed into a practical tool for the numerical analysis of macroscopic flow in geometrically complex composite moldings. Encouraging agreement between the presented analytical solutions and published experimental results is demonstrated.

Simplified control-volume finite-element method

Localized vector algebra treatment of nonorthogonality is applied to two-dimensional quadrilateral control volumes using Cartesian base vectors in a primitive variable formulation of the Navier-Stokes equations for steady incompressible laminar flow. With optional grid-aligned, locally analytic interpolation, a simplified control-volume finite-element scheme is presented. Discretization of source terms, determination of interface convection-diffusion fluxes, pressure correction factors, and geometric quantities are described briefly. Results of three test cases provide useful initial insights into the performance of the method. The conclusion is reached that a simple finite-volume-based approach to nonorthogonality has been achieved.

NUMERICAL MODELING OF INTERIOR BOUNDARIES

The numerical modeling of interior boundaries of finite and infinitesimal volume (area) is described for finite volume numerical methods. The treatment of passive structures such as solid obstacles and infinitesimally thin porous and nonporous walls, as well as hydro-dynamically active structures such as pumps and fans, are discussed. Properties peculiar to the pressure-velocity coupling are stressed, while more generally applicable techniques for other dependencies are shown. Heal transfer and turbulence effects complementary to the hydrodynamics are discussed. An example is presented showing application of the techniques to the flow of air about a very large directly air-cooled heat exchanger.

Modelling And Analysis Of Permeability Of Anisotropic Compressed Non‐Woven Filters

An existing geometrical pore‐scale model for flow through isotropic spongelike media is adapted to predict flow through anisotropic non‐woven glass fibre filters. Model predictions are compared to experimental results for the permeability obtained for a filter under different stages of compression to demonstrate the capability of the model to adjust to changes in porosity. The experimental data used are for a glass fibre paper with a uniform fibre diameter. The input parameters of the pore‐scale model are the porosity, fibre diameter and some measure of the anisotropy between the in‐plane and normal directions to the paper. Correlation between the predictions and the experimental results is satisfactory and provides confidence in the modelling procedure. It is shown that the permeability is very sensitive to changes in the level of anisotropy, i.e. the level of compression of the nonwoven material.

Derivation of a modified hybrid approximation

To assess the present algorithm as compared with the BTDMIA, we give, in Table 1, their storage needs and operation counts per grid point, when both are applied, without and with bordering, to the inverse boundary-layer problem described earlier. For fair comparison when m > m\ the two methods should operate in opposite directions. The forward sweep of the present algorithm is to be carried out starting from the side with m conditions, whereas its counterpart of the most efficient form of the BTDMIA is to be carried out starting from the other side. The conclusion is twofold: non-bordering is superior to bordering. The present algorithm is more efficient than the BTDMIA.

NUMERICAL MODELING OF ATMOSPHERIC BOUNDARIES

The half cell method, a practical method of computing the flow pattern on an atmospheric boundary where both inflow and outflow conditions can occur, is described. The method is useful when most of the boundary conditions, even the velocity and pressure, are given in terms of gradients. The treatment of the velocity and pressure boundaries for the discrete momentum and continuity equations is described for the SIMPLE family of methods. As an example, the flow of air about a directly air-cooled heat exchanger is discussed. The method makes it possible to restrict the computational domain for this problem to manageable limits.

An Adaptable Analytical Ergun‐Type Equation For High Porosity Spongelike Porous Media

An analytical Ergun‐type equation for spongelike media is introduced in which developing flow in the short ducts of high porosity metallic foams are accounted for. Instead of the customary procedure of adjusting the empirical coefficients of the Ergun equation to apply to consolidated spongelike media, a pore scale model is introduced and the physical flow conditions remodelled. The pore‐scale linear dimensions are expressed as a function of porosity and the dependence of the form drag coefficient on porosity is incorporated into the model which leads to satisfactory predictions for the inertial coefficient. The model predictions are compared to experimental data from the literature and the satisfactory correspondence provides confidence in the physical adaptability of the model.

NUMERICAL MODELLING OF AIR FLOW IN CONFINED TAPERED DUCT INLETS

A study is made of the flow pattern and the corresponding pressure loss at the entrance to a confined tapered duct. This problem is of relevance in the design of natural draught cooling towers. A semi-implicit finite difference numerical procedure is used to model axi-symmetric air flow at the duct inlet. Inlet velocity profiles are obtained iteratively by means of a separate solution of a potential flow velocity field which takes into account the region of flow separation predicted by the viscous solution. Numerically predicted velocity fields and pressure drop characteristics are presented. The latter is shown to compare favourably with analytical results for pure potential flow with separation introduced by the definition of a free streamline boundary for the latter region.