Analysis of Darcy Flow in Confined Porous Media Including Wall Effect

2011 ◽  
Author(s):  
Nihad Dukhan
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Heat Exchange ◽  
Velocity Field ◽  
Darcy Number ◽  
Momentum Transport ◽  
Mean Velocity ◽  
Viscous Shear ◽  
The Mean

Effective cooling lies at the heart of reactor design and safe operation. Materials for cooling systems include solid porous media (e.g. metal foam). This is due to the large surface area per unit volume and the random internal structure of such porous medium. The former promotes heat exchange rates by providing large surface area, while the latter enhances it by providing vigorous mixing of the working fluid, which gives rise to what is called dispersion (an added mechanism of heat transfer). Hence, momentum transport in porous media is critical for heat transfer analysis, computation and design. Porous media are also used as storage of nuclear waste. In such applications, the porous medium is confined by solid boundaries. These impermeable boundaries give rise to shear stress and boundary layers, which strongly influence the velocity field and the pressure drop inside the porous medium. The velocity field directly influence the heat transfer rate, while the pressure drop determines the required pumping power. The Brinkman-extended Darcy equation describes the momentum transport due to fully developed Newtonian fluid flow in confined porous media. This equation is an extension of the famous Darcy equation, and it contains the viscous shear at the boundaries as well as the viscous shear on the internal surface of the porous medium. The equation is solved analytically inside and outside the boundary layer in a cylindrical porous-media system. As, expected, the volume-averaged velocity decays as the distance from the boundary increases. The mean and maximum velocities are obtained and their behavior is investigated in terms of the Darcy number and the ratio of the effective to the actual fluid viscosity. The friction factor is defined based on the mean velocity and is found to be inversely proportional to the Reynolds number, the Darcy number and the mean velocity. The analytical velocity can be directly substituted in the governing convection heat transfer equation to assess the heat transfer performance of confined cylindrical heat exchange systems.

Analysis of Brinkman-Extended Darcy Flow in Porous Media and Experimental Verification Using Metal Foam

2012 ◽  
Vol 134 (7) ◽  
Author(s):  
Nihad Dukhan
Keyword(s):  
Porous Media ◽  
Metal Foam ◽  
Waste Heat ◽  
Darcy Number ◽  
Momentum Transport ◽  
Flow In Porous Media ◽  
Mean Velocity ◽  
Ground Water Pollution ◽  
The Mean ◽  
Media Flows

Momentum transport in porous media exists in numerous engineering and process applications, e.g., ground water pollution, storage of nuclear waste, heat exchangers, and chemical reactors. In many of such applications, the porous medium is confined by solid boundaries. These impermeable boundaries give rise to shear stress and boundary layers. The Brinkman-extended Darcy equation describes the momentum transport due to Newtonian fluid flow in confined porous media. This equation is solved analytically in a cylindrical system, employing an existing fully-developed boundary-layer concept particular to porous media flows. The volume-averaged velocity increases as the distance from the boundary increases reaching a maximum at the center. The mean and maximum velocities are obtained and their behavior is investigated in terms of pertinent flow parameters. The friction factor is defined based on the mean velocity and is found to be inversely proportional to the Reynolds number, the Darcy number, and the mean velocity. The analytical results are verified by experiments using two types of metal foam. In the Darcy regime, reasonably good agreement is found between the analytical and the experimental friction factors for the 20-pore-per-inch foam, while a poor agreement is found for the 10-pore-per-inch foam.

Natural Convection in Enclosed Porous Media With Rectangular Boundaries

1970 ◽  
Vol 92 (1) ◽  
pp. 21-27 ◽  
Author(s):  
B. K. C. Chan ◽  
C. M. Ivey ◽  
J. M. Barry
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Natural Convection ◽  
Transfer Rate ◽  
Darcy Number ◽  
Field Equations ◽  
Geometric Aspect ◽  
Lumped Parameter ◽  
Different Temperatures

Numerical methods are used to solve the field equations for heat transfer in a porous medium filled with gas and bounded by plane rectangular surfaces at different temperatures. The results are presented in terms of theoretical streamlines and isotherms. From these the relative increases in heat transfer rate, corresponding to natural convection, are obtained as functions of three-dimensionless parameters: the Darcy number Da, the Rayleigh number Ra, and a geometric aspect ratio L/D. A possible correlation using the lumped parameter Da Ra is proposed for Da Ra greater than about 40.

Numerical Analysis of Wooden Porous Media Effects on Heat Transfer From a Staggered Tube Bundle

2008 ◽  
Vol 130 (1) ◽  
Author(s):  
Mohammad Layeghi
Keyword(s):  
Heat Transfer ◽  
Thermal Conductivity ◽  
Porous Medium ◽  
Porous Media ◽  
Numerical Analysis ◽  
Pressure Drop ◽  
Tube Bundle ◽  
Numerical Data ◽  
Darcy Number ◽  
Finite Volume Approach

A numerical analysis of forced convective heat transfer from a staggered tube bundle with various low conductivity wooden porous media inserts at maximum Reynolds numbers 100 and 300, Prandtl number 0.7, and Darcy number 0.25 is presented. The tubes are at constant temperature. The extended Darcy–Brinkman–Forchheimer equations and corresponding energy equation are solved numerically using finite volume approach. Parametric studies are done for the analysis of porous medium thermal conductivity and Reynolds number on the local Nusselt number distribution. Three different porous media with various solid to fluid thermal conductivity ratios 2.5, 5, and 7.5 are used in the numerical analysis. The results are compared with the numerical data for tube bundles without porous media insert and show that the presence of wooden porous media can increase the heat transfer from a tube bundle significantly (more than 50% in some cases). It is shown that high conductivity porous media are more effective than the others for the heat transfer enhancement from a staggered tube bundle. However, the presence of a porous medium increases the pressure drop. Therefore, careful attention is needed for the selection of a porous material with good heat transfer characteristics and acceptable pressure drop.

Passage Flow Structure and Its Influence on Endwall Heat Transfer in a 90 deg Turning Duct: Mean Flow and High Resolution Endwall Heat Transfer Experiments

1997 ◽  
Vol 119 (1) ◽  
pp. 39-50 ◽  
Author(s):  
B. G. Wiedner ◽  
C. Camci
Keyword(s):  
Heat Transfer ◽  
High Resolution ◽  
Velocity Field ◽  
Cross Section ◽  
Three Dimensional ◽  
Mean Flow ◽  
Mean Velocity ◽  
Pressure Fields ◽  
The Mean ◽  
Three Components

Three-dimensional measurements of the mean velocity field have been made in a square-cross-sectional, strongly curved, 90 deg turbulent duct flow. The mean radius to duct width ratio was 2.3. The study was performed as part of an overall investigation of the physics of endwall convective heat transfer. All three components of the velocity vector and the static and total pressure fields were measured using a five-hole probe at four duct cross sections: inlet, 0, 45, and 90 deg. Preliminary turbulence measurements using a single sensor hot wire at the inlet cross section were also obtained to provide streamwise fluctuation levels through the boundary layer. The endwall heat transfer coefficient distribution was determined using a steady-state measurement technique and liquid crystal thermography. A high-resolution heat transfer map of the endwall surface from far upstream of the curve through the 90 deg cross section is presented. The three-dimensional velocity field measurements indicate that a highly symmetric, strong secondary flow develops in the duct with a significant transfer of streamwise momentum to the transverse directions. The cross-stream vorticity components within the measurement plane were estimated using the five-hole probe data and an inviscid from of the incompressible momentum equation. The development of the total and static pressure fields, the three-dimensional mean velocity field, and all three components of the vorticity field are discussed. The endwall heat transfer distribution is interpreted with respect to the measured mean flow quantities.

scholarly journals Data-Driven Reduced-Order Modeling of Convective Heat Transfer in Porous Media

Fluids ◽  
2021 ◽  
Vol 6 (8) ◽  
pp. 266
Author(s):  
Péter German ◽  
Mauricio E. Tano ◽  
Carlo Fiorina ◽  
Jean C. Ragusa
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Porous Media ◽  
Steady State ◽  
Convective Heat Transfer ◽  
Convective Heat ◽  
Data Driven ◽  
Reduced Order ◽  
Medium Formulation ◽  
Flow Resistances

This work presents a data-driven Reduced-Order Model (ROM) for parametric convective heat transfer problems in porous media. The intrusive Proper Orthogonal Decomposition aided Reduced-Basis (POD-RB) technique is employed to reduce the porous medium formulation of the incompressible Reynolds-Averaged Navier–Stokes (RANS) equations coupled with heat transfer. Instead of resolving the exact flow configuration with high fidelity, the porous medium formulation solves a homogenized flow in which the fluid-structure interactions are captured via volumetric flow resistances with nonlinear, semi-empirical friction correlations. A supremizer approach is implemented for the stabilization of the reduced fluid dynamics equations. The reduced nonlinear flow resistances are treated using the Discrete Empirical Interpolation Method (DEIM), while the turbulent eddy viscosity and diffusivity are approximated by adopting a Radial Basis Function (RBF) interpolation-based approach. The proposed method is tested using a 2D numerical model of the Molten Salt Fast Reactor (MSFR), which involves the simulation of both clean and porous medium regions in the same domain. For the steady-state example, five model parameters are considered to be uncertain: the magnitude of the pumping force, the external coolant temperature, the heat transfer coefficient, the thermal expansion coefficient, and the Prandtl number. For transient scenarios, on the other hand, the coastdown-time of the pump is the only uncertain parameter. The results indicate that the POD-RB-ROMs are suitable for the reduction of similar problems. The relative L2 errors are below 3.34% for every field of interest for all cases analyzed, while the speedup factors vary between 54 (transient) and 40,000 (steady-state).

scholarly journals Darcy-Brinkman free convection about a wedge and a cone subjected to a mixed thermal boundary condition

1992 ◽  
Vol 15 (4) ◽  
pp. 789-794 ◽  
Author(s):  
G. Ramanaiah ◽  
V. Kumaran
Keyword(s):  
Heat Transfer ◽  
Boundary Condition ◽  
Heat Flux ◽  
Porous Medium ◽  
Heat Transfer Coefficient ◽  
Free Convection ◽  
Transformation Group ◽  
Darcy Number ◽  
Thermal Boundary Condition ◽  
High Porosity

The Darcy-Brinkman free convection near a wedge and a cone in a porous medium with high porosity has been considered. The surfaces are subjected to a mixed thermal boundary condition characterized by a parameterm;m=0,1,∞correspond to the cases of prescribed temperature, prescribed heat flux and prescribed heat transfer coefficient respectively. It is shown that the solutions for differentmare dependent and a transformation group has been found, through which one can get solution for anymprovided solution for a particular value ofmis known. The effects of Darcy number on skin friction and rate of heat transfer are analyzed.

scholarly journals Global energy fluxes in turbulent channels with flow control

2018 ◽  
Vol 857 ◽  
pp. 345-373 ◽  
Author(s):  
Davide Gatti ◽  
Andrea Cimarelli ◽  
Yosuke Hasegawa ◽  
Bettina Frohnapfel ◽  
Maurizio Quadrio
Keyword(s):  
Reynolds Number ◽  
Shear Stress ◽  
Energy Dissipation ◽  
Velocity Field ◽  
Flow Control ◽  
Reynolds Shear Stress ◽  
Power Input ◽  
Energy Fluxes ◽  
Mean Velocity ◽  
The Mean

This paper addresses the integral energy fluxes in natural and controlled turbulent channel flows, where active skin-friction drag reduction techniques allow a more efficient use of the available power. We study whether the increased efficiency shows any general trend in how energy is dissipated by the mean velocity field (mean dissipation) and by the fluctuating velocity field (turbulent dissipation). Direct numerical simulations (DNS) of different control strategies are performed at constant power input (CPI), so that at statistical equilibrium, each flow (either uncontrolled or controlled by different means) has the same power input, hence the same global energy flux and, by definition, the same total energy dissipation rate. The simulations reveal that changes in mean and turbulent energy dissipation rates can be of either sign in a successfully controlled flow. A quantitative description of these changes is made possible by a new decomposition of the total dissipation, stemming from an extended Reynolds decomposition, where the mean velocity is split into a laminar component and a deviation from it. Thanks to the analytical expressions of the laminar quantities, exact relationships are derived that link the achieved flow rate increase and all energy fluxes in the flow system with two wall-normal integrals of the Reynolds shear stress and the Reynolds number. The dependence of the energy fluxes on the Reynolds number is elucidated with a simple model in which the control-dependent changes of the Reynolds shear stress are accounted for via a modification of the mean velocity profile. The physical meaning of the energy fluxes stemming from the new decomposition unveils their inter-relations and connection to flow control, so that a clear target for flow control can be identified.

scholarly journals A Novel Simplification of the Reynolds-Chou-Navier-Stokes Turbulence Equations of Incompressible Flow

2018 ◽  
Author(s):  
Bohua Sun
Keyword(s):  
Velocity Field ◽  
Stress Tensor ◽  
Reynolds Stress ◽  
Incompressible Flow ◽  
Velocity Fluctuation ◽  
Explicit Representation ◽  
Navier Stokes ◽  
Mean Velocity ◽  
The Mean ◽  
Turbulence Formulations

Based on author's previous work [Sun, B. The Reynolds Navier-Stokes Turbulence Equations of Incompressible Flow Are Closed Rather Than Unclosed. Preprints 2018, 2018060461 (doi: 10.20944/preprints201806.0461.v1)], this paper proposed an explicit representation of velocity fluctuation and formulated the Reynolds stress tensor in terms of the mean velocity field. The proposed closed Reynolds Navier-Stokes turbulence formulations reveal that the mean vorticity is the key source of producing turbulence.

Natural Convection Inside a Bidisperse Porous Medium Enclosure

2009 ◽  
Vol 132 (1) ◽  
Author(s):  
Arunn Narasimhan ◽  
B. V. K. Reddy
Keyword(s):  
Heat Transfer ◽  
Porous Medium ◽  
Natural Convection ◽  
Volume Fraction ◽  
Solid Phase ◽  
Darcy Number ◽  
The Core ◽  
Side Walls ◽  
Porous Blocks ◽  
Rayleigh Numbers

Bidisperse porous medium (BDPM) consists of a macroporous medium whose solid phase is replaced with a microporous medium. This study investigates using numerical simulations, steady natural convection inside a square BDPM enclosure made from uniformly spaced, disconnected square porous blocks that form the microporous medium. The side walls are subjected to differential heating, while the top and bottom ones are kept adiabatic. The bidispersion effect is generated by varying the number of blocks (N2), macropore volume fraction (ϕE), and internal Darcy number (DaI) for several enclosure Rayleigh numbers (Ra). Their effect on the BDPM heat transfer (Nu) is investigated. When Ra is fixed, the Nu increases with an increase in both DaI and DaE. At low Ra values, Nu is strongly affected by both DaI and ϕE. When N2 is fixed, at high Ra values, the porous blocks in the core region have negligible effect on the Nu. A correlation is proposed to evaluate the heat transfer from the BDPM enclosure, Nu, as a function of Raϕ, DaE, DaI, and N2. It predicts the numerical results of Nu within ±15% and ±9% in two successive ranges of modified Rayleigh number, RaϕDaE.

Nanofluid Treatment in Existence of Magnetic Field Using Non-Darcy Model for Porous Media

2018 ◽  
pp. 456-555
Keyword(s):  
Magnetic Field ◽  
Heat Transfer ◽  
Finite Element Method ◽  
Porous Media ◽  
Control Volume ◽  
Convection Heat Transfer ◽  
Darcy Number ◽  
Convection Heat ◽  
Darcy Model ◽  
Flow And Heat Transfer

In this chapter, the non-Darcy model is employed for porous media filled with nanofluid. Both natural and forced convection heat transfer can be analyzed with this model. The governing equations in forms of vorticity stream function are derived and then they are solved via control volume-based finite element method (CVFEM). The effect of Darcy number on nanofluid flow and heat transfer is examined.

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