Transport Phenomena in Porous Media

Derek B Ingham, I. Pop

• Publisher: Pergamon
• Year: 1998
• Print ISBN: 9780080428437
• eBook ISBN: 9780080543192
• Topic: Tecnología e ingeniería
• Subtopic: Ingeniería química y bioquímica

Research into thermal convection in porous media has substantially increased during recent years due to its numerous practical applications. These problems have attracted the attention of industrialists, engineers and scientists from many very diversified disciplines, such as applied mathematics, chemical, civil, environmental, mechanical and nuclear engineering, geothermal physics and food science. Thus, there is a wealth of information now available on convective processes in porous media and it is therefore appropriate and timely to undertake a new critical evaluation of this contemporary information. Transport Phenomena in Porous Media contains 17 chapters and represents the collective work of 27 of the world's leading experts, from 12 countries, in heat transfer in porous media. The recent intensive research in this area has substantially raised the expectations for numerous new practical applications and this makes the book a most timely addition to the existing literature. It includes recent major developments in both the fundamentals and applications, and provides valuable information to researchers dealing with practical problems in thermal convection in porous media. Each chapter of the book describes recent developments in the highly advanced analytical, numerical and experimental techniques which are currently being employed and discussions of possible future developments are provided. Such reviews not only result in the consolidation of the currently available information, but also facilitate the identification of new industrial applications and research topics which merit further work.

THE FUNDAMENTAL THEORY OF FLOW THROUGH PERMEABLE MEDIA FROM DARCY TO TURBULENCE^†

J.L. LAGE, Mechanical Engineering Department, Southern Methodist University, P. O. Box 750337, Dallas, TX 75275-0337, USA

INTRODUCTION

Professor Arthur Bergles began his keynote speech during the 1996 ASME National Heat Transfer Conference in Houston saying: “Old is good!”. It is indeed the responsibility of the new generation to show interest and recognition for the old. But it is also our responsibility to challenge existing ideas, and to question and seek confirmation of old dogmas, i.e. myths hampering the progress of knowledge.

During the past years I have observed several dogmas in the area of flow through permeable media like the writing of D’Arcy as the last name of Henry Darcy, the pioneer of hydraulics in saturated porous media, or the writing of Forchheimer as Forschheimer (notice the ‘s’). Although irrelevant, these errors are nevertheless disturbing: if it can happen with names, it can very well happen with model equations, mathematical solutions, experimental data or with the interpretation of physical phenomena.

Writing this invited review chapter on the theory of incompressible flow through permeable media was the opportunity to seek answers to some more fundamental questions (dogmas). In my journey I travelled back to the 19th century, when Darcy reported his theory, and follow the historic events leading to the most recent (1997) ideas on modeling turbulent flow through a permeable medium.

The topic of flow through permeable media extends over several areas of science and engineering. The amount of knowledge available is huge. To select what to include and what to leave out was to trust my own instincts (scholarship). All of it was written in my home-office, strategically located in our games room where my two children and I learn that working (at what we like) and playing are not dissimilar – literally!

DARCY EQUATION

It was in 1856 that Henry Philibert Gaspard Darcy, then the Inspecteur Général de Ponts et Chaussées, or “Dean of the School of Bridges and Roadways,” had his report on the public fountains of Dijon, a middle size city in the central-eastern region of France, published [1]. After obtaining his education in Paris at the École Polytechnique and at the École de Ponts et Chaussées, Darcy became a member of the French Imperial Corps of Bridges, Roadways and Mines, returning to Dijon, his birthplace, where he worked on hydraulics until his premature death by a neurological disease in 1858 [2].

Darcy’s major contribution is undoubtedly in the area of filter hydraulics with the discovery of an empirical law that bears his name. His report on the public fountains of Dijon [1] presented the principles to follow and the formulae to use in the design and construction of water distribution systems, water filtering, sewer systems, and the manufacturing of valves.

The French term fontaines in the main title of his report has induced some English-speaking researchers to erroneously relate Darcy’s work to water springs. In his report, Darcy discussed in detail experiments on water flow in a conduit, water flow in the hydraulic network of Dijon, and the effect of trapped air on the flow rate of a hydraulic network. It is clear that the objective of his study was very broad, including administrative and judicial aspects of water distribution in a city as related to the well being of the people living in it. It is fascinating to read his witness to the seriousness with which water quality was treated in a small French city back in 1856.

The more technical aspects of his essay were detailed in seven appendices. This fact should not be interpreted, in any way, as an indication of reduced importance. In fact, Darcy stresses the significance of these appendices in the introduction of his report, acknowledging the original aspect (based on his own discernment) of the information included there.

One of the appendices, appendix D (pp. 559–603), contains a description of his studies on filtration leading to his cornerstone equation of flow through a porous medium. It is worth noting that appendix D takes up only 44 pages out of the 647 pages of his book.

In appendix D, Darcy brings to light some generic concepts of filtration. He describes the two main procedures used at the time to clear up water (i.e. to remove solid particles making the water clear), specifically, precipitation and filtering. He remarks on the practical difficulty in using precipitation, including the required extended time, the bacterial growth effect because of the long exposure time, the size of the tanks necessary to supply water to a city, etc. He indicates that precipitation might be used as a preliminary technique for treating water to be supplied to a city, but should not be the only one.

He then explains a very simple filtration technique, i.e. to make liquid flow through passageways small enough to retain solid particulates to be eliminated from the water without severely obstructing the flow of liquid. This technique can be applied in two ways: artificially – water flows under the influence of variable pressure through layers of fine sand, gravel, and small stones; naturally – water flows through the alluvium accumulated naturally along a river bed, for instance.

Darcy mentions the existence of self-acting filters, i.e. filters through which water flows spontaneously by gravity through sand layers. He points out the experimental results from six filters in France and England. These filters were composed of sand supported by gravel, with a filtering flow of 3–13 m³/m² per 24 hours. He notices the difficulty in obtaining a general law of filtering because the sand used in each experiment was of different origin, the loads (ratio of pressure to flow rate) were different, and because the water arrived at the filters with different degrees of cleanness.

Having decided to investigate the phenomenon of water filtering, Darcy describes in his report a simple and ingenious apparatus. He then presents the results of carefully performed experiments done by himself and a fellow engineer in Dijon. The same experiments were later on performed by a chief engineer, his acquaintance, for independent confirmation.

The experimental apparatus used by Darcy and his colleagues was a 3.5 m high vertical column – a circular duct of interior diameter 0.35 m – closed at the extremities with a screwed plate. Inside, at 0.20 m from the bottom, there was a horizontal separator to support the sand layer, dividing the column into two chambers. This separator was formed, from bottom to top, of an iron grid of 7 mm prismatic bars spaced by 7 mm, a grid of 5 mm cylindrical bars spaced by 5 mm (the prisms were placed perpendicular to the cylinders), and finally a metallic screen of 2 mm thickness.

Water was fed through a pipe extending from the hydraulic network of the hospital in Dijon, were his laboratory was located. The water flow rate was controlled by a valve placed along the feeding pipe connected to the top of the column.

The lower section of the column led to a one meter wide reservoir collecting the water for measuring the volumetric flow rate. The pressures above and below the sand layer were indicated via two U-shape mercury manometers equipped with diaphragms (piezometers). Finally, the top chamber had an air bleeding valve for charging the column with water.

The experiments were performed with silica (quartz) sand from the Saône river, with the following composition: 58 percent of sand with grains smaller than 0.77 mm diameter, 13 percent of sand with 1.1 mm diameter grains, 12 percent of sand with 2 mm grains, and 17 percent of gravel and shell fragments of various siz...

Table of contents

• Cover image
• Title page
• Table of Contents
• Inside Front Cover
• Copyright
• PREFACE
• Chapter 1: THE FUNDAMENTAL THEORY OF FLOW THROUGH PERMEABLE MEDIA FROM DARCY TO TURBULENCE
• Chapter 2: TRANSPORT PHENOMENA IN ENCLOSED POROUS CAVITIES
• Chapter 3: HEAT CONDUCTION
• Chapter 4: ONSET OF OSCILLATORY CONVECTION IN A POROUS MEDIUM
• Chapter 5: THERMAL NONEQUILIBRIUM FORCED CONVECTION IN POROUS MEDIA
• Chapter 6: MATHEMATICAL MODELS FOR HEAT AND MASS TRANSPORT IN GEOTHERMAL SYSTEMS
• Chapter 7: NATURAL CONVECTION IN A HORIZONTAL POROUS ANNULUS
• Chapter 8: A UNIFIED TREATMENT OF DARCY-FORCHHEIMER BOUNDARY-LAYER FLOWS
• Chapter 9: TRANSIENT CONVECTION HEAT TRANSFER IN A POROUS MEDIUM: EXTERNAL FLOWS
• Chapter 10: THERMAL BOUNDARY–LAYER INSTABILITIES IN POROUS MEDIA: A CRITICAL REVIEW
• Chapter 11: EFFECTS OF ANISOTROPY ON CONVECTIVE FLOW THROUGH POROUS MEDIA
• Chapter 12: FREE CONVECTION IN ROTATING POROUS MEDIA
• Chapter 13: NON-DARCIAN EFFECTS IN CONFINED FORCED CONVECTIVE FLOWS
• Chapter 14: NATURAL CONVECTION IN ENCLOSURES FILLED WITH ANISOTROPIC POROUS MEDIA
• Chapter 15: INTERNAL NATURAL, FORCED AND MIXED CONVECTION IN FLUID-SATURATED POROUS MEDIUM
• Chapter 16: MODELING MULTIPHASE FLOW AND TRANSPORT IN POROUS MEDIA
• Chapter 17: CONVECTIVE HEAT FLOW FROM SUDDENLY HEATED SURFACES EMBEDDED IN POROUS MEDIA
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