Divided Solids Mechanics

By Jean-Paul Duroudier

Divided Solids Mechanics, part of the Industrial Equipment for Chemical Engineering set, defines how to perform the selection and calculation of equipment needed in the basic operations of process engineering, offering reliable and simple methods, with this volume providing a comprehensive focus divided solids mechanics.

Throughout these concise and easy-to-use books, the author uses his vast practical experience and precision knowledge of global research to present an in-depth study of a variety of aspects within the field of chemical engineering.

• Presents a guide that is particularly innovative in this field of study
• Contains measurements of the mechanical properties of divided solids
• Includes methods of discrete elements (of distinct particles)
• Provides the properties of powders for pressing

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 14, 2016
• ISBN: 9780081017777

Jean-Paul Duroudier

Jean-Paul Duroudier is an engineer from Ecole centrale de Paris, France. He has devoted his professional life to the study of materials in chemical engineering.

Book preview

Divided Solids Mechanics

Jean-Paul Duroudier

Industrial Equipment for Chemical Engineering Set

coordinated by

Jean-Paul Duroudier

Table of Contents

Cover image

Title page

Dedication

Copyright

Preface

1: Mechanical Characteristics of Divided Solids

Abstract

1.1 Two simple properties

1.2 The mechanics of continuous media

1.3 Flow of divided solids

1.4 Current identities

1.5 Measurement of the mechanical properties of divided solids

1.6 Stockpiling

2: Stresses in Hoppers and Silos: Filling, Emptying and Content Homogeneity

Abstract

2.1 Stresses on the walls

2.2 Variation in the stresses with ensiling and desiling. Homogeneity, heterogeneity of content

3: Draining of Hoppers and Silos: Stresses and Flow Rate

Abstract

3.1 General information

3.2 Flow types and flow regimes

3.3 Criteria for mass flow

3.4 Flow with dead zone

3.5 Arching or doming and its prevention

3.6 Rate of emptying

3.7 Withdrawal of fine products

3.8 The kinematic theory of flow in a hopper

3.9 Activation of the emptying

3.10 Caking

4: Mechanics of Divided Solids

Abstract

4.1 Static limit of divided solids: method of characteristics

4.2 The dynamic of D.S. according to Bagnold [BAG 54]

4.3 Dynamics of divided solids: method of discrete elements (of distinct particles)

4.4 Surface dynamics of a D.S.

4.5 Experimental studies

5: Densification of Powders: Tablets and Granules

Abstract

5.1 Useful properties of powders for pressing

5.2 The pressing operation of powders

5.3 The physics of rolling-granulation

5.4 Granulating equipment

5.5 Resistance of the granular

6: Mechanics and Thermics of Gaseous Fluidized Beds

Abstract

6.1 Mechanics of gaseous fluidized beds

6.2 Flow thresholds

6.3 Morphology of a fluidized bed

6.4 Plugging

6.5 Heat transfer

6.6 Applications of fluidization

Appendix 1: Apparent Mass Density of Bulk Divided Solids (kg.m- 3)

A1.1 Vegetable products

A1.2 Natural inorganic products

A1.3 Manufactured products

Appendix 2: Simple Results of Analytical Geometry

A2.1 Product of two vectors V1 and V2

A2.2 The rotation vector Ω and the velocity vector

A2.3 Normal to a plane

Appendix 3: Mohs’ Scale

Bibliography

Index

Dedication

There are no such things as applied sciences, only applications of science.

Louis Pasteur (11 September 1871)

Dedicated to my wife, Anne, without whose unwavering support, none of this would have been possible.

Copyright

First published 2016 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

ISTE Press Ltd

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London SW19 4EU

UK

www.iste.co.uk

Elsevier Ltd

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Kidlington, Oxford, OX5 1GB

UK

www.elsevier.com

Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

For information on all our publications visit our website at http://store.elsevier.com/

© ISTE Press Ltd 2016

The rights of Jean-Paul Duroudier to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-187-1

Printed and bound in the UK and US

Preface

The observation is often made that, in creating a chemical installation, the time spent on the recipient where the reaction takes place (the reactor) accounts for no more than 5% of the total time spent on the project. This series of books deals with the remaining 95% (with the exception of oil-fired furnaces).

It is conceivable that humans will never understand all the truths of the world. What is certain, though, is that we can and indeed must understand what we and other humans have done and created, and, in particular, the tools we have designed.

Even two thousand years ago, the saying existed: faber fit fabricando, which, loosely translated, means: "c’est en forgeant que l’on devient forgeron" (a popular French adage: one becomes a smith by smithing), or, still more freely translated into English, practice makes perfect. The artisan (faber) of the 21st Century is really the engineer who devises or describes models of thought. It is precisely that which this series of books investigates, the author having long combined industrial practice and reflection about world research.

Scientific and technical research in the 20th century was characterized by a veritable explosion of results. Undeniably, some of the techniques discussed herein date back a very long way (for instance, the mixture of water and ethanol has been being distilled for over a millennium). Today, though, computers are needed to simulate the operation of the atmospheric distillation column of an oil refinery. The laws used may be simple statistical correlations but, sometimes, simple reasoning is enough to account for a phenomenon.

Since our very beginnings on this planet, humans have had to deal with the four primordial elements as they were known in the ancient world: earth, water, air and fire (and a fifth: aether). Today, we speak of gases, liquids, minerals and vegetables, and finally energy.

The unit operation expressing the behavior of matter are described in thirteen volumes.

It would be pointless, as popular wisdom has it, to try to reinvent the wheel – i.e. go through prior results. Indeed, we well know that all human reflection is based on memory, and it has been said for centuries that every generation is standing on the shoulders of the previous one.

Therefore, exploiting numerous references taken from all over the world, this series of books describes the operation, the advantages, the drawbacks and, especially, the choices needing to be made for the various pieces of equipment used in tens of elementary operations in industry. It presents simple calculations but also sophisticated logics which will help businesses avoid lengthy and costly testing and trial-and-error.

Herein, readers will find the methods needed for the understanding the machinery, even if, sometimes, we must not shy away from complicated calculations. Fortunately, engineers are trained in computer science, and highly-accurate machines are available on the market, which enables the operator or designer to, themselves, build the programs they need. Indeed, we have to be careful in using commercial programs with obscure internal logic which are not necessarily well suited to the problem at hand.

The copies of all the publications used in this book were provided by the Institut National d’Information Scientifique et Technique at Vandœuvre-lès-Nancy.

The books published in France can be consulted at the Bibliothèque Nationale de France; those from elsewhere are available at the British Library in London.

In the in-chapter bibliographies, the name of the author is specified so as to give each researcher his/her due. By consulting these works, readers may gain more in-depth knowledge about each subject if he/she so desires. In a reflection of today’s multilingual world, the references to which this series points are in German, French and English.

The problems of optimization of costs have not been touched upon. However, when armed with a good knowledge of the devices’ operating parameters, there is no problem with using the method of steepest descent so as to minimize the sum of the investment and operating expenditure.

1

Mechanical Characteristics of Divided Solids

Abstract

In the following, we will use the abbreviation D.S. for divided solid.

Keywords

Angle of repose; Compressibility; Consolidation stress; Divided solids; Energy dissipation; Hysteresis; Identities; Mohr’s circles; Stockpiling; Stress tensor symmetry

1.1 Two simple properties

1.1.1 The size of particles

In the following, we will use the abbreviation D.S. for divided solid.

In terms of particle size alone, we distinguish:

– powdered solids, that is powders consisting of:

- ultrafine (dp < 20 μm) talc, flour, some pigments and coloring agents. These products are a result of fine grindings;

- fine (20 μm < dp < 100 μm);

– granules consisting of artificial granules or natural particles such that:

These particles are called granules as opposed to fine;

– intermediate solids often as a result of current grindings or usual crystallizations in a mother liquor:

The expression granular solid is to be avoided.

In [WIE 75], Wiegbardt reviews the different properties of D.S. as well as the measurement of these latter.

1.1.2 Compressibility

If the diameter of the powdered solids is inferior to 60 or 100 μm, van der Waals forces of attraction are predominant compared to forces of gravity.

If the porosity drops locally below a certain value, due to interactions between particles, bonds are formed, which lead to clusters.

If we try to fluidize a powder, the gas will flow in preferential passages separated by high solid density regions.

If we want to transport a powder pneumatically and in a dense phase, any local drop in the gas velocity will lead to the formation of a solid plug.

During the emptying of a hopper, the flow of powder takes a pulsed behavior, because a local increase in solid density is sufficient to lead to a solid plug, which almost provokes the stop of the flow (arching, doming).

In order to define this property of powders, some authors suggest we use Hausner’s index, which is defined as the ratio between the apparent mass density ρa of the solid after compression and density in the aerated state, that is loose or movable:

The aerated mass density ρa is explained in Appendix 1 for several D.S.

Ergun [ERG 51] provides a method to determine the true (intrinsic) mass density of a D.S:

The compression is obtained by repeated shocks (tapping) and the aerated state is that of the powder collected in a cylindrical container located under a vibrated sieve such that the flow into the container is uniform on the area.

The shocks are supposed to simulate the ease of appearance of a strong solid density during a flow.

In their publication, Harnby et al. [HAR 87] specify the use of this concept using an example. However, in flow studies, Hausner’s index is not convenient and Jenike suggests that:

where σℓc is the major principal contact stress. The contact stresses only act on the solid skeleton.

Remember that if we consider a D.S. as a continuous medium, the tensor matrix of the stresses can be written as:

P: pressure of the interstitial fluid (Pa)

[I]: unit matrix (matrix of pressure stresses)

σ: stress in the continuous medium (Pa)

ε: porosity of the medium

[Σ]: matrix of stresses

[Σ2]: matrix of the contact stresses.

The compressibility β is different depending on the products and whether the product is immobile or flowing.

Note

We sometimes exert pressure on divided solids saturated with liquid to drive out the interstitial liquor. The permitted variation law is similar to the previous one:

where Vo and Vm are the initial volume and the minimum volume (for P → ∞). The parameter e is the base of natural logarithms and P* is also a parameter that characterizes the compressibility of the product. The pressure P is measured in rel. bar.

Note

A D.S. can be:

– soft (loose) if it results from a reduced speed finish;

– compact if it has been subjected to pressure. This pressure is said to be of consolidation. To us, it will always be less than 50,000 Pa.

Note

The reader interested in the microscopic behavior of D.S. can refer to Cambou and Jean [CAM 01].

1.2 The mechanics of continuous media

1.2.1 Notion of stress in a solid

. We write:

tangent to the surface dS.

is directed toward the exterior of the solid field. We say that the unit vector of the perpendicular stress is that of the outgoing perpendicular vector.

On the contrary, in soil mechanics and in the mechanics of divided solids, the opposite convention is used, the compressions are considered positive, that is throughout the entering .

and let ox be an axis of this plane.

changes sign depending on the observer standing on one side or the other of the plane (P).

Figure 1.1 Stress at the point M

Note

At a given point, we can only split down the stresses into vector components if they are exerted on the same surface area. If it is not the case, we can only split forces into vector components, which is the product of stresses and the surface elements of different orientations.

1.2.2 Equilibrium of forces and tensor of the stresses in plane coordinates

Consider the right-angled triangle PAB in the xoy plane.

.

exerted on side AB of the triangle PAB.

be the unit vector of the entering normal vector to AB. The compression stress will be:

(on the Figure 1.2, this angle is such that: π < β < 3 π/2).

Figure 1.2 Equilibrium of forces

The force exerted by