Vacuum Engineering Calculations, Formulas, and Solved Exercises Armand Berman National Physical Laboratory of Israel Hebrew University Givat Ram Jerusalem Academic Press, Inc. Harcourt Brace Jovanovich, Publishers San Diego New York Boston London Sydney Tokyo Toronto This book is printed on acid-free paper. 0 Copyright © 1992 by ACADEMIC PRESS, INC. All Rights Reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopy, recording, or any information storage and retrieval system, without permission in writing from the publisher. Academic Press, Inc. 1250 Sixth Avenue, San Diego, California 92101-4311 United Kingdom Edition published by Academic Press Limited 24-28 Oval Road, London NW1 7DX Armand Berman P.O. Box 2243, Bat-Yam 59121 Israel Library of Congress Cataloging-in-Publication Data Berman, A. (Armand) Vacuum engineering calculations, formulas, and solved exercises / Armand Berman. p. cm. Includes index. ISBN 0-12-092455-2 1. Vacuum-Handbooks, Manuals, etc. 2. Vacuum-Problems, exercises, etc. 3. Vacuum technology-Handbooks, manuals, etc. 4. Vacuum technology—Problems, exercises, etc. I. Title. QC166.B47 1992 621.5'5-dc20 92-4140 CIP PRINTED IN THE UNITED STATES OF AMERICA 92 93 94 95 96 97 QW 9 8 7 6 5 4 3 2 1 Preface Vacuum is now required for many scientific and industrial purposes, particularly those associated with semiconductors, lighting bulbs, and material-coating industries. The need to manufacture in highly characterized and controlled vacuum brings continual progress in the design, operation, and maintenance of vacuum systems, challenging the people involved in these activities. To many of these people, the use and application of vacuum engineering formulas rather than vacuum's theoretical basis are prime objectives. This book is written for those who can devote little time for a full mathematical treatment of the many problems encountered in the vacuum practice but who have a reasonable knowledge of the essentials of vacuum together with elementary physics and mathematics. The text, which is intended to be a systematic answer to the manifold situations arising in practice, was written bearing in mind two main objectives: (1) to summarize and organize the vast material of the vacuum technology in sets of useful formulas, and (2) to frame a collection of worked-out exercises exemplifying how to use these formulas for solving technological problems. The book starts with chapters on ideal (perfect) and real gases (Chapters 1 and 2) and the kinetic theory of gases (Chapter 3), which are intended to give the reader enough technique and experience with calculations. Next, Chapter 4 deals with the flow rate of gases, throughput, and impedance of interconnected vacuum components. There follow three chapters on the steady flow of gas in the viscous (Chapter 5), molecular (Chapter 6), and transition (Chapter 7) ranges, with emphasis on the viscous-chocked flow and molecular flow through short pipes, channels, and vacuum components of complex geometry. The extent of Chapter 7 on steady flow of gas in the transition range has been kept reduced, since for many practical vacuum systems it is sufficient to limit calculations to either steady viscous or molecular flow. Chapter 8 on gas load examines the sources of gas in vacuum systems and evaluates the gas load. The treatment of gas adsorption and diffusion has been omitted, taking into account the aim of the book. There then follows Chapter 9, devoted to vacuum pumps, and chapters dealing with the removal from a vacuum system of either the original gas under transient pressure (Chapter 10) or the residual gas under a pressure associated with the gas load divided by the available pumping speed (Chapter 11 ). xi xii Preface The derivation of the formulas is not given, since there are many authorita- tive textbooks where the basic theory is extensively discussed. For the reader interested in studying the subject in more detail, several references are listed at the end of each chapter. All the equations are written in a form that permits the reader to use any system of units. In equations in which a combination of numerical quantities and symbols occurs, the units of measurement are stated. For the sake of clarity, the units of measurement have been written even in the intermediate steps of numerical calculations. Although not universally adopted, the self-consistent system SI of units has been used throughout the text. To facilitate understanding for readers not yet accustomed to this system, other coherent and incoherent systems of units have been used. The Torr has been retained as a pressure unit, since it is very largely spread among vacuum workers. Solved exercises, appended at the end of each chapter, were not chosen at random but rather have been carefully selected so as to include at least one representative example of each type of technological problem. Many exercises require equations from more than one chapter to permit greater flexibility in the coverage of those points having importance and usefulness in practice. The reader is strongly advised to check the accuracy of intermediary and final results of the exercises by using a dimensional equation. The book is intended for the engineer, physicist, graduate student, or advanced technician engaged in purchasing, setting up, operating, or maintain- ing vacuum systems and equipment. Armand Berman Acknowledgments I gratefully acknowledge constant encouragement and helpful suggestions and criticism from the late Professor A. Roth. I should like to thank the authors, journals, and publishers cited in the text for permission to reproduce figures and tables. Many thanks are due to the staff of Academic Press for their kind help and cooperation in publishing this book. And to my wife, Ilane, who has tolerated preoccupations, shared disappointments, read many drafts, and still found nice words to say, I express my gratitude and devotion. List of Symbols A Area a Linear dimension; constant a Amount of gas in mass units kg a Amount of gas in molecule units m a Amount of gas in mole units mol a Amount of gas in pressure-volume units pv B Second virial coefficient B Volume of gas impinging on unit area per unit time B Volume of gas escaping through an orifice per unit time 0 b Linear dimension; constant C Conductance to gas flow Q Conductance to laminar flow C Conductance to chocked laminar flow kh C Conductance to nonchocked laminar flow lnc C Conductance to molecular gas flow m C Conductance to molecular gas flow of an aperture ma C Conductance to molecular gas flow of a diaphragm md C Conductance to molecular gas flow of a pipe, end effects mp disregarded C Conductance to molecular gas flow of rectangular slits ms C Conductance to molecular gas flow of a pipe, end effects mT accounted for C Compression ratio of a mechanical pump r C Sutherland's constant s C Conductance of vacuum components connected in parallel T C Conductance to transition flow tr Q Mole fraction of a gas component in a mixture of gases c Specific heat capacity at constant pressure p c Specific heat capacity at constant volume v D Coefficient of diffusion d\ d Diameter 0 E Total translational energy of molecules in random motion E Average energy transferred per molecule E Average translational energy per molecule in random motion E Energy transferred by molecules in the molecular range of flow m E Energy transferred by molecules in the viscous range of flow v XV xvi List of Symbols F Force F, Correction factor in laminar gas flow U Fractional number of molecules G Mass of gas incident per unit area per unit time g Acceleration due to gravity H Coefficient of heat conductivity v h Height L Correction factor in molecular gas flow Κ; Κ'; Κ,; 0 0 Κ ; Κ Correction factors in molecular gas flow 2 = Permeability coefficient ^Per Κ' Knudsen's number k Boltzmann's constant L Length M Molar mass of a substance (also known as molar weight) Ma Mach number m Mass per molecule N Total number of particles in a volume N Avogadro's constant (also known as Avogadro's " number") A n Number density of molecules »M Mole amount o Orifice coefficient in chocked laminar flow ch 0 Perimeter P Total pressure of gas P Average pressure of gas Pa...V Partial pressure of gas P....Î Po Pressure at the start of pumpdown Pc Critical pressure f>D Pressure due to outgassing of materials Pressure due to evaporation of materials PE Pressure due to gas load PG Pressure at the port of the pump PP Pressure due to permeation Pper pr; Jj'r Molecular transmission probability of a vacuum component P Steady-state pressure s Pt Pressure at a certain time t Pu Transition pressure Pu Ultimate pressure of a vacuum pump Pus Ultimate pressure of a vacuum system Q Throughput List of Symbols Q Gas flow rate due to outgassing D Q Gas flow rate due to outgassing of elastomers De Q Gas flow rate due to outgassing of metals Dm Q Gas flow rate due to evaporation of materials E Q Gas load G Q Gas flow rate due to leakage L Q Gas flow rate due to true leaks Lt Q Gas flow rate due to virtual leaks Lv Q ! Throughput in laminar flow of gas Q Throughput in chocked laminar flow of gas lch Q Throughput in nonchocked laminar flow of gas lnc Q Throughput in molecular flow of gas m Q Molecular throughput of gas through an aperture ma Q Molecular throughput of a pipe mp Q Gas flow rate of residual gas r Q Throughput in turbulent flow of gas t Q Throughput in transition flow of gas tr Q Gas flow rate returned from the pump u q Specific outgassing rate of elastomers Oe q Specific outgassing rate of glass and ceramics Dgs q Specific outgassing rate of metals Om ^Dpoiy Specific outgassing rate of polymers q Specific evaporation rate of a material E q Effusion rate of molecules e q Mass flow rate of gas kg q Molecular flow rate of gas m q Molar flow rate of gas mol R Molar constant (per "mole basis"; also known as universal gas constant) Re Reynold's number r; R' Radius r Critical pressure ratio c S Pumping speed—volume rate of flow S Pumping speed of a backing pump b S Molecular pumping speed m S Molecular pumping speed of an aperture ma S Molecular pumping speed of a diaphragm md S Net (effective) pumping speed n S Pumping speed at the port of a pump P S Theoretical pumping speed of a pump th s Sticking coefficient f xviii List of Symbols T Temperature T Limiting temperature to which the equation of state can be applied B to high pressures T Critical temperature c t Time t Time to form a monolayer m U Energy V Volume V Volume of gas at a specified temperature V Critical volume c V Molar volume M V Volume of a mixture of gases m v Velocity v Arithmetic average velocity of molecules a v Most probable velocity of molecules v Mean-square velocity of molecules r v Velocity of sound (acoustic velocity) in gas s W Electric power W Rate of evaporation of a material W Mass of substance t x Mole concentration per unit volume ( V Correction factor for conductance in laminar flow Z Impedance to gas flow Z Impedance to gas flow of vacuum components connected in series T a Accommodation coefficient S Diameter (apparent) of molecule m ε Collision frequency per molecule per unit time η Coefficient of viscosity (dynamic) Θ Slope on a log-log plot of outgassing rate versus time Λ Mean free path of molecules in gas A Mean free path of electrons in gas e A Mean free path of ions in gas i Λ Coefficient of heat conductivity in gas 0 v Ratio of specific heat capacities p Density of gas r Time constant of a vessel φ Rate at which molecules at steady state strike unit area per unit time Terminology Accommodation coefficient The ratio of the energy actually transferred be- tween impinging gas molecules and the energy that theoretically would be transferred if the impinging molecules reached complete thermal equilibrium with the surface. Backing pressure (also forepressure) The total pressure measured downstream from the outlet or foreline of a vacuum pump. Backing pump (forepump) A vacuum pump for maintaining the backing pressure (forepressure) of another pump below its critical value. Bakeout Degassing by heating materials under vacuum. Calibration factor The ratio of the pressure indicated by a measuring gauge in a vacuum system to the pressure measured by a primary (or secondary) gauge in the same system. Cehius The designation of the degree on the International Temperature Scale. Also used to indicate the temperature scale. Chocked flow Situation encountered in the viscous flow region when the velocity of the gas through the vacuum element reaches the velocity of sound in that gas. A further reduction of the pressure downstream from the element cannot be sensed at the upstream side. Compression ratio The ratio of pressures at the discharge port to the inlet port of the pump, at zero flow through the pump. Critical pressure The pressure under which a gas may exist in equilibrium with its liquid phase at the critical temperature. Critical temperature The temperature above which a gas cannot be liquefied by pressure alone. Crossover forepressure The forepressure at which the inlet pressure of a pump becomes equal to the forepressure, so that the pump can efficiently overtake pumping. Crossover pressure The pressure at which the inlet pressure of a vacuum pump becomes equal to the forepressure, or the back pressure. Degassing Desorption of gas from material exposed to vacuum, as a result of supplying the material with energy such as thermal, or impact with atoms or molecules, electrons, ions, or photons. Desorption The process of removing adsorbed neutral or ionized atoms and/or molecules from a material. Evaporation rate The amount of material evaporated from a surface per unit area per unit time. xix