Simultaneous Mass Transfer and Chemical Reactions in Engineering Science E-Book

Bertram K. C. Chan

• Herausgeber: Wiley-VCH GmbH
• Kategorie: Fachliteratur
• Sprache: Englisch

Seitenzahl: 859

Veröffentlichungsjahr: 2023

Beschreibung

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science

A comprehensive look at the basic science of diffusional process and mass transfer

Mass transfer as a principle is an essential part of numerous unit operations in biomolecular, chemical, and process engineering; crystallization, distillation, and membrane separation processes, for example, use this important method. Given this significance – particularly in engineering design where these processes occur – understanding the design and analysis of such unit operations must begin with a basic understanding of how simultaneous mass transfer and the chemical reactions that influence these occurrences. It is also vital to be aware of the most up-to-date technologies for analyzing and predicting the phenomena.

Given the significance of this process, Simultaneous Mass Transfer and Chemical Reactions in Engineering Science is an important resource as it introduces the reader to the complex subject of simultaneous mass transfer with biochemical and chemical reactions and gives them the tools to develop an applicable design. Analyzing the systems of simultaneous mass transfer and reactions is at the core of this book, as all known design approaches are carefully examined and compared. The volume also provides the reader with a working knowledge of the latest technologies – with a special focus on the open-sourced computer programming language R – and how these tools are an essential resource in quantitative assessment in analysis models.

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science provides a working knowledge of the latest information on simultaneous mass transfer and reactions by focusing on the analysis of this process, as well as discussing the existence and distinctive quality of the solutions to the Simultaneous Mass Transfer and Chemical Reactions in Engineering Science readers will also find:

• A theoretical basis of each design model that is carefully stated, compared, and assessed
• Carefully developed and established Existence and Uniqueness Theorems for a general design model
• Comprehensive coverage of how the programming language R may be used to analyze models
• Numerous examples and case studies that provide a working knowledge of simultaneous mass transfer and reactions

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science is a useful reference for students in chemical engineering, biotechnology, or chemistry, as well as professional process and chemical engineers.

Leseprobe

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science

Bertram K. C. Chan

Author

Bertram K. C. Chan

1534 Orillia Court

CA

United States

Cover Image: © Pattadis Walarput/Getty Images

All books published by WILEY‐VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

Library of Congress Card No.: applied for

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The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at <http://dnb.d-nb.de>.

© 2023 WILEY‐VCH GmbH, Boschstr. 12, 69469 Weinheim, Germany

All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law.

Print ISBN: 978‐3‐527‐34665‐3

ePDF ISBN: 978‐3‐527‐82350‐5

ePub ISBN: 978‐3‐527‐82351‐2

oBook ISBN: 978‐3‐527‐82352‐9

Dedicated to

the glory of God,

my better half Marie Nashed Yacoub Chan, and

the fond memories of my high school physical science teacher, the Rev. Brother Vincent Cotter, BSc, at the De La Salle Catholic College, Cronulla, Sydney, New South Wales, Australia, as well as my former professors in Chemical Engineering in Australia, including:

at the University of New South Wales:

Professor Geoffrey Harold Roper and Visiting Professor Thomas Hamilton Chilton, from the University of Delaware, USA

and at the University of Sydney:

Professor Thomas Girvan Hunter and Professor Rudolf George Herman Prince.

Preface

This book aims to provide a comprehensive theoretical reference for students, professors, design and practicing engineers in the chemical, biomolecular, and process engineering industries a thorough and modern scientific approach to the design of major equipment for processes involving simultaneous mass transfer and chemical reactions.

Key Features

Presents the basic scientific and computational models of diffusional processes involving mass transfer with simultaneous chemical reactions.

Provides a vigorous theoretical and computational approach to processes involving simultaneous mass transfer and chemical reactions.

Involves the use of the open‐sourced computer programming language

R

, for quantitative assessment in the analysis of models for simultaneous mass transfer and chemical reactions.

What Problems Does this Book Solve?

This book is a complete resource for

A fundamental description of the scientific basis for diffusional processes and mass transfer operations in the presence of simultaneous chemical reactions. Several models are presented, assessed, and showcased for engineering design applications.

Based on a vigorous assessment of several theoretical models for mass transfer, a selected preferred methodology is demonstrated and recommended as a firm basis for engineering design.

Author Biography

Bertram K.C. Chan, PhD, PE (California, USA), Life Member–IEEE, Registered Professional Chemical Engineer in the State of California, completed his secondary education in the De La Salle College, Cronulla, Sydney, New South Wales, Australia, having passed the New South Wales State Leaving Certificate Examination (viz the state‐wide university matriculation public examination) with excellent results, particularly in pure and applied mathematics, and in Honors Physics and Honors Chemistry. He then completed both a Bachelor of Science degree in Chemical Engineering with First Class Honors, and a Master of Engineering Science degree in Nuclear Engineering at the University of New South Wales, followed by a PhD degree in Chemical and Biomolecular Engineering at the University of Sydney, both universities are in Sydney, New South Wales, Australia.

This was followed by two years of working as a Research Engineering Scientist (in Nuclear Engineering) at the Australian Atomic Energy Commission Research Establishment, Lucas Heights, New South Wales, and two years of a Canadian Atomic Energy Commission Post‐doctoral Fellowship (in Chemical and Nuclear Engineering) at the University of Waterloo, Waterloo, Ontario, Canada.

He had undertaken additional graduate studies at the University of New South Wales, at the American University of Beirut, and at Stanford University, in mathematical statistics, computer science, and pure and applied mathematics (abstract algebra, automata theory, numerical analysis, etc.), and in electronics, and electromagnetic engineering.

His professional career includes over 10 years of full‐time, and 10 years of part‐time, university‐level teaching and research experiences in several academic and industrial institutions, including a Research Associateship in Biomedical and Statistical Analysis, Perinatal Biology Section, ObGyn Department, University of Southern California Medical School, teaching at Loma Linda University, Middle East College (now University), and San Jose State University, and had held full‐time industrial research staff positions, in the Silicon Valley, California, for 27 years – at Lockheed Missile & Space Company (10 years), Apple Computer (7 years), Hewlett‐Packard (3 years), and as a research and design electromagnetic compatibility engineer at a start‐up company: Foundry Networks (7 years).

In recent years:

He supported the biostatistical work of the Adventist Health Studies II research program at the Loma Linda University Health (LLUH) School of Medicine, California, and consulted as a forum lecturer for several years in the LLUH School of Public Health (Biostatistics, Epidemiology, and Population Medicine). In these lectures, Dr. Chan introduced the use of the open‐sourced programming language

R

and designed these lectures for the biostatistical elements for courses in the MPH, MsPH, DrPH, and PhD programs, with special reference to epidemiology and biostatistics in particular, and public health and population medicine in general.

Dr. Chan had been granted three US patents in electromagnetic engineering, had published over 30 engineering research papers, and authored a 16‐book set in educational mathematics

[1]

, as well as 5 monographs entitled: “Biostatistics for Epidemiology and Public Health Using

R

”

[2]

, “Applied Probabilistic Calculus for Financial Engineering: An Introduction Using

R

”

[3]

, “Biostatistics for Human Genetic Epidemiology”

[4]

, “Simultaneous Mass Transfer and Chemical Reactions in Engineering Science – Solution Methods and Chemical Engineering Applications”

[5]

, and “Fundamental System Design Principles for Simultaneous Mass Transfer and Chemical Reactions in Chemical Engineering Science – including a Computational Approach with

R

”

[6]

.

He is a registered Professional Chemical Engineer (

PE

) in the State of California, USA, as well as a Life Member of the Institute of Electrical and Electronic Engineers (

IEEE

).

References

1

Chan, B. (1978).

A New School Mathematics for Hong Kong

. Hong Kong: Ling Kee Publishing Co. 10 Volumes of Texts: 1A, 1B, 2A, 2B, 3A, 3B, 4A, 4B, 5A, 5B. 6 Volumes of Workbooks: 1A, 1B, 2A, 2B, 3A, 3B.

2

Chan, B.K.C. (2016).

Biostatistics for Epidemiology and Public Health Using

R

. New York: Springer Publishing Company (with additional materials on the Publisher's website).

3

Chan, B.K.C. (2017).

Applied Probability Calculus for Financial Engineering: An Introduction Using

R

. Hoboken, NJ: Wiley.

4

Chan, B.K.C. (2018).

Biostatistics for Human Genetic Epidemiology

. New York/Cham, Switzerland: Springer International Publishing AG.

5

Chan, B.K.C. (2020).

Simultaneous Mass Transfer and Chemical Reactions in Engineering Science – Solution Methods and Chemical Engineering Applications

. Cambridge, MA/Amsterdam, Holland, The Netherlands: Elsevier.

6

Chan, B.K.C. (2021).

Fundamental System Design Principles for Simultaneous Mass Transfer and Chemical Reactions in Chemical Engineering Science – Including a Computational Approach with

R

. Cambridge, MA/Amsterdam, Holland, The Netherlands: Elsevier.

1Introduction to Simultaneous Mass Transfer and Chemical Reactions in Engineering Science

In many biochemical, biomedical, and chemical processes, in both the chemical industry and in physiological systems, including environmental sciences, mass transfer, accompanied by reversible, complex biochemical, or in chemical reactions in gas–liquid systems, is frequently found. From the viewpoint of biochemical and/or chemical design purposes, it is very important that the absorption rates of the transferred reactants may be estimated accurately.

Moreover, the mass transfer phenomena can also affect substantially important process variables like selectivity and yield. Considerable research effort has been expended in describing these processes and in the development of mathematical models that may be used for the computation of the mass transfer rates and other parameters.

For example, the description of the absorption of a gas followed by a single first‐order reversible reaction is simple and straightforward. For all mass transfer models, e.g. film, surface renewal, and penetration, this process may be analytically solved. For other processes, however, only for a limited number of special cases analytical solutions are possible, and therefore numerical techniques must be used for the description of these phenomena. Besides numerically solved absorption models, the mass transfer rates often may be calculated, with sufficient accuracy by simplifying the actual process by means of approximations and/or linearizations. In this book, an overview will be given of the absorption models that are available for the calculation of the mass transfer rates in gas–liquid systems with (complex) reversible reactions. Both numerically solved and approximated models will be treated and conclusions on the applicability and restrictions will be presented.

1.1 Gas–Liquid Reactions

It is well known that many biochemical and chemical processes involve mass transfer of one or more species from the gas phase into the liquid phase. In the liquid phase, the species from the gas phase are converted by one or more (possibly irreversible) biochemical or chemical reactions with certain species present in the liquid phase.

Typical of such examples are provided in Sections 1.1.1.1 and 1.1.1.2.

1.1.1 Simultaneous Biomolecular Reactions and Mass Transfer

1.1.1.1 The Biomedical Environment

In epidemiologic investigations, occurrences of simultaneous biomolecular reactions and mass transfer are common in many biomedical environments. Some typical examples are:

Intestinal Drug Absorption Involving Bio‐transporters and Metabolic Reactions with Enzymes

[1]

: The absorption of drugs via the oral route is a subject of on‐going and serious investigations in the pharmaceutical industry since good bio‐availability implies that the drug is able to reach the systemic circulation via the oral path. Oral absorption depends on both the

drug properties

and the physiology of the gastrointestinal tract, or

patient properties

, including drug dissolution, drug interaction with the aqueous environment and membrane, permeation across membrane, and irreversible removal by organs such as the liver, intestines, and the lung.

Oxygen Transport via Metal Complexes

[1]

: On average, an adult at rest consumes 250 ml of pure oxygen per minute to provide energy for all the tissues and organs of the body, even when the body is at rest. During strenuous activities, such as exercising, the oxygen needs increase dramatically. The oxygen is transported in the blood from the lungs to the tissues where it is consumed. However, only about 1.5% of the oxygen transported in the blood is dissolved directly in the blood plasma. Transporting the large amount of oxygen required by the body, and allowing it to leave the blood when it reaches the tissues that demand the most oxygen, require a more sophisticated mechanism than simply dissolving the gas in the blood. To meet this challenge, the body is equipped with a finely tuned transport system that centers on the metal complex

heme

. The metal ions bind and then release ligands in some processes, and to oxidize and reduce in other processes, making them ideal for use in biological systems. The most common metal used in the body is iron, and it plays a central role in almost all living cells. For example, iron complexes are used in the transport of oxygen in the blood and tissues. Metal–ion complexes consist of a metal ion that is bonded via “coordinate‐covalent bonds” to a small number of anions or neutral molecules called ligands. For example, the ammonia (NH

3

) ligand is a mono‐dentate ligand; i.e. each mono‐dentate ligand in a metal–ion complex possesses a single electron‐pair‐donor atom and occupies only one site in the coordination sphere of a metal ion. Some ligands have two or more electron‐pair‐donor atoms that can simultaneously coordinate to a metal ion and occupy two or more coordination sites; these ligands are called polydentate ligands. They are also known as chelating (Greek word for “claw”) agents because they appear to grasp the metal ion between two or more electron‐pair‐donor atoms. The coordination number of a metal refers to the total number of occupied coordination sites around the central metal ion (i.e. the total number of metal–ligand bonds in the complex). This process is another important example of biomolecular reaction and transport.

Carotenoid Transport in the Lipid Transporters SR‐BI, NPC1L1, and ABCA1

: The intestinal absorption of carotenoids in vivo involves several crucial steps:

release from the food matrix in the lumen

solubilization into mixed micelles

uptake by intestinal mucosal cells

incorporation into chylomicrons

secretion into the lymph.

Research has shown that:

EZ is an inhibitor of the intestinal absorption of carotenoids, an effect that decreased with increasing polarity of the carotenoid molecule

SR‐BI is involved in intestinal carotenoid transport

EZ acts not only by interacting physically with cholesterol transporters as previously suggested but also by downregulating the gene expression of three proteins involved in cholesterol transport in the enterocyte, the transporters SR‐BI, NPC1L1, and ABCA1.

The intestinal transport of carotenoid is thus a facilitated process resembling that of cholesterol; therefore, carotenoid transport in intestinal cells may also involve more than one transporter.

Hence, the study of biomolecular reaction and transport is an area of importance in biomedical processes and their occurrences in epidemiologic investigations.

In this section, one applies the facilities available in the R environment to solve problems arisen from these processes. This study is being approached from two directions:

• Using the

R

environment as a

support

to numerical analytical schemes that may be developed to solve this class of problems.

• Applying the

R

functions in the CRAN package

ReacTran

[2]

.

1.1.1.2 The Industrial Chemistry and Chemical Engineering Environment

Typical examples of industrial chemical and chemical engineering processes in which this phenomenon occurs include chlorination, gas purification, hydrogenation, and oxidation processes. To undertake the process and equipment design of new reactors and the optimization of existing reactors, applicable theoretical models for reactors are helpful and most likely needed. In general, models of liquid–gas contactors consist of two main parts: the micro model and the macro model:

the

micro

model describes the interphase mass transfer between the gas phase and the liquid phase,

the

macro

model describes the mixing behavior in both phases.

Both parts of the overall model may be solved sequentially, but solving micro and macro models simultaneously is preferred because of optimization of computational time.

Gas–liquid mass transfer modeling has been well studied. The Whitman stagnant film model was first described in 1923 by W.G. Whitman, and it was concluded that some phenomena of gas–liquid mass transfer may be regarded as nearly incompletely explained. Moreover, the Higbie penetration model has been used as a basis for the development of some new reactor models. The influence of the bulk liquid on the mass transfer process has been studied in some detail. More attention has been paid to the dynamical behavior and stability of gas–liquid reactors and the influence of mass transfer limitations on the dynamics. Also, some important differences between the results of the Higbie penetration model and the Whitman stagnant film model are found.

Analytical solution of micro models for mass transfer (accompanied by chemical reactions) is restricted to asymptotic cases in which many simplifying assumptions had to be made (e.g. reaction kinetics are simple and the rate of the reaction is either very fast or very slow compared to the mass transfer). For all other situations, numerical–computational techniques are required for solving the coupled mass balances of the micro model.

In general, it seems that mostly numerical solution techniques have been applied. Wherever possible, analytical solutions of asymptotic cases have been used to check the validity of the numerical solution method.

For example, by modifying one of the boundary conditions of the Higbie penetration model, it had been found that the mass transfer may be affected by the presence of the bulk liquid. For example, in a packed column, the liquid flows down the column as a thin layer over the packings. It has been examined whether or not the penetration model may be applied for these configurations. Both physical absorption and absorption accompanied by first‐ and second‐order chemical reactions have been investigated.

From model calculations, it is concluded that the original penetration theory, by assuming the presence of a well‐mixed liquid bulk, may be applied also to systems where no liquid bulk is present, provided that the liquid layer is sufficiently thick!

For packed columns this means, in terms of the Sherwood number,

N

Sh

 = 4, for both physical absorption and absorption accompanied by a first‐order reaction.

In case of a second‐order 1,1‐reaction, a second criterion:

N

Sh

 ≥ 4√(

D

b

/

D

a

) has to be fulfilled.

For very thin liquid layers (

N

Sh

 < 4, or

N

Sh

 < 4√(

D

b

/

D

a

)), the original penetration model may give erroneous results, depending on the exact physical and chemical parameters, and a modified model is required.

Analytical solution of models for gas–liquid reactors is restricted to a few asymptotic cases, while most numerical models make use of the physically less realistic stagnant film model – this is relatively simplistic and easy to apply using the “hinterland model.” The hinterland model assumes the reaction phase to consist of ONLY a stagnant film and a well‐mixed bulk. Inflow and outflow of species to and from the reactor proceeds via the non‐reaction phase or via the bulk of the reaction phase, but never via the stagnant film. (“Hinterland” is a German word meaning “the land behind” [a port, a city, …] in geographic usages!)

By modifying one of the boundary conditions of the Higbie penetration model, it illustrated how the mass transfer may be affected by the presence of the liquid bulk. Thus, for example, in a packed column, the liquid flows as a thin layer over the structured or dumped packing. It has been examined whether or not the penetration model can be applied for these situations. Both physical absorption and absorption accompanied by first‐ and second‐order chemical reactions have been investigated.

From model calculations, it is concluded that the original penetration theory, which assumes the presence of a well‐mixed liquid bulk, may be applied also to systems where no liquid bulk is present, provided that the liquid layer has sufficient thickness.

For packed columns, this means, in terms of Sherwood number, Sh > 4 for both physical absorption and absorption accompanied by a first‐order reaction. In case of a second‐order 1,1‐reaction a second criterion Sh ≥ 4√(Db/Da) has to be fulfilled. For very thin liquid layers, Sh < 4 or Sh < √(Db/Da), the original penetration model may give erroneous results, depending on the exact physical and chemical parameters, and the modified model is required.

Most numerical models of gas–liquid reactors make use of the physically less realistic stagnant film model because implementation of the stagnant film model is relatively easy using the hinterland concept. The combination of stagnant film model and Hinterland concept may successfully predict many phenomena of gas–liquid reactors.

The Higbie penetration model is however preferred as a micro model because it is physically more realistic. Direct implementation of the hinterland concept is not possible with the Higbie penetration model. Nevertheless, numerical techniques have been applied to develop a new model that implements the Higbie penetration model for the phenomenon mass transfer accompanied by chemical reaction in well‐mixed two‐phase reactors: assuming the stagnant film.

A model was developed that simulates the dynamic behavior of gas–liquid tank reactors by simultaneously solving the Higbie penetration model for the phenomenon of mass transfer accompanied by chemical reaction and the dynamic gas and liquid phase component balances. The model makes it possible to implement an alternative for the well‐known hinterland concept, which is usually used together with the stagnant film model. In contrast to many other numerical and analytical models, the present model can be used for a wide range of conditions, the entire range of Hatta numbers, (semi‐)batch reactors, multiple complex reactions, and equilibrium reactions, components with different diffusion coefficients, and also for systems with more than one gas phase component. By comparing the model results with analytical asymptotic solutions, it was concluded that the model predicts the dynamic behavior of the reactor satisfactorily. It had been shown that under some circumstances, substantial differences exist between the exact numerical and existing approximate results. It is also known that for some special cases, differences can exist between the results obtained using the stagnant film model with hinterland concept and the implementation of the Higbie penetration model.

Analytical solution of models for gas–liquid reactors is restricted to a few asymptotic cases, while most numerical models make use of the physically less realistic stagnant film model.

1.1.2 Conclusions

The penetration model is preferred for the phenomenon mass transfer accompanied by chemical reaction in well‐mixed two‐phase reactors.

By comparing the model results with analytical asymptotic solutions, it is concluded that the model predicts the reactor satisfactorily. It is shown that for many asymptotic cases, the results of this new model coincide with the results of the stagnant film model with hinterland concept.

For some special conditions, differences may exist between the results obtained using the stagnant film model with hinterland concept and the implementation of the Higbie penetration model.

An important result is that for 1,1‐reactions, the saturation of the liquid phase with gas phase species does not approach zero with increasing reaction rate (increasing Hatta number), contrary to what is predicted by the film model with Hinterland concept. Another important deviation may be found at the specific conditions of a so‐called instantaneous reaction in combination with the absence of chemical enhancement of mass transfer.

Application of the penetration model does not provide any numerical difficulties, while application of the stagnant film model would lead to a discontinuity in the concentration gradient.

Another disadvantage of the hinterland concept is that it can strictly only be applied to isothermal systems, whereas in the systems investigated in this thesis the reaction enthalpy is an important parameter that may significantly influence the phenomena of gas–liquid mass transfer.

A rigorous model may be developed that simulates the dynamic behavior of stirred nonisothermal gas–liquid reactors by simultaneously solving the Higbie penetration model for the phenomenon mass transfer accompanied by chemical reaction and the dynamic gas and liquid phase component and heat balances. This is achieved by coupling the ordinary differential equations of the macro model mass and heat balances to the partial differential equations of the penetration model. This model is not yet published!

Using the newly developed rigorous reactor model, it is shown that dynamic instability (limit cycles) can occur in gas–liquid reactors. The influence of mass transfer limitations on these limit cycles has been studied, and it has been found that mass transfer limitations make the process more stable.

1.1.3 Summary

Although the rigorous model is believed to be a very accurate model, it has the disadvantage that owing to the complex numerical methods applied it is a rather time‐consuming model. On behalf of a more efficient prediction of the possible occurrence of limit cycles, the reactor model was simplified. The simplified model is suited for the prediction of limit cycles using a stability analysis. A stability analysis is a very efficient method to predict the dynamic behavior and stability of a system of ordinary differential equations by linearization of the governing nonlinear ODEs in the neighborhood of the steady state and analyzing the Eigenvalues. This method is very powerful for attaining design rules for stable operation of stirred gas–liquid reactors. The influence of mass transfer limitations on the limit cycles is predicted very well using the simplified model, though small discrepancies are found with the more accurate rigorous model.

The developed reactor models have been used to model the dynamics of a new, to‐be‐developed, industrial hydro‐formylation reactor. At a certain design of the reactor, the model predicts serious and undesired limit cycles. These conditions have to be avoided by an appropriate reactor design. Hydro‐formylation reactions are often characterized by a negative reaction order in carbon monoxide. Model calculations showed that this may lead to interesting phenomena: at certain process conditions, an improvement of the mass transfer (higher kLa, for example, owing to improved mixing) may give rise to a less stable reactor, without increasing the conversion. This unusual phenomenon is explained by the negative reaction order of carbon monoxide. Apparently, the increasing hydrogen and carbon monoxide concentrations cancel each other out and the overall reaction rate remains unchanged. The increasing hydrogen and carbon monoxide concentrations do however make the process more sensitive for the occurrence of limit cycles.

Finally, a start has been made with studying the influence of macro‐mixing on the dynamical behavior of gas–liquid reactors. For this purpose, a cascade of two reactors in series is compared to a single reactor. Initial results indicate that a cascade of reactors in series provides a dynamically more stable design. The total required cooling surface to prevent the occurrence of temperature–concentration limit cycles decreases significantly with increasing number of reactors in series. The first reactor in the cascade is the one with the highest risk of dynamic instability.

1.2 The Modeling of Gas–Liquid Reactions

This process has evolved through a number of theoretical processes, including:

1.2.1 Film Theory of Mass Transfer

In typical industrial absorption processes, one should consider the absorption of gases into liquids which are agitated such that the dissolved gas is transported from the surface to the interior by convective motions. The agitation may occur in various ways, including:

The gas, or vapor, may be blown through the liquid as a stream of bubbles – as, for example, on a perforated plate or in a sparged vessel.

The liquid may be running in a layer over an inclines or vertical surface, and the flow may be turbulent (as, for example, in a wetted‐wall cylindrical column operating at a sufficiently high Reynolds number), or ripples may develop and enhance the absorption rate by convective motion. Discontinuities on the surface may cause periodic mixing of the liquid in the course of its flow, or strings of discs or of spheres.

The liquid may be advantageously agitated by a mechanical stirrer, which may also entrain bubbles of gases into the liquid.

The liquid may be sprayed through the gas as jets or drops. First consider a steady‐state situation in which the composition of the liquid and gas, averaged over a specified region and also with respect to any temporal fluctuations, are statistically constant. For example, one may consider an agitated vessel through which liquid and gas flow steadily, both being so thoroughly mixed that their time‐average compositions are the same at all points; or one may consider a short vertical section of a packed column (or sphere or disc or wetted‐wall column) operating at steady state, such that the average compositions of the liquid and gas in the element remain

constant

with time.

Clearly, the situation is a complicated one: the concentrations of the various species are not uniform or constant when measured over short length and time scales. Diffusion, convection, and reaction proceed simultaneously. The nature of the convective movements of liquid and gas is difficult to define: any attempt to describe them completely would encounter considerable complications. Thus, to obtain useful predictions about the behavior of such systems for practical purposes, it is necessary to use simplified models which simulate the situation sufficiently well, without introducing a large number of unknown parameters. This approach may take a number of simplifying steps, as follows:

Physical Absorption

[2]

Consider first physical absorption, in which the gas dissolves in the liquid without any reaction; it is found experimentally that the rate of absorption of the gas is given by

(1.1)

in which A* is the concentration of dissolved gas at the interface between gas and liquid, assuming this partial pressure to be uniform throughout the element of space under discussion. The area of interface between the gas and liquid, per unit volume of the system, is a and kL is the “physical mass‐transfer coefficient.” R is the rate of transfer which may vary from point to point and from time to time. R is the average rate of transfer of gas per unit area; the actual rate of transfer may vary from point to point, and from time to time. A0 is the average concentration of dissolved gas in the bulk of the liquid.

It is usually not possible to determine kL and a separately, by measurements of physical absorption. For example, in a packed column, the fraction of the surface of packings which is effectively wetted is unknown, and in a system containing bubbles, the interfacial area is not generally known! Thus, the quantity directly measurable by physical absorption measurements is the combined quantity kLa.

Hence, the validity of Eq. (1.1) has been established in numerous experimental studies, and an expression of this form would be predicted from first principles, provided that certain conditions are met. The chief of these are that the temperature and diffusivity at the surface (where the concentration is A*) should be very different from those in the bulk of the liquid; and that no chemical reaction occurs, so that all molecules of dissolved gas are in the same condition. It is sometimes difficult to decide whether a solute reacts chemically with a liquid or merely interacts with it physically. From the present purpose, “physical” solution means that it molecules are indistinguishable.

1.2.2 Surface Renewal Theory of Mass Transfer

Models evolved under this theory take as their basis the replacement at time intervals of liquid at the surface by liquid from the interior which has the local mean bulk composition. While the liquid element is at the surface and is exposed to the gas, it absorbs gas as though it were infinitely deep and quiescent: the rate of absorption, R, is then a function of the exposure time of the liquid element and will be described by a suitable expression such as those to be described by the reaction kinetics of the system. In general, the rate of absorption is fast or infinite initially, decreasing with time. The replacement of liquid at the surface by fresh liquid of the bulk composition may be due to the turbulent motion of the body of the liquid. Moreover, when liquid runs over the surface of a packing, it may be in a state of undisturbed laminar flow at the top of each piece of packing, except at the discontinuities between pieces of packing, where it may mix thoroughly: at the top of each piece of packing, a fresh surface would then be developed and moved discontinuity, when it would then be replaced again by fresh liquid.

With this scenario, the surface‐renewal models propose that the surface of an agitated liquid, or a liquid flowing over a packing, is a collection of elements which have been exposed to the gas for different durations of time, and which may well be, in general, absorbing at different specific rates. Thus, different versions of the model will lead to different specific rates. Moreover, different versions of the model will lead to different distributions of surface ages about the mean value. The form of the surface‐renewal model first proposed by Higbie, in 1935, assumed that every element of surface is exposed to the gas for the same duration of time, θ, before being replaced by liquid of the bulk composition. During this time, the element of liquid absorbs the same amount Q of gas per unit area as though it were infinitely deep and stagnant. The average rate of absorption is therefore Q/θ, and this is also the rate of absorption R per unit area averaged over the interface in a representative region of a steady‐state absorption system in which the bulk composition is statistically uniform – e.g. in a small, but representative, volume element of a packed column.

The exposure‐time θ may be determined by the hydrodynamic properties of the system, and is the only parameter required to account for their effect on the transfer coefficient kL. The relation between θ and kL is derived herebelow – in physical absorption.

Under such circumstances, the variation in time and space of the concentration a of dissolved gas in the liquid in the absence of reaction is governed by the diffusion

(1.2)

And the rate of transfer of dissolved gas initial concentration passage of gas across the interface, then the concentration of the surface might vary with time. For the present, it is assumed that the diffusion of dissolved gas into the latter. This assumption generally holds when the solubility of the gas is not very great, so that A* represents a mole fraction much less than unity.

It would not be true, for example, if ammonia at atmospheric pressure were diffusing into pure water (in which there will be a substantial temperature rise). Under these conditions, the variation in time and space of the concentration, a, of dissolved gas in the liquid in the absence of reaction as governed by the diffusion from bubbles or absorption by wetted‐wall columns, the mass transfer surface is formed instantaneously and transient diffusion of the material takes place. Assuming that a bubble is rising in a pool of liquid (where the liquid elements are swept on its surface) and remains in contact with it during their motion and finally detached at the bottom.

The basic assumptions of the penetration theory are:

Unsteady‐state mass transfer occurring to a liquid element as long as it is in contact with the gas bubbles

Equilibrium existing at gas–liquid interface

Each liquid element staying in contact with the gas for same period of time. (The liquid elements are moving at the same rate, and there is not a velocity gradient within the liquid.)

Under these assumptions, the convective terms in the diffusion may be neglected and the unsteady‐state mass transfer of gas (penetration) into the liquid element may be written from Fick's second law for unsteady‐state diffusion as (Figure 1.1)

(1.3)

and the boundary conditions are

and

where

c

Ab

=

The concentration of solute A at an infinite distance from the surface (viz. the bulk concentration)

c

Ai

=

The interfacial concentration of solute A at the surface

Figure 1.1 Schematic of the penetration model.

On solving the above partial differential equation, one obtains:

(1.4)

If the process of mass transfer is a unidirectional diffusion and the surface concentration is very low: i.e. cAb ≈ 0; then the mass flux of solute A, given by NA (kg/m2 s) may be estimated by the following equation:

(1.5)

From the above two expressions, the rate of mass transfer at time t is given by the following equation:

(1.6)

And the mass transfer coefficient is given by

(1.7)

Moreover, the average mass transfer coefficient during a time interval tc(t) may be obtained by integrating Eq. (1.4) as

(1.8)

Thus, from the above equation, the mass transfer coefficient is proportional to the square root of the diffusivity. This was first proposed by R. Higbie in 1935 and the theory is called the Higbie penetration theory.

1.2.3 Absorption into a Quiescent Liquid

First, consider the case in which no chemical reaction occurring between the dissolved gas and the liquid [2]. The surface of the liquid first contacts the gas at time t = 0, and it may be assumed that, from that instance onward, the concentration in the plane of the surface is uniformly equal to A* – this concentration corresponds to the solubility of the gas at the prevailing partial pressure above the surface of the liquid – and is assumed to be constant. If this gas were mixed with another gas of different solubility, or if there were a resistance the passage of gas across the interface, then the concentration at the surface may vary with time.

Further, it is assumed that the diffusion of dissolved gas into the liquid does not appreciably affect the temperature, or other physical properties of the latter. This is likely to be true only when the solubility of the gas is not very great, so that A* represents a mole fraction much less than unity.

Under these special circumstances, the variation in time and space of the concentration a of dissolved gas in the liquid in the absence of reaction is governed by the diffusion equation:

(1.9)

and the rate of transfer of dissolved gas across unit area, Rx of any plane parallel to the surface is

(1.10)

Here, x is the distance measured from the surface, where x = 0, and DA is the diffusivity or diffusion coefficient of the dissolved gas. Hence, the rate of absorption of gas at any time is

(1.11)

The term (∂a/∂x)x=0 is the concentration gradient at the surface and is a function of time.

Let the initial concentration of A be uniformly equal to A0, and its concentration remote from the surface remains A0. Then the solution of Eq. (1.9) with boundary conditions:

(1.12)

is

(1.13)

Giving the distribution of concentration in the case where the initial concentration is A0, and the function

(1.14)

is the error function of x/2√(DAt), and is defined by

(1.15)

Values of the error function may be found in standard mathematical tables, etc. From Eqs. (1.11) and (1.13), it follows that

(1.16)

Thus, the rate of absorption is infinite when the gas and liquid are first in contact, decreasing with time, and the amount Q absorbed by a unit area of surface, in time t, is given by

(1.17)

1.2.3.1 Absorption Accompanied by Chemical Reactions

If the dissolved gas reacts with the liquid, or with a substance dissolved in the liquid [2], then Eq. (1.9) should be replaced by

(1.18)

in which r(x, t) is the rate per unit volume of liquid at which the reaction is using up the solute gas at time t and at a distance x below the surface. This rate will depend on the local concentration of the gas, and of any other solute with which it reacts. For some cases, numerical and/or analytical solutions of the diffusion reaction equations are available.

It is assumed throughout that the temperature and the values of such as physico‐chemical quantities as diffusivities, reaction‐rate constants, and solubilities remain constant and uniform. Moreover, enhancement factors E may be computed which are the ratios of the amount which would be absorbed if there were no reaction, viz.

1.2.3.2 Irreversible Reactions

1.2.3.2.1 First‐Order Reactions

Here

(1.19)

in which k1 is the first‐order rate constant for the reaction. The rate of reaction of dissolved gas at any point is directly proportional to its concentration. Under these circumstances, the solution to Eq. (1.18), with boundary conditions (1.12), and with A0 = 0, is [2, 4]:

(1.20)

so that

(1.21)

and

(1.22)

Thus, when k1t is large, the distribution of concentration and absorption rate tend to limiting values and no longer change with time:

(1.23)

(1.24)

and

(1.25)

The error in Eq. (1.25) is less than 3% when k1t > 2.

When k1t is large,

(1.26)

to within 5% when k1t > 10.

For short times of exposures, for k1t ≪ 1

(1.27)

(1.28)

to within 5% when k1t < 0.5.

The above equations form the basis of methods for measuring k1 and A*√DA.

In practice, truly first‐order irreversible reactions are seldom found. However, when the solute gas undergoes a reaction with a dissolved reactant, which is first order with respect to the concentration of the dissolved gas, then, under certain circumstances, the concentration of the reactant may be uniform and the reaction rate of the dissolved w order and the above equations do apply.

If the product of an irreversible first‐order gas reaction has the same diffusivity as the dissolved gas, then its concentration P* at the surface is given by Danckwerts (1967):

(1.29)

where y moles of product arise from the reaction of one mole of dissolved gas.

1.2.3.2.2 Instantaneous Reactions

In this case, the dissolved gas reacts instantaneously with a dissolved reactant. There is a plane underneath the surface where the concentration of both is zero, and the rate at which the two substances can diffuse to the reaction plane. The actual kinetics of the reaction are immaterial. The initial concentration of the reactant is uniformly B0, and z moles of it react with each mole of dissolved gas.

The solution of the equations governing this case (and similar phenomena involving a moving boundary) has been given by Danckwerts [5]:

(1.30)

(1.31)

(1.32)

(1.33)

(1.34)

(1.35)

where

(1.36)

and β is defined by

(1.37)

Thus, the factor Ei is a function of DB/DA and B0/zA*. Here a, b are the local concentrations of dissolved gas and reactant, respectively, and DA, DBtheir diffusivities. The reaction plane is at a depth of 2β√t beneath the surface. The quantity Ei is the factor by which the reaction increases the amount absorbed in a given time, as compared to absorption without reaction.

The concentration p of the product (assuming y moles to be formed from each mole of reacting gas) is

(1.38)

(1.39)

where DP is the diffusivity of the product.

When the diffusivities are equal:

(1.40)

If, in addition, one mole of reactant reacts with one mole of gas to produce one mole of product, then y = z = 1, and the concentration of product at the surface is the same as that of the reactant in the bulk.

When Ei is much greater than unity, then

(1.41)

the error being of order 1/2Ei. When DA = DB,

(1.42)

for all values of the variables.

Under these conditions, the rate of absorption is equal to that of physical absorption of a gas of solubility (A* + B0/z).

Remarks: In many cases, the reactions of the dissolved gases are so fast that they may be considered as instantaneous under all circumstances, for example:

H

2

S + OH

−

 → HS

−

 + H

2

O

H

2

S + RNH

2

 → RNH

3

+

 + HS

−

NH

3

 + HCl → NH

4

+

 + Cl

−

Other reactions may have finite rates which may control the rate of absorption under certain circumstances, for example:

CO

2

 + H

2

O → HCO

3

−

 + H

+

CO

2

 + OH

−

 → HCO

3

−

CO

2

 + 2RNH

2

 → RNHCOO

−

 + RNH

3

+

4Cu

+

 + 4H

+

 + O

2

 → 4Cu

++

 + 2H

2

O

In all these reactions, in which the reaction rates may be controlled by diffusion under some circumstances, and then the chemical steps in the reaction may be treated as though they were instantaneous. In general, irreversible reactions may be treated as instantaneous if the following condition is fulfilled:

(1.43)

in which Q0 is the amount of gas which would be absorbed in time t if there were no depletion of reactant B in the neighborhood of the surface; if the condition is fulfilled, then the reactant B is entirely depleted in the neighborhood of the surface and the rate of reaction is only controlled by diffusion.

Worked Example 1.1

H2S at 1 atm is absorbed into quiescent water and into a 0.1 M solution of monoethanolamine (MEA) at 25 °C. The reaction

may be taken to be instantaneous and irreversible. Calculate the amount of H2S absorbed per cm2 of surface in 0.1 s in water.

The diffusivity of H2S in water is 1.48 × 10−5 cm2