BIBLIOTHEEK TU Delft P 1989 5394
FUNDAMENTAL ASPECTS
OF
FLUIDISED BED COATING
PROEFSCHRIFT
TER VERKRIJGING VAN DE GRAAD VAN DOCTOR IN DE TECHNISCHE WETENSCHAPPEN AAN DE TECHNISCHE HOGESCHOOL DELFT, OP GEZAG VAN DE RECTOR MAGNIFICUS DR. IR. C. J. D. M. VERHAGEN, HOOGLERAAR IN DE AFDELING DER TECHNISCHE NATUURKUNDE, VOOR EEN COMMISSIE UIT DE SENAAT TE VERDEDIGEN
OP WOENSDAG 25 JUNI 1969 TE 14.00 UUR
DOOR
MUHARREM ELMAS
Master of Science in Theoretical Physical Chemistry
DELFTSCHE UITGEVERS MAATSCHAPPIJ N.V. DELFT 1969 1
Dit proefschrift is goedgekeurd door de promotor PROF. DR. IR. W. J. BEEK.
The author appreciates the collahoration and cherishes the memory of his students:
H.T. van Asselt J. Bac
H.J.M. Beaujean P. Bongers S.E. den Broeder H.W. Grootendorst G.J. van Hasselt J.H.O. Hazewinkel L.J. Korteweg A. van Melle H.A. Stork S. Strijbos B.L. van der Ven E.D. Wilschut
Anneme, agateyime ve karima
v i l
CONTENTS
Summary ix
Introduction 1
1.1 The Problem o f Fluidised Bed Coating 1 1.2 The History and t h e Present State of A r t s 3
1.3 Set-up o f t h e Research 1|
Fluidisation 7 2.1 T h e U s e s o f Fluidised Systems 7
2.2 Basic Definitions in Fluidised Systems 8
2.3 Mechanics of Fluidisation 10
2.k Recent Developments in Fluidisation 11
Particle Motion in a Fluidised Bed I7
3.1 Introduction 1 7
3.2 Particle Motion in the Bed I7 3.3 The Results and thier Interpretations 19
3.^ Conclusions 21 Thermal Properties of the Fluidised Bed 23
U.l Introduction 23
k,2 Factors Affecting the Mechanism of Coating 23 it.3 Experimental Apparatus and Procedure 26
k.k Interpretation of Results 29
I+.5 Conclusions 37 The Mechanism of the Coating Process 39
5.1 Introduction 39
5.2 Experimental Set-up and Procedure U O ^.3 Experimental Results and t h e i r Interpretations
for Polymers w i t h Negligible Heat of Fusion
(polystyrene) kO
1 D i p - c o a t i n g e x p e r i m e n t s ; simplified treatment Ul 2 Experiments with constant flux heating;
sim-plified treatment h6
V l l l
Contents (cont.)
5.U Experimental Results and their Interpretations
for Polymers with Heat of Fusion (polyethylene) 51 1 Dip-coating experiments; simplified treatment 51 2 Experiments with constant heat flux;
simpli-fied treatment 55
5.5 Conclusions 57 _6 Coating of Continuously Moving Axially Symmetric
Objects 59 6.1 Introduction 59
6.2 Experimental Apparatus and Procedure 59 6.3 Theoretical Predictions and the Experimental
Results ror the Coating Thickness on a
Continu-ously Moving Wire 60
6,k Conclusions 63
7_ Some Practical Considerations and Conclusions 65
7.1 Introduction 65 7.2 Post Heating of the Coating '65
7.3 Considerations of Optimum Conditions of Coating
' Operations 68 7.U Conclusions 72
References 73 Nomenclature 77
IX
Summary
FUNDAMENTAL ASPECTS OF FLUIDISED BED COATING
During the last twenty years fluidised bed coating of objects for the purpose of insulating or protecting them from their surroundings nas been developed in many branches of industry. However, the understanding of the process contrary to its applications has lagged behind and so far, in the de-sign and operation of such plants empirical rules have domi-nated.
This research was undertaken ito give an explanation of the mechanism of the coating process and to try to find out the most important parameters influencing the way the coating process proceeds.
The study is divided into seven chapters.
In the first chapter the problem of fluidised bed coating together with the state of 'the art will be introduced.
Chapter 2 gives a short introduction to the theory of fluidisation and the necessary relationships between the gas velocity, particle diameter and the bed porosity to allow one to appreciate the problems involved in the application and the design of fluidised systems. This chapter ends with some general remarks on the present state of fluidisation the-ory.
In chapter 3 an attempt has been made to obtain some in-formation over the particle motion in a gas-solid fluidised bed. The results indicate that under homogeneous fluidisation conditions there is no observable particle motion in times comparable with the coating process time. At fluidisation velocities of 1.U times the minimum fluidisation velocity and about ten particle diameters above the porous plate, the particles begin to oscillate around their fixed positions and the amplitude of oscillation is strongly dependent on the po-sition of the particle above the porous plate. The resiilts also indicate that the cavities formed as an outcome of par-ticle oscillations (i.e. porosity fluctuations) travel upwards in the bed and gain the maximum velocities equal to those pre-dicted for bubbles in fluidised beds. Furthermore the initial stage of the formation of bubbles (i.e. cavities) has been experimentally observed and semiquantitatively treated for
X
the first time. This chapter ends with the conclusion that in the coating process in a homogeneously fluidised bed the particle mechanics can not play an important role; an expla-nation for the mechanism of the process may be advanced as-suming that the bed is a homogeneous medium with effective physical properties.
In chapter k the effective physical properties of the homogeneously fluidised bed consisting of two representa-tive polymers (polystyrene and polyethylene) have been evalu-ated. The measured thermal diffusion coefficients are of the order of 10"^ m^/s. This agrees well with its predicted va-lues.
In chapter 5 the experimental results for the dip-coating of flat plates with polystyrene and polyethylene together with their interpretations are presented. The predictions made for the coating thickness by means of the theoretical relation-ships agree well with the experimental results. The theore-tical relationships have also been generalised so as to be applicable to the coating with all kinds of polymers; i.e. polymers with a sharp melting temperature and a definite heat of fusion as well as those without. The chapter ends with the conclusion that the theory of the non-stationary conduction of heat is sufficient to account for the mechanism of the coating process.
In chapter 6 we have extended the theoretical concepts presented in chapter 5 to the coating of continuously moving objects such as wires. Our own measurements as well as those of others give an effective viscosity of 10 poise for a homo-geneously fluidised bed of object-to-particle diameter ratio of 100. This bed does not mechanically impede the motion of objects in it. When the reported viscosity is combined with the very low thermal diffusion coefficient reported in chap-ter U, one obtains a large, effective Prandtl number. Thus the concept of a quasi-fluid vith a large Prandtl number can be used together with the theory developed in chapter 5 to predict the coating thicknesses obtainable on continuously moving wires. In this chapter the relations derived for the
coating thicknesses differ from the ones in chapter 5 only with respect to the correction factors introduced to take in-to account the large curvatiore of thin wires. Again the ex-perimental results agree well with their predicted counter-parts.
In chapter 7 an initial attempt is made to cure the coating in an oven after the fluidised bed process. An
ap-XL
proach based on a more general theory of sintering of powders indicates that the c\aring time is a function of the surface tension and the viscosity of the molten polymer. The relation-ship derived between the cinre-time, the porosity of the coating, the viscosity and the surface tension satisfactorily explains our experimental findings of the c\aring stage as well as the observations of other research workers. The chapter concludes this research with some preliminary considerations of optimum conditions under which a fluidised bed should be operated to obtain higher production rates and to reduce un-wanted heat losses.
§1.1 1
Chapter 1 INTRODUCTION
1.1 The Problem of Fluidised-Bed Coating
Fluidised bed coating is one of the recent innovations in the coatings field. It is a rapid, convenient and econom-ic method for applying a solventless (100 percent solids) coating on objects of metal, glass, ceramics, plastics and other materials capable of withstanding the heating tempera-tures involved in the process. It also makes possible the utilisation of insoluble resins that can not be readily applied as a coating by ordinary methods. The technique is particularly suitable for coating small objects and applying thick, uniform coating even to complex or undercut shapes. The method is applicable both to clear and pigmented coatings.
In the fluidised-bed coating, the coating material in the form of a fine (20 - 200 y) dry powder is fluidised by suspending the coating particles in a controlled upstream of the gas (Fig. 1-1), usually compressed air or, in the case of inflammable powders, nitrogen. This is accomplished in a vessel that is divided into an upper and lower chamber by a porous plate that retains the powder while permitting the passage of the gas. The coating particles must not be too
irregular in shape in order to obtain a homogeneous fluidisa-tion to avoid agglomerafluidisa-tions. Both powder and gas should be dry to ensure the mobility and the free flowing of the pow-der. The generation of ribo-electricity due to relative mo-tion between the gas and the particles sometimes does prevent unwanted agglomeration. However, such an effect does not seem to be reproducible in practice. The use of a vibro-fluidised bed can materially increase the fluidisation quality, thus permitting free flow of suspended particles to enhance the uniform coating of irregular shaped objects.
In the process of dip-aoating , the articles to be coated are preheated to a temperature above the melting temperature of the resin and dipped into the fluidised bed, where it acquires a uniform coating of the resin material as the particles melt onto it. Upon withdrawal, the coating congeals into a more or less continuous film which may be
2 § 1 . 1 ,
r^
1 t 1 I fs : //y^otiitc\ ^ o b j » c t t o be coated - — o i r ous plate or N j Figure 1-1 A typical fluidised-bed set-up for the coating processfurther annealed in an oven to ensure gooa levelling and a pinhole-free coating and to com-plete the curing of the resin. In the continuous coating pro-cess, i.e. the coating of a wire for insulation purposes, the wire is heated either in an oven before it enters the flui-dised bed or by means of a high-frequency induction coil in the bed. In view of the small heat capacity of wire, in general, the latter method is used to reduce the unwanted heat losses between the oven and the bed. As in the case of dip-coating the wire can be afterwards passed through an annealing oven to cure the resin.
The proposed and actual applications of fluidised bed coating are numerous. The followings are representative applications. Certainly there are many more. Transformer parts are now being coated by this technique and: valves, refrigerator parts, refinery equipment, impellers, pump parts, wire screens, washing machine parts, rollers, conveyors, hand tools, etc.
From the above it must not be concluded that the whole process of coating is a simple one. So far, the studies made in this field consist of obtaining empirical relations be-tween various parameters, such as the rate of coating, the
fluidising gas velocity, the resin properties etc.; they do not shed light on the mechanism of fluidised-bed coating. As yet there is no general theory available in predicting the rate of coating in order to obtain optimum conditions of the process in terms of the physical and thermal proper-ties of both the substrate and the coating material.
In order to endeavour to find answers to the many
questions raised above and to give a logical account of the process this research was imdertaken. The availability of a model allows one the possibility of manipulating the process at will. In the following pages we intend to give a methodic study of the subject matter and consider the applications of some of the findings to industrial applications as well as to compare, in the light of the model found in this research, the pros and cons of fLuidised-bed coating technique with other well-established methods of coating.
§1.2
3
1.2 The History and the Present State of Arts
An interesting discussion of fluidisation (1) indicates that the fluid bed processing was described as early as 1956 as a device for purifying ores. In more modern times it was utilised in Germany shortly after World War I for producing water gas i.e. a mixture of hydrogen and carbon monoxide -by forcing air and steam through a fluid bed of brown coal. Some thirty years ago work started at the Massachusett's Institute of Technology aimed at utilising fluid catalysts beds to facilitate cracking of petroleum fractions. The petroleum industry now utilizes fluidised-bed of catalysts widely for the so-called catalytic reforming operation. The metallurgical and food processing industries greatly use the technique of fluidised-bed at one stage or another of their production plants. A great deal of research is \mder way in many areas of the chemical processing industry to apply flu-idisation to good advantage wherever contact of solids and fluids are necessary. From the above discussion it is seen that the concept of the fluidised-bed of fine solid particles was initiated in the chemical processing industries in order to bring a solid into intimate contact with a gas or liquid. By fluidising the solid, the maximum amount of surface area was presented to the gas. Since the very small particles were in vigorous motion, maximum contact and mixing between the gas and the solid could be achieved.
Although in the examination of the literatiire, it is interesting to note that the idea of using a gaseous disper-sion of particles as a coating medium was conceived over seventy years ago (2), nevertheless its description as an industrial process was first done by Gemmev (3) in 1957. How-ever there is some disagreement as to who began to coat ob-jects in fluidised-beds. According to at least one reference
(4), the use of thermoplastic fluidised-bed coatings was developed in 19^0 in the laboratories of the Telegraph Con-struction and Maintenance Company in England who devised a method for producing powdered polyethylene. This powder commercially became available in 19^6 and has been since utilised in coating of various objects. Since then numerous companies, chiefly in the United States, have helped to pioneer the fluidised-bed process of coating.
However, contrary to its applications, the mechanism of coating process has not been cleared up. Earliest work on the mechanism of the process has been done by Gaynor (5), who has
k
§1.2presented an excellent discussion and survey of the coating process. However, in this discussion and the numerous reviews that followed, one only finds the description of the process and not the mechanistic explanation of it. Later workers, chiefly van der Hoeven (6) has attacked the problem in a dif-ferent style and produced a large amount of systematic data such as coating thickness, gas velocity etc. Yet the above study did not answer the principle question, also posed by the users of the process: what is the mechanism of coating?
Since the work of van der Hoeven one does not encounter further fundamental research into the mechanism of the pro-cess. Recently Landrock {?) reviewed the research in the field and the state of arts in a useful monograph.
1.3 Set-up of the Research
In the previous sections we have given a short account of the present state of the fluidised bed coating. There we have seen that tne worlt so far done in predicting the rate of coating process is far from satisfactory. Many of the workers in the field have tried many types of polymers and fluidised bed conditions (.not infrequently the regimes of flu-idisation with various operational parameters hardly used in the process) to try to gain insight into the fluidised bed coating.
This research was also undertaken in order to find a mechanism for the coating process. However we shall in the following pages take a different line. Obviously, there are many interacting parameters involved in the process. It would be only natural to try to single out these parameters and
con-sider them with respect to their influences on the process. Following this train of thought, we have chosen only two sorts of polymers. One with a definite heat of fusion and the other with a negligible heat of fusion. In this way it was possible to obtain a general formulation to account for the effects of the varying thermal properties of the polymers on the coating process. Further we have tried to use homogeneous fluidisation conditions. Again, in this way it was possible to calculate the thermal properties of the fluidised bed on the basis of a simple model, i.e. that of a loosely packed bed.
The set-up of the research is as follows:
In chapter 2 we have tried to give some general background information and the problems presented in any application of the fluidised bed. The chapter ends with the review of some
§1.3
5
of the recent developments in fluidisation. In chapter 3 we examine the motion of particles, for different gas flow rates, by means of an X-ray technique. Chapter h begins with consider-ation of the most important parameters affecting the coating process. In this chapter, the effective thermal diffusivity and conductivity of the bed used for the coating process have been measured and compared with their calculated values based upon the consideration of some simple models, such as loosely packed bed systems. In chapter 5» the dip-coating measurements are reported and the experimental results are compared with the theoretical predictions. The established model accounting for the mechanism of the coating is tested for various object forms and boundary conditions. In chapter 6 the findings of the last chapter are applied to the coating of continuously moving axially symmetrical bodies. In this way a unified model explaining the mechanism of the dip-coating and continu-ous coating will be established. Chapter 7 ends this research, aimed at a basic approach to the fluidised bed coating, with some preliminary investigations of post heating (or curing) of the coating layer, as well as some remarks on optimum operational conditions under which the fluidised bed coating shoiild be performed in order to make the applications of the process an economically viable proposition with respect to the already established methods of coating.
§2.1 7
Chapter 2 FLUIDISATION
2.1- The Uses of Fluidised Systems
If a bed of solid particles is supported on a horizontal gauze in a vertical tube and gas or liquid is then forced to flow upwards thro\igh the gauze, and so through the particle bed, a pressure drop across the bed occ\irs. When this pres-sure drop is sufficient to support the weight of the particles in the bed (and the dry friction between the particles), then the bed is said to be "incipiently fluidised". Further in-crease in the fluid velocity causes the bed to expand to accom-modate the increase. This state of suspended particles in a gas stream thus formed has many of the characteristics of a liquid; its upper surface remains horizontal when tilted, it hardly impedes the movement of objects immersed into it, but it has no surface tension (unlike a liquid). Thus a fluidised system has a number of highly useful properties, the most im-portant being concerned with temperature control, heat trans-fer and continuity of operation. Due to the fact that a flu-idised system does not impede the movement of objects, it can also be used as a device for solventless dry painting or coat-ing, which is the subject of this research.
It would be constructive to outline briefly some of the useful properties of a fluidised system as well as some of its limitations in chemical processing.
i) Because of high degree of mixing there is practically no global temperatvire gradients in the bed. Thus it is most useful for catalytic reactions where close temperature control is required. Since the heat transfer coefficient is high, fluidised beds can also be used as constant temperature baths.
ii) Because in a fluidised system the suspension of solid particles behaves like a pseudo-liquid, it can be used to design continuous processes. Thus greatly enabling one to add or remove solids from the process equipment. Hence, coating, which is the subject of this research, can be regarded as the removal of the solid particles
6
§2.1from the fluidised system by a desirable surface process. iii) Because a fluidised system is one of the best systems
known to bring a gas into intimate contact with a solid, it is therefore well suited to catalytic and physical-mechanical processes.
However in spite of many possibilities in employing flu-idised systems in chemical and physical processes, neverthe-less there are considerable difficulties. These difficulties arise either from the lack of knowledge of the behaviour of such systems under varying process conditions or from certain limitations, such as the ones listed below, which are inherent to the processes themselves. There can be many more reasons for the inadequacy of fluidised systems. It will suffice here to list a few:
i) Certain chemical reactions proceed along a reaction path where a temperature gradient is prevalent. Thus the quick equilibration of temperature in a fluidised system is not suitable.
ii) The fluidisability of catalyst particles is essential for a good fluidised system.
iii) The inherent instability of gas-solid fluidised systems are such that the generation of bubbles or lack of them can create problems of efficiency as mixing or reactor devices. Mechanical forces generated by bubbles can give rise to serious problems, i.e. in fluidised bed drying of paper.
From these general considerations of the pros and cons of the fluidised bed technique it is not surprising to see that the fluidised beds have foiind their ways into the realm of coating and dry-painting of objects for the purpose of insula-tion or protecinsula-tion against their environment. In the following
sections we proceed with some basic aspects of fluidisation as well as its recent developments. In this way it is hoped that the magnitude of the problems encountered in fluidised bed coating would be more easily appreciated.
2.2 Basic Definitions in Fluidised Systems
As the subject of this research is the coating process in a gas-solid fluidised bed, we shall, by fluidised bed, always mean a gas-solid system, unless otherwise indicated.
In order to be able to follow the arguments to be pre-sented in the course of this work, it would be instructive
§ 2 . 2 9 < M 2 -0 4 -0 «mf o,« experiments extrapolated ( d , •0.4 mm)
• i " l[gmcni ] o,« experiments
. p , , ( d , . 0 4 m m )
mki. Hiüd.. Umi
-Ug [ c m / , ]
Figure 2-1 The pressure drop and the porosity as functions of superficial gas velocity
to define certain general terms used in fluidisation. In Fig. 2-1, there is a typical result for a gas fluidised bed; the pressure drop across the bed and its porosity are plotted versus the superficial gas velocity. There is a marked hyster-esis, so that for the slowly increasing flow the curves i and for the slowly decreasing flow the curves ii are generated. This is due to the wedging action between the particles. An increase in the gas velocity frees the particles and the point K is reached. At this stage the pressure drop becomes just enough to support the weight of the bed and this point K is usually called the point of "incipient fluidisation"; the superficial gas velocity being Uj^f and the voidage frac-tion, Einf. If the gas velocity is further increased above Emf then some of the gas may pass through the bed as bubbles. This regime is known as "aggregative fluidisation" and com-monly occurs when solids are fluidised by gases. In the case of liquid-solid fluidised systems, where p = p]_, the in-crease of liquid velocity above U^f does not materially change the appearance of the fluidisation. This is known as "particulate fluidisation". Although aggregative fluidisation is normally found with gas-solid systems and particvilate be-haviour with liquid-solid systems, there are exceptions to this generalisation. For example Kwauk and Wilhelm (8) found that a bed of lead shot when fluidised by water behaved
ag-10 §2.2
gregatively, i.e. water bubbles were present, and we have observed particulate fluidisation when light resin particles were fluidised with air under atmospheric pressure. It should be emphasized that a proper design of the bed can have much influence on the fluidisation behaviour of the system. Thus the proper selection of the bed height-to-bed diameter ratio, the distributor pore size and interpore distance-to-,particle diameter ratio, the pressure drop across the distributor and the particle shape can do more to enhance a good fluidisation regime. The particle size, its distribution and shape is one of the most important parameters in designing a fluidised bed. One should distinguish between the particles larger than
1.5 nun and those of a size smaller than 1 mm. In the latter case it is very probable to obtain, under considerable gas velocity variations, a reasonably well fluidised bed, while in the former case the mode of fluidisation is such that the regime of the two-phase system can be better described as a "teeter-bed" than as a fluidised-bed.
2.3 Mechanics of Fluidisation
As the considerable development and the vast applica-tions of the fluidisation technique are achieved, further advancement is possible only when a more fundamental under-standing of the fluidised system is reached. Although numerous studies on the mechanics of fluidised-beds have been made there is no generalised method to predict the fluidisation behaviour of the system once fluidised under varying experi-mental conditions.
However, for the sake of the completeness of the subject matter, here we shall give generally accepted semi-quantita-tive relations of the basic parameters defined in the pre-vious section; i.e. the incipient fluidisation velocity, ^^ft
the pressure drop through the bed, Ap and so on.
When a spherical particle is allowed to travel under an external force in a viscous medium there eire three main forces acting on it: the external force, in our case, the gravity, the buoyance force and the drag force due to the surrounding medium. When these forces are in equilibrium particles move with a constant speed; in case of the fall of small particles under gravity through the fluid-medium, this velocity is defined as the terminal velocity, U , and is given by:
§2.3 11
3ïïiiU d = ( p ^ - p )gTrd3/6 , ( 2 - 1 )
o s S g 5
where the left hand side gives the drag force given by Stokes law and the right hand side is the effective gravity force. Thus we have:
U+ = (p= - P )gd2/l8M . (2-2) t s g s
The equation (2-2) is valid, if the Reynolds nimiber,
(p^U^dg/y), is less than 1 and, if so, it is also applicable to the case when one particle is stationary, while the fluid
flows upwards over it with velocity U-^. Thus as a first ap-proximation in the absence of interparticle forces in a multi-particle system, U^ can be taken very roughly as being equal to minimum fluidisation velocity, ^-^f. However in practice particles do influence each other, and the prediction of Uj^f
is complicated further by the uncertainty in the values of voidage ^mf, at minimimi fluidisation conditions. Naturally there is no doubt the best way to determine Uj^f is to measure it. The method is to measure the pressure drop through the bed of particles as a function of the slowly increasing and then slowly decreasing gas flow rates. The results give
curves of the kind shown in Fig. 2-1, and Umf is the velocity at the point K, though this point is not always well defined. Nevertheless, it is useful to be able to estimate U^f from first principles, both for design purposes, and because the plant conditions may be difficult to simulate in the labora-tory.
We, first, briefly describe the general method used in predicting Lbf from the pressure drop considerations. At the incipient fluidisation the pressure drop is given by:
Ap/h = (1 - ej^f)(p3 - Pg)g . (2-3) Since, the bed at minimum fluidisation can be regarded as a
loosely packed bed, then the pressure drop can be approximated by Ergim's equation:
Ap/h = ,50 . - Ü ^ Ü . ^ • 1.75 J ^ % .
i2.k)
8 S
From the last two equations we obtain:
12 §2.3
71
• liquid fluidisation r -i y o gas ['•«'•9] , / ^ ^ Remt(experimentaO where Ga (= —^ (Pg-P )g'is' •) a n d -) are the 1Ö' lö' Figiire 2-2 Comparison of Reiiif reported in the lit-erature (9) and calculated from equation (2-5)ml p
Galileo and Reynolds numbers respectively. In case of non spherical particles it is ne-cessary to include a shape factor. The equation (2-5) is a dimensionless equation for the minimum fluidisation velocity based on the pressiire drop consideration and should be applicable for Remf up to 3000. In Fig.' 2-2 some measured values of Rej^f and those cal-culated from equation (2-5) are compared. As is observed, the theoretical and empirical results match well.
2.U Recent Developments in Fluidisation
About four years ago Botterill (W) and Reuter (11)
independently and at the same time reviewed over two hundred papers on the subject. Since then there has been no slacken-ing of interest in this field as Rowe (12) noted in his opening address at the international symposium held at Eind-hoven ( 1967). Indeed, at this meeting alone, there were 58 original contributions. It is not, however, surprising that interest in fluidisation should be sustained at such a level. In the first place, it offers much potential promise for the solution of processing problems where a solid phase has to be handled. Nevertheless, applications are still, in large part, limited by problems of mechanical nature: distributor design, solids loss from the bed by elutriation, particle size distribution control when growth or breakage occurs, particle/particle interaction forces and the like.
Theoretical studies have progressed but there is still no adequate theory that can describe how bubbles are formed in a gas fluidised bed. It was Davidson (13) who first made a useful approach to bubble phenomenon from an engineering viewpoint in I96I. He adopted a continuous approach; in this
§2.i+ 13
model potential flows for both the fluid and particulate phases and a fully developed circular bubble completely de-void of particles were assumed. Notable contributions to a more rigorous treatment of fluidised bed behaviour have been made by Murray (14) and Molerus (15), but it is still not possible to develop a model which predicts bubble fonna-tion and growth in gas fluidised systems. Murray's model based on equations for mass and momentum conservation de-scribes many features of stable bubble behaviour that have been studied by Rowe, et al. (16) in particular. Murray's
description involves a drag coefficient which could be re-lated to bed viscosity. First extensive studies of effective viscosity were done by Schugerl (17). Recent measurements of effective bed viscosity using a torsion pendulum are
re-ported by Hagyard and Saaerdote (18).
Molerus gave a solution in one dimension based on Hinze's
equations (19) for the conservation of mass and momentum for dispersions which predicts that there will be certain stable and unstable regimes for both gas and liquid fluidised systems. He also showed that a consequence of these equations was that impressed fluid vibrations should have a stabilising effect. In other studies of bed stability Pigford and Baron (20)
reached the conclusion that the bed will always be unstable and Chappelear (21) tested their treatment further to see if particle inertia would tend to stabilise perturbations of short wavelength but, with their model, concluded that they would not. Hiby (22) showed theoretically and practically that spontaneous periodic vertical fluctuations develop with-in shallow gas fluidised beds and are present at the base of deeper beds. Ruakenstein and Muntean (23) developed a model describing the way bubbles could grow at the distributor until the buoyancy effect acting on the bubbles overcomes opposing growth and inertial forces when they will rise. The periodic motion of particles and the generation of initial bubbles have much in common. In chapter 3 of this work we give some of our own measurements of periodic particle motion with re-spect to coating process, which were made by an X-ray tech-nique. Besides Murray's theoretical treatment of stable
bubbling behaviour Lockett and Harrison (24) measured the
voidage variation close to a bubble. Gabor (25) reported the effect of the constraining column walls on particle movement and Collins (26) on how the residence time of some gas ele-ments may be increased as bubbles rise within a small column. The disturbance to fluidisation caused by an immersed body
l i t §2.1+
within a gas-fluidised bed has been briefly described. In liquid fluidised systems the behaviour of the bed is much simpler, although small scale disturbances, i.e.
stratification, may still occur. Carlos and Richardson (27) determined the particle speed distribution. Also our own measurements of porosity fluctuations in solid-liquid systems are not conclusive. Other measurements of local porosity have been made by Coeuret and Le Goff (28). Comment has been made on the tendency for preferred low resistance paths to develop and for voidage fluctuations to exhibit a Gaussian distribu-tion.
There is now much evidence that particles within the con-tinuous phase of gas fluidised beds do touch each other even if only intermittently. Many processes use beds of electri-cally conducting particles for direct resistance heating (29).
Large current densities obtained in such processes warrant some sort of direct contact between particles.
That fluidised beds have an "apparent viscosity" is some-times agreed upon. Our own measurements of bed viscosity
support the value of about 10 poises reported in the litera-ture.
The natiire and the dynamics of fluidised beds have large influence on the heat and mass transfer. The difference in heat transfer behaviour between gas and liquid-fluidised beds lies in the fact that solid/fluid thermal properties are more nearly similar in liquid systems than in gaseous systems. For the gaseous system, the fluid only has very small heat capacity, so it is the circulating particles that have the capacity to transfer heat. Botterill (30) and Ziegler (31)
studied the residence time of particles on a surface intro-duced into the bed, as well as the relative soUd/gas thermal
properties. Gorelik (32)^ Borodulya (33) and Tamarin (34) have measured effective bed thermal diffusivities. Unfortu-nately they did not measure such diffusivities at minimum fluidisation so that it is not possible to compare our own measurements reported in chapter 1+ of this work. Heat trans-fer measurements in liquid fluidised systems have been per-formed by Wasmund and Smith (35) and many others. In these studies, the authors did not report systematic circulations in the bed; their measured heat transfer coefficients dis-play a maximum in the porosity range 0.6 - 0.7. The same maximum is observed, in the same range of porosities, in the mass transfer coefficient studies of Le Goff.
§2.1+ 15
receive much attention either because the users of the pro-cess were alien to the field of fluidisation or because of the complicated nature of the process alienated those who tried to formulate a mechanism for it. Instead a large amount of work has been devoted to powder formulation, physical properties of resins and description of the apparatus. These and other aspects of fluidised bed coating are reviewed by Pettigrew (36)^ Sherwood (37) and Landrock (38). As far as
the author knows our work is the only available study aimed at the mechanism of coating. Thus the rest of the work is concerned with the study of the process.
§3.1 17 Chapter 3
PARTICLE MOTION IN A FLUIDISED BED
3.1 Introduction
The fluidised bed is a multiphase system involving the interaction of fluids and solid particles. The motion of particles and their concentration, the flow rate of the gas and with it the formation of bubbles or their coalescence and break-up have important bearings on transport processes taking place in the bed.
Starting with these considerations, we intend to examine in this chapter the particle motion in the bed. These prelim-inary studies were found necessary, for: the coating process involves heat transfer from an object in the bed to the poly-mer particles, the eventual adherence and melting of the particles on the object surface and the final formation of a polymeric coating layer. Naturally, the motion of the parti-cles during this heat transfer and melting process may great-ly affect the progress of the coating mechanism. Thus it would be of interest to measure particle motion.
3.2 Particle Motion in the Bed In this section we in-tend to look at the micro-scopic properties of the par-ticulate phase of our system i.e. the particle motion in the fluidised bed. To accom-plish this we resorted to a X-ray technique. The method consists of filming the image of tracer particles formed on a fluorescent screen.
fluidisation section
The experimental appa-ratus is shown in Fig. 3-1 and consists of a cylindri-cal perspex tube of 5.55 cm internal diameter separated by a porous plate into a
lead glass
Fig. 3-1 The fluidised bed and the experimental arrangement
18 §3.2 - e.oo cm 5.55 cm - .^-Imoit, X - r a y bundle Sid* vlaw "^fluorascant ! j ^ scroan 1 J — r 1 1 h«lght of tha bad. h [cm]
....I
.•" .-0 .. •». y.„...•*. 0 0 0 : * " ^ - - ^ - " ^ - " * - . - . - . . - . A - . . x . o ^ . •Ó* 0 0 00
0
- J É 1 _ - ^ number of frames [ - ] 4 0Fig. 3-2 The cross-section view of the bed as projected on the fluorescent screen
Fig. 3-3 The positions of tracer particles in the bed; "O" indicates the position of the cavity centre (1 frame = 1/29 sec; Ug= 1.1+ Uj^f). calming and a fluidisation section. The fluidisation section was placed between a fluorescent screen and the X-ray appara-tus of low energy ('^ 15 kW). As a protection against radiation the bed was surrounded by a lead-box with two windows, one allowing the radiation to enter while the other was sealed with the fluorescent screen on which the image of the moving particles could be followed. The particles used were of poly-ethylene with diameters of 1.0 + 0.10 mm and specific mass of 960 kg/m^. In order to minimize the electrostatic charges be-ing generated while the fluidisation is in progress, the par-ticles were coated with a very thin layer of graphite powder.
The tracers were Al203-particles with the same diameter and density as the polyethylene particles. In order to make them opaque against X-rays they were painted with colloidal
silver paint. As the deposited silver layer was very thin, the physical properties of the tracer particles after this opaque-ing process were not changed. The air flow rate was measured by a rotameter and controlled by a needle valve. Other experi-mental arrangements are seen in Fig. 3-2. The cinefilm
§3.2
19
Plate 3-1 The X-ray image of the bed and the tracer particle as seen on the fluorescent screen
(500 ASA) with 90 x l6 mm frames and speed, of 29 frames sec"'^ was used to record the image of the tracer particles formed on the fluorescent screen.
3.3 The Results and their Interpretations
Into the fluidisation section 71^.6 gm of polyethylene particles were put; the height of the bed proved to be 5 cm, giving a packed bed porosity of about 0.1+0. The minimum flu-idisation velocity observed agreed well with its calculated value of 30 cm sec"-' given by equation (2-5).
A typical X-ray image of the bed with several tracer particles taken from the film sequence is shown in plate 3-1. At minimum fluidisation the bed was homogeneous and no
ob-servable motion of the tracers co\ild be seen. Upon varying the gas flow rate above that of Umf the bed still persisted
in its homogeneity until a flow rate of ^ .h U^f was reached. At this gas velocity the tracer particle motion as well as periodically rising cavities (voids), formed about 1 cm above the porous plate, could be observed. The motion of the tracer particles was strongly coupled with the formation and the po-sition of these cavities.
20 §3.3
^ height in the bed. h[cm] 1 I
Fig. 3-1+ The magnitude of the amplitude of the particle oscillation as a function of the vertical position of the tracer particle in the bed (Ug= 1.1+ Unif)
Fig. 3-5 The velocity of a cavity as it moves up the bed
(x, direct measurements, o, obtained from figure 3-3, calculated using the relation 0 . 7 Y ^ ; U„= 1.1+ Ujnf)
frame by frame. For the fluidisation velocity up to 1.1+ U^f the tracer particles were stationary. Above this critical ve-locity the position of the tracer particles as function of frame number (1 frame = 1/29 second) could be directly mea-sured. The results are shown in Fig. 3-3. In the same figure is plotted the motion of the cavity centres. From the figure we see that the particles do not migrate through the bed but they oscillate around their fixed positions in the bed; the maximum deviation from their positions occurs when the cavity passes. Further analysis of the amplitude of the oscillation of the particles and the motion of the cavities are shown in figures 3-1+ and 3-5- The velocity of the cavity seems to in-crease as it ascends in the bed. Leung and Harrison (39), for the velocity of a gas void rising through the fluidised bed, gave 0.7 /gR, where R is the radius of curvature of the nose of the void. It will be supposed now, that the amplitude, A, of the tracer particle motion is proportional to the radi-us, R, of the cavity. From the figure 3-3 we see that the re-corded values of the amplitude of the oscillating particles are a function of h, the height in the bed: this is analysed separately in figure 3-1+. So, the radius of the cavity will
§3.3 21
also be a function of its height in the bed. Since we supposed that R is proportional to the amplitude, say R = y^A, the re-lation given by Harrison and Lockett applied to our situation becomes:
V^a,r = 0.7 Y ^2gA(h) . (3-1)
C o . V
The values of V^^Y thus calculated with the help of the equa-tion (3-1) are compared with the direct measiirements by ana-lysing the cinefilm of figure 3-3. The predicted values of ^cav ^^s i^ fair agreement with the actual observations of Vcavj if Y is taken to be 2.
Our observation of the formation of cavities arising from fluctuations in the porosity and the vibrationary particle motions are in qualitative agreement with the findings of Hiby
et al. (40).
3.1+ Conclusions
The aim of the study presented in this chapter was to look into the particle motion in the gas-solid fluidised bed. As was mentioned in the introduction of this chapter and also forseeing the results of the next chapters the mode of the particle motion directly influences the heat transport mechanism between an object and the fluidised bed. As the
economics of fluidised bed coating indicate that heat losses to the outside should be minimised (see also the next chapter). the homogeneous fluidisation regime will be used in coating processes. With respect to this, the conclusion of the work presented in this chapter can be summarised as follows:
i) Under homogeneous fluidisation conditions there is no migration of the particles, at least not in a time com-parable with the coating-time which is, in general, of the order of a few seconds, and
ii) up to Ug= 1.1+ Uuif , the particles only vibrate arotmd their fixed positions in the bed; consequently, the heat transfer process from the object to be coated to the bed can be examined further with the justified assimiption that there is no convection of heat due to particle motion.
§ 1 + . 1 23
Chapter 1+
THERMAL PROPERTIES OF THE FLUIDISED BED
1+. 1 Introduction
An object heated to temperatures higher than the melting temperature of the fluidised polymer powder, when immersed into it, will cause the melting of the powder, thus forming a coating layer. It is obvious that the main process path is the transport of the heat from the object into the bed. Before attempting to look into the mechanism of coating it was
thought necessary to try to evaluate the thermal character-istics of the bed used for the actual coating process measure-ments. Therefore, in this chapter we intend to evaluate the thermal properties of the bed by measuring the heat transfer rate from the article to be coated into the bed. This is done by measuring the heat transfer coefficient for varying flu-idising gas flow rates and varying residence times of the article in the bed. By interpreting the heat transfer coeffi-cients, we are able to obtain the information on thermal con-ductivity, the thermal diffusivity and the volumetric heat capacity of the bed. The measured thermal properties of the fluidised bed are compared with the ones directly calculated from the known values of the thermal properties of the gas and the polymer resin under fluidisation conditions. 1+.2 Factors Affecting the Mechanism of Coating
As in every process in coating too there are overriding parameters that have great influence on the mechanism of the process. It is our aim now to predict these main parameters and design experimental procedures to single them out for further analysis of the process.
The coating itself is a result of the melting of poly-meric powder (fluidised by means of air or nitrogen),
contact-ing the surface of the object to be coated. Thus, basically, the main parameters, affecting the overall thickness of coating, are the thermal transport characteristics of the
system consisting of the article and the bed. These character-istics are heat capacities, thermal diffusivities and
conduc-2l+ §l+.2
thermal properties of the article
heat supplied / geometrical properties by the object v of the article
thermal properties of the coating \ growth r a t e of coating heat absorbed by the coating thermal p r o p e r t i e s of the polymer average temperature d i s t r i b u t i o n in t h e coating t h i c k n e s s of coating progress of the growth rate
heat loss (or heat stored) into the bed
heat content of the — article / \ thermal properties of the bed the the heat bed content of influence of fluidi-sation on the thermal properties of the bed
thermal and physical properties of the article
final temperature of the article at the end of the coating process
Table 1+-1 Summary of the various parameters likely to influ-ence the thickness of coating
tivities. Other physical properties, such as densities, geo-metrical shapes of both the article and the polymer powder should also be included into the mechanistic model of the process sought. Thus we can summarise the main parameters in the following way:
thermal and geometrical characteristics of the object to be coated,
Fig. l+-1a The thermal prop-^ erties of the polystyrene particles used
Fig. l+-1b The thermal prop-erties of the polyethylene particles used
- thermal and physical properties of the polymeric powder used as coating,
- overall state of fluidised bed, i.e. its effective ther-mal and physical properties.
It is seen that the main factors affecting the overall coating process are the heat transfer characteristics of the fluidised bed, the polymer and the article. We have also tabulated these in table 1+-1. Before proceeding further it would be constructive to mention some of the important ther-mal characteristics of polymers so far as they are of rele-vance to subject matter under study.
In general polymers do not have sharp melting tempera-tures. In the choice of model polymers to be used in our coating experiments this fact was taken into account. Two different polymers were chosen: one with a not well-defined melting temperature: polystyrene, and the other with a
reasonably well-defined melting temperature: low density poly-ethylene. The physical and the thermal properties of both of these polymers are shown in figures l+-1a and l+-1b. The polysty-rene particles were spherical in shape and their size distribu-tion is given in table 1+-2. Whereas the maximimi-particle size for polyethylene was of the order of 10"^ m and furthermore the particle shape was irregular.
Another important parameter which will be used in the description of the coating process is the "sticking tempera-ture". Tg; it is defined as the temperature of the polymer particles at which they begin to fuse on to the surface of the articles being coated. In case of polymers with
well-26
§l+.2^sE^]
> 5 X lO""* 1+.2 - 5 X 10-'+ 2 . 9 - 1+.2 X 10"'* 2 . 5 - 2 . 9 X 10-'+ < 2 . 5 X 10-'* w e i g h t % 30 53 11+ 3 0defined melting temperature Tg is the melting tempera-ture itself. For example, in the case of polyethy-lene: Tg is 105 C, which is its melting temperature. However, in case of poly-mers without a sharp melt-ing temperature, Tg may be determined as follows: the object to be coated is Table 1+-2 Particle size heated to different temper-distribution of polystyrene atures and dipped into the fluidised bed. The minimum object temperature at which the particles begin to stick to its surface can be taken as the sticking temperature, Tg, of the polymer. For polystyrene Tg was found to be lUO C.
In the following sections we attempt to design experi-ments to measure and to calculate the physical and thermal properties of the fluidised bed of coating polymer. On evalu-ation of these properties we proceed with the coating measure-ments and compare the experimental results thus obtained with the predictions of our mechanistic model of the process. 1+.3 Experimental Apparatus and Procedure
The main part of the apparatus was a fluidised bed and with it were associated a gas heater, control valves and a dryer unit for the gas (see Fig. 1+-2). The bed itself was a perspex tube of 20 cm diameter, at the bottom separated from the "calming" section by a porous plate. As was mentioned in previous chapters the influence of porous plate on the mode of fluidisation is very large. Hence its choice as well as its positioning with respect to the column is very im-portant in the design and operation of any fluidised bed. The porous plate used was made of bronze and the pressure drop across it, with 180 litre/minute gas flow rate, was of the order of 2 atmospheres. Also as it is generally agreed the pressure drop across the porous plate should at least be equal to the weight of the bed, with the choice of the bed height-diameter ratios of about 2, the above empirical condition to obtain a stable fluidised bed was satisfied. Indeed upon varying the gas flow in the range 1 Uj^f-1.1+ Umf
§l+.3
27
3D
-Xt
1 2 3 4 Thcrmoflask Object heater Plate Fluidised bed 5 Air heater 6 Rotameter 7 Air d r y e r 1 1 1 ' ' » 6 ! ^ in-i 11
< 7 • ^ ;t
r 1 1 1 Li""'U
• . .Figure 1+-2 Experimental set-up
a homogeneous fluidisation regime could always be obtained. As we were, to begin with, interested in the mechanism of the process it was natural to use optimum conditions i.e. regularly shaped monodisperse particles, minimum fluidisation conditions and substrates with simple geometrical shapes. In table 1+-1 we have postulated that the process determining parameters are the heat transport characteristics of the bed and those of the objects. In the practical applications care should be taken to reduce the unwanted heat leakages. However, the heat transfer rate in a fluidised bed increases monoto-nically as the fluidising gas velocity increases. Thus it was obvious that in order to make the coating process an economi-cally attractive proposition, the minimum fluidisation regime shoxold be used. In this way not only was it possible to di-minish the unnecessary heat losses to the bed, but also the process rendered itself to analysis. And furthermore, under minimum fluidisation conditions the bed does not impede the movement of the article being coated in it.
Dry air was used as fluidising medium. Upon incorpora-ting an air pressure regulaincorpora-ting valve with air inlet pipe
28 §l+.3 bed height [ m ] p o r o s i t y
[-J
a i r flow r a t e [ l i t r e / m i n ] a t 293°K s u p e r f i c i a l a i r v e l o c i t y i n t h e bed [m s e c - ^ J i n t e r s t i t i a l a i r v e l o c i t y fm s e c - ^ ] packed bed 20x10-2 0 . 3 7 m i n . f l u i d . 21x10-2 0.1+0 110 5 . 8 x 1 0 - 2 ll+.5x10-2 h o m . f l u i d . 2 1 . 7 x 1 0 - 2 0.1+2 131+ 7 . 1 x 1 0 - 2 17x10"2 l i m i t of h o m . f l u i d 22.7x10~2 0.1+5 l6o 8 . 5 x 1 0 - 2 19x10-2Table 1+-3 Various values of air flow rates, porosities and bed heights used during the experiments
the minor fluctuations in the flow of gas which may have been present in the pipe lines were compensated for. The range of fluidisation as well as bed expansions used are tabulated in table 4-3. The determination of the porosities v a s made by using the following definition of the fluidised bed bulk density* ; (1+-1) as <p> = (l-e)Pg + ep p << p g s then e 'V' 1 _ l£-> = p„
w
_
!
_
V p„ (1+-2)where W and V are the total mass of polystyrene in the bed and the volume of the fluidised bed under experimental con-ditions respectively.
The articles used to determine the thermal properties of the bed are shown in figure 1+-3, while their physical
§l+.3
29
properties are tabulated in table 1+-1+. The material of the articles used was copper. Fiirther, as the rate of heat transfer from the arti-cles into the bed woiild be dependent on the geometrical shape of the arti-cles, in order to unify the diverse geometrical shapes and sizes a new constant, C^, the "article constant" was defined: the ratio of the heat capacity of the article to the sur-face area of the article, i.e.
(pc V)„/A„. The temperature of the •p V V
articles was measured by means of a chromel-alumel thermocouple incor-porated with a compensator (see figure 1+-3). The residence or dip-time of the article in the fluidised bed was determined by a simple mechanism which
short-circuits the recorder when the article enters or leaves the bed, in this way the dip-time (residence t i m e ) , with less than 10 percent error in its measiirement, could be determined directly.
t o thermoflasK
thermocouple
QScrn
Fig. i+-3 Copper plates used for heat transfer measure-ments and coatings
article number
.r r 31 A r 21 r l ^ l rJouOel « (pcr.v)v r Joule 1 dnnens.^ V, [m^ A, [m^ ] p , [ ^ ] c J ^ ^ - ^ * - - ^ [ j , , ^ ]
1 6>'U«0.5 11.5«10-' se.SMO"* a.gé'iio' 386 T.I^IO'
2 6..U«0.3 T.'OltxlO-s 53.a3«10-'* 8.96x10' 386 U.57x10'
Table 1+-1+ The physical and thermal properties of the articles used in heat transfer and coating measurements
1+.1+ Interpretation of the Results
In chapter 3 we have observed the particle motion in the fluidisation range of 1 U^f - 1.1+ Umf. In the fluidisation regime of interest there was no appreciable motion of the particles during the times comparable with the dip-time en-countered in the actual coating process or in the heat trans-fer measurements; therefore we may assume that the fluidised
30 §1+.1+ ioo
'-m
TVoM a 270 « 370 o 420 A O o t[5]Figure 1+-1+ Measured heat transfer coefficient shown as fiinc-tion of residence time
bed behaves as if it were a stagnant, homogeneous medium. By defining a heat transfer coefficient a the heat losses into the surroiinding of the objects can be accounted for as follows:
The heat balance
V) ^ = gives: PC
p V dt A a(T V V ^b) (U-3)
or defining:
1
<a> = r ƒ a(t)dt , we obtain:
T - T, vo b
= exp ( <a>t ) . (1+-1+: As the thermal diffusivity of article material is very large,
it can be assumed that the article has a uniform temperature during the cooling and that this temperature is equal to its surface temperature, which then is measured by the thermocou-ple built in. Thus, from the measured temperature of the article, while cooling in the fluidised bed, and with the aid of equation (1+-1+), it is possible to calculate the heat transfer coefficient between the article and its surroundings. From the figure 1+-1+ we see the heat transfer coefficient cal-culated as a function of time. It is apparent that the heat transfer coefficient, a, changes with the residence time,t, as
§1+.1+ 31 article no 1 2 1 2 gas flow rate
r 1 1
.
^^^
J
110 131+ T vo 270°C ll+0 150 130 11+5 370°C 160 -150 -1+20°C 175 190 170 180 c, average _ 1 _ W seclm2°C^ J
150+20 170+20 160+20 165+20Table 1+-5 The values of constant c, for various Ty , initial object temperatures calculated from the equation
(^-5)
The values of c are tabulated in table 1+-5; they prove to be reasonably constant.
Equation (1+-5) suggests a non-stationary heat transfer p r o c e s s , associated with the diffusion of heat into the bed surrounding the object to be coated. In order to analyse this process it might be assumed that the structure of the coating layer and of the bed itself are not too different. Therefore, it will be attempted to interprete the equation (1+5) for a in terms of the heat transfer coefficient c a l -culated from the penetration theory, namely:
= \A
X2ïïat {h-6)
Comparing the equations (1+-5) and (1+-6) we see that if the penetration theory should be a good approximation to the heat transfer taking place., in our system then the., value of con-stant, c C w sec2/m2°C2j» must equal {X^/va.)^, where a and \
are the thermal properties of the fluidised bed. The value of x2/a (= TTc2) also determines the temperature history of the object as the following analysis w i l l show.
Assuming that the penetration theory can account for the transfer of the thermal energy into the b e d , we can calculate the temperature distribution in the bed as follows:
The transient temperature distribution in the bed satis-fies the equation:
32 §1+.1+
3T
3t = <a> 92T
with the following initial and boundary conditions t = 0 t > 0 T = T T = T, 8T ( P C V ) ^ - : ^ = p V dt a s X -*• «> 3T 8x X = 0
(M)
(1+-8) (1^-9) (^-10) The solution of equation (1+-7) is:^A>2 ^ . „, <x>^ C'^<a> V t) erfc <X>^ V
« c ^ * )
2(<a>t) 2(<a>t)'•}
(1^-11)where < >indicates the effective thermal properties of the bed and 6 = (T -T^)/(T„ - T ^ ) .
b ^o b
In order to be able to calculate e(x,t) from the equa-tion (1+-11) we have to know the values of <x2>/<a>. We can either, directly use the values obtained from the experiments and tabulated in the table 1+-6 or try to calculate them from semi-empirical relations for packed beds. Below we shall at-tempt, with a help of a simplified model, to predict the average value of <x2>/<a>for a homogeneously fluidised bed. This procedure is justified, for our fluidised bed under homo-geneous conditions (i.e. in the absence of bubbles) behaves as if it were a packed bed.
$ l / m i n l 110 131+ __^Vo 270 370 1+20 270 370 1+20
[°C
1 <x2>/<a> [w2sec/m'+°c2] 8.1+xio'+ 10.3x10'* 12.3x10"+ 7.8 10"+ 8.9 10"+ 11.3 10"+Table 1+-6 The measured values of <X^>/<a>
The effective density of the bed is given by the equation (1+-1) while the effective volumetric thermal capacity of the
§1+.1+
33
bed by definition is represented by: <pc > = (l-£)(pc ) + e(pc ) . P P s P S << (pc P s' (1+-12) the equation (1+-13) As (pc ) P g (1+-12) gives: <pc > = (l-e)(pc ) P P s
However, <X> has a rather complex fimctional form. In order to simplify the matter we can assume that it consists of two parts. One is the effective ther-mal conductivity, <XQ>of a packed bed with stagnant interstitial fluid and the other is the contri-bution due to the fluid flowing through the packing, <X,>. The former can be represented by:
<X > X T, T,>T, V7y solid > TTTTT, r A A A A A n "9- T T J T, T,>T, • < » o > o A A A r o A A A / ^ <^> Fig. 1+-5 A simplified model to calculate the effective thermal con-ductivity of a packed bed in the presence of a stationary inter-stitial fluid
(1+-11+)
where X and X are the heat conductivities of gas and solids respectively.
There are many ways of calculating the effective thermal conductivity of a packed bed with a stationary fluid occupy-ing the interstitial void. Recently Krupicka (41) gave an ana-lytical model for its calculation and compared it with experi-mental results available in the literature. However, his mathematical relations are cumbersome. Maxwell (42) in his treaty on "Electricity and Magnetism" gave a simple model to predict the thermal conductivity of a granular medium. His re-lation reads:
< V 3X^^1-e) + (2X + X^)e X 3X (l-e) + (2X + X )e '
g g g s
In the derivation of the above relationship it is assumed that the flow of heat from particle to particle by direct contact
is negligible. Consequently for ^r, ~ ^i "t^^ above relation pre-dicts <XQ> = 0, which, strictly speaking,can not be true. Therefore, we will present another simple but constructive model to calculate < X Q > .
31+ §1+.1+ < » o > -S7 curve E 1 0 3 2 0 4 3 Q 5 4 0 6 - — According to Maxwell — According to Eqn ( 4 - 1 7 ) / ' 1 1
f'
1 1 / / ^ / y^ / y^ y / y^ y^ -X y"^ -- '^ y^^''' y ^ ^ = -1 -1Consider two idealised different geometrical arrangements of the particles illustrated in Fig. 1+-5 and subjected to a tempera-ture gradient. In the same fig-ure we have also shown the elec-trical analogue for the two cases separately. For the case
(a), the following relationship is derived for the effective heat conductivity of the compo-sition:
<X'> = (X - X )(l-£) + X whereas for the case (b):
g (lt-15) <X"> = o Fig. 1+-6 Comparison of <Xo>/<Xg> according to Maxwell^s correlation and to the equation (I+-I7)
;x /x -1 K s 1-e + 1 (U-16) <X > o = m<X'> + n<X"> o
The actual packing arrangements of the particles is neither of the above cases but a combina-tion of them:
(U-17) where m and n are constants. For a random packing of isometric particles (i.e. of spheres and cubes it can be shown that m and n must have the values of 2/3 and 1/3 respectively. In fig-ure 1+-6 we have calculated and compared the values of <XQ> given by equation (1+-17) and by Maxwell's relation. For Xg/X„ < 10, the agreement between the two predictions on one hand and the reported experimental results on the other hand is fair. However, at large values of Xg/X the two predictions differ greatly and the reported experimental measurements fall between the two predictions. In our experiments X^/Xg % 10 and for the sake of simplicity we will use eq. (1+-177 to pre-dict <XQ>.
On the other hand the contribution to the total effective conductivity due to the motion of the interstitial fluid can be simply obtained either by dimensional analysis or by means of the simplified model shown in Fig. 1+-7. If Ug, ag and dg are the fluid velocity, the thermal diffusivity of the fluid and the particle diameter respectively, then the following
§1+.1+
35
I <'>A, ^ '• , 1 1 — • y • ,•• calcuiatsd from data of Kunii «t al. [43]
— tv.'
1 1 200 400 600
Fig. 1+-7 A simple model for the prediction of the functional form of <X^>/X^
Fig. 1+-8 Correlation of experi-mental data plotted according to the equation (1+-20) (for a packed bed of spherical particles, e =0.1+0
heat balance can be written:
<X > —' ^ t d
s
(pc ) U (Ti - T2) P g g
which upon rearrangement reads:
= ci Pe , ^\'
(1+-18)
(1+-19) g
where the Peclet-number is defined as Pe =Ud^/a and ci is a constant.
Using the reported experimental results from the litera-ture we plot <X>IX vs. Pe. The results are shown in Fig. 1+-8. We, then, immediately see that cj - 0.12. These findings are in complete agreement with those reported by Kunii et al.
(43). Thus for the overall effective thermal conductivity of our bed we obtain:
<X> X U d = 0.12-S-^ <X > o (1+-20) g
With the help of figures 1+-2,1+-6 and the equation (1+-20) we have calculated the values of <X^>/<a> and compared them with the experimental values in table 1+-7. The agreement is good.
As the thermal conductivity of the Cu-article is very large its surface temperature is the same as the temperature measured by the thermocouple. The surface temperature ob-tained from the equation (I+-II) is given by:
36
§l+.l+ l2 $ T ( ) ' v o ^ < a > ^ ^ e a s . r ^ / • 1 fo^ 1 fW^sec • [mVminJ [ cJ [ ^ ^ r ^ J 270 110x10-3 370 1+20 270 I3I+XIO-3 370 1+20 8.1+x10'+ 10.3x10"+ 12.3x10"+ 7.8x10"+ 8.9x10"+ 11.3x10*+ <X> ., <a> , c a l c . c a l c .r w ]
rm2
1
.m ° c j L s e c , 0.090 0.096 0 . 1 0 0 0.089 0.090 0.095 8 . 6 2 x 1 0 - 8 9 . 2 2 x 1 0 - 8 9.52x10~8 8 . 8 3 x 1 0 - 8 7 . 8 x i o - 8 8.1+ x l O - 8 ( < ^ ^ ' / < - > ^ c a l c . ^ < ^ > ^ / < - > ^ m e a s . 1.118 0.972 O.85I+ 1.150 1.019 1.008Table U-7 Comparison of experimental and calculated values of <X>2/<a>
(0,t) = e x p j — ^ ^ t } e r f c { ( - 4 7 ^ t) } . (1+-21)
The calculated values of with the measurements in basic conclusions can be the transport of heat int
• measurements — eqn. ( 4 - 1 1 )
Fig. 1+-9 Comparison of Oy, given by equation (4_21) with measurements
the surface temperature are compared figure 1+-9. From the figure two drawn. The first conclusion is that o the fluidised bed, operating under
coating conditions (Ty > T g ) , takes place according to the pene-tration theory and the second con-clusion is that the temperature in the heart of the object is nearly the same as the temperature at its surface, which can be pre-dicted by the equation (1+-21).
Further we have also com-pared the calculated and the measured heat transfer coeffi-cients. From the equation (1+-3) we obtain:
a = - C
dT (x.t)/dx
^\ - V
x=0 (1+-22)§1+.U 37
t [ 5 ]
I i i i i i — i 1 1 i 1 i i I i I i
1 2 4 6 8 10 20 40 60 80 100
Figure 1+-10 Comparison of the measured heat transfer coef-ficients and those given by equation (1+-23) the last equation becomes:
« = ^ ^ V ( \ ^ i i ^ » p ( 5 J i l \ ) .rf=(\^ii7,) - .1 . (^-23)
In figure It-10 we have compared the measured values of a with its values calculated by means of equation (U-23). Except for long times, the agreement is reasonably good.
1+.5 Conclusions
Starting with the postulate (as summarised in table 1+-1) that the most important parameters in determining the coating process are the heat characteristics of the fluidised bed, we have meas\ired the heat transfer coefficient between the bed and an immersed plate. The plate temperature was higher than the melting temperature of the polymer and a sintering process took place at the vicinity of the plate. However, the porosity of this sintered layer formed on the plate was found to oe about the same as that of the homogeneously fluidised bed. Conse-quently the measured values of effective thermal character-istics of the bed do not depend strongly on the article temper-ature as can be seen from table 1+-7.
In chapter 3 it was concluded that in homogeneous fluid-isation regime there was no appreciable particle motion. The heat transfer coefficient measured in this chapter has shown
