HEAT TRANSFER IN FLUIDISED BEDS.
One area of particular interest to me has been fluidised bed technology. My particular interest is heat and mass transfer within such systems and for those wishing to delve a bit deeper thepages begin the story of understanding the behaviour of these fascinating chaos sytems.
Beds of small particles can be fluidised by either passing a gas through them,or by vibration. The vibrated bed has the unique property of allowing heat transfer control in a vacuum. The linearity of the fluidised bed offers good possibilities of accurate temperature control over a wide range. In addition the independence of the particle material offers control of heat transfer by external stimulation rather than material properties. The addition of phase change materials to the bed (solid to solid) has also be investigated.
The first stages of assembling a theoretical model of the behaviour of such systems was one of the hardest steps and the following may be of interest if you are about to travel that road.
Quite clearly any use of these corelations must be entirely at your own risk and I would warn against making assumptions about the suitability of particular equations without first confirming their applicability for the purpose.
Any references in the following description can probably be tracked down from the name of the author and the pages devoted to the topic on the References page. Also associated with these are two others covering a brief summary of the history of fluidised beds and the history of working with iron and steel.
Probably the most important property of a fluidised bed is known as it's Minimum Fluidisation Velocity (UMF). This value represents the speed of the upflowing gas required to just hold up the particles against gravity and thus force them into motion. It will be seen that many other properties of the bed are very dependent upon its agitation and the best measure of that is the flow rate relative to UMF. Thus a bed running at 4 times UMF is considerably more active than one running at 2 times UMF.
It is thus vital for any predictive model to be able to predict UMF for any particle size or material and any gas used for fluidisation.
Procedures and expressions for finding the pressure drop of a gas flow through a bed of packed particles have been suggested by Perry based on the work of Chilton and Colburn and by both Carmen and Brown based upon the work of Bramall and Katz. The more recent, and simpler correlation suggested by Ergun is used by Kunii and Levenspiel to develop the following equations; and has been found by them to represent the data for granular materials to ± 25%. The solution of equating the frictional pressure drop to the bed weight then results in a quadratic for U_{mf }
Where:
and
Notably it is d_{p} which is used in these correlations, being the particle diameter estimated from screening, and not d_{eff} which is the diameter of a sphere estimated to give the same behaviour. Several workers have observed that the constants of the above equation remain very similar across a wide range of conditions, where
and
Consequently it would seem reasonable that first estimates of U_{mf} can safely be derived using these equations and thus by way of example for a bed of alpha alumina particles (effective diameter 180mm ) in an air driven bed at ambient 20°C and using the correlations with the values of k1 and k2 suggested by Wen and Yu:
Shirai reports the initiation of entrainment for sand particles in the range 100300mm commencing at gas velocities of 10U_{mf}. Using this estimate it becomes possible to assess the maximum air demand, air metering requirements and air heating power required to run a bed below entrainment velocities.
Having evaluated the minimum fluidisation velocity it becomes possible to consider how any selected gas flow will behave in the bed. An amount very approximately equal to that required to produce a flow of Umf will be used to fluidise the bed and the excess will appear as bubbles. Having established the flow regime in the bed one can move on to consider the issues of both heat and mass transfer.
Bubbles in bubbling fluidised beds.
The initial bubble size at the base of the bed is suggested by Kunii and Levenspiel as
This correlation provides a reasonable fit with the data from several other investigators presented in the above reference. Thus for the particles used as a example (diameter 180 um) in an air driven bed at 20°C running at 4Umf (U0=0.16 m/s) then using this corelation the bubble diameter at the base of the bed is 0.004 m.
Toomey and Johnstone introduced the simple two phase model by assuming that all gas flowing through the bed in excess of Umf flows as bubbles which grow as they move up the bed. Hilligardt and Werther observed that significant amounts of gas does pass through the emulsion phase and the simple two phase model can produce errors of 50%. They proposed that the gas velocity within the emulsion is increased above Umf and that it can be reasonably correlated for a three dimensional bed by:
To estimate the bubble size at any position within the bed correlations with the data can be obtained by assuming that the final bubble size is a function of excess gas flow and relating the transition from initial size to final size with an exponential growth. On this basis Mori and Wen propose that the maximum bubbles size is given by
and the actual bubble size at various heights in the bed are given by:
Using these corelations for the exampled bed at a height of 0.1 m gives the maximum bubble size as 0.07 m and the size at a height of 0.1 m of 0.013 m.
For Geldart type A and B particles the correlations recommended by Kunii and Levenspiel for bubble speed were derived from their analysis of the data reported by Werther for bed smaller than 1m diameter.
and
where as proposed by Davidson and Harrison for a single bubble on the basis of the two phase theory. Hence selecting type B as most appropriate for the alpha alumina particles gives
The bubble fraction is taken from the model proposed by Kunii and Levenspiel, which for a well fluidised bed where suggests
The frequency of bubbles passing any point will be a function of their size, speed of rising and the relative amount of the bed occupied by such bubbles. No convincing investigations were found for this very important parameter, and no proposals appear to have been made as to how the bubble frequency is affected by the fluidisation velocity, bed pressure or bed temperature. Kunii and Levenspiel noted that the results from several other workers suggest the data forms a narrow band primarily dependent upon the height in the bed as illustrated in Figure 52 and from which a frequency of 6 Hz has been estimated at a height of 0.1 m. Clearly a range of approximately 3.5 to 6.5 is possible from this data, but no literature was found to suggest the parameters upon which a selection should be based. However for Geldart "A" particles the bubbling does not occur immediately at minimum fluidisation and for such particles Geldart and Abrahamsen recommend that bubble frequency must be suppressed until where . They do not however suggest an increase in bubble frequency once bubbling commences.
Having established the regime in the bed, which now containsregions of emulsion (i.e. just fluidised particles) and gas (bubbles) we can move on to consider what the heat transfer is going to be in each case
The various contributory factor to heat transfer.
The total rate of heat transfer in the bed can be estimated as a combination of that taking place within the bubbles i.e. convection and radiation plus that in the emulsion i.e. viewed as conduction and radiation. Heat transfer within the bubbles is evaluated using the convection correlations suggested by Incropera and DeWitt taking velocity as bubble speed and the characteristic dimension as bubble diameter whilst assuming laminar flow across a flat plate:
The inaccuracy in such a simple set of assumptions is recognised, but because the final contribution of this element to the total heat transfer is relatively small, it is considered justified. Indeed several texts suggest that bubble contribution may be ignored completely.
ii) To estimate the effect of emulsion upon heat transfer coefficient
The analysis now proceeds, having established the population and activity of bubbles, with the assumption that the remainder of the bed is "emulsion", and that this is a state of "just fluidised" particles. The effective conductivity within a packet of emulsion at a surface is defined in terms of the contributions of the gas in the voidage and the solid occupying the other space. If the heat flow were in parallel paths of either gas or solid then the effective conductivity could be expressed as:
The use of such a simplified model gives predictions several orders of magnitude greater than experimental results; however Kunii and Smith modified Equation 14 to allow for the contact geometry between particles which were assumed spherical, and thus being less well packed at a surface, to give:
Where at a flat wall and in the bulk .
Similarly represents the effective gas thickness of the film around the particle and drops from 0.33 as the ratio of the solid and gas conductivity rises. At the wall it drops more slowly than in the bed because the particle contacts are less densely packed being up against a presumed flat surface. This factor is taken from the table below which was extracted from Kunii and Levenspiel. For the air at 50°C and the alumina particles the thermal conductivities are 0.026 and 10 W/m.K respectively and thus:
Table 1 Effective gas thickness around particles 







1 
10 
100 
500 
1000 
10000 

.33 
.18 
.08 
.07 
.05 
.04 

.33 
.25 
.12 
.1 
.08 
.06 
Giving approximately values for and .
It also becomes necessary to estimate and a value of 60% is used , derived from experimental results on minimum fluidisation.
In addition, as has been mentioned above, some of the additional gas flow in excess of Umf is known to flow through the emulsion. Abrahamson and Geldart propose the emulsion voidage can be correlated to the emulsion fluidisation velocity by:
Thus using these equations .
Mickly and Fairbanks proposed a "renewal model" where a "packet" of emulsion bathes the surface while it passes; to be replaced by another while the original packet moves away into the bed bulk. The instantaneous heat transfer coefficient for such a "packet" is proposed as:
and if this is integrated over time on the basis that the bubble fraction represents the time which emulsion is not bathing the surface together with the assumption that bubble size is not changed at a surface then and the emulsion heat transfer can be evaluated at a wall as:
iii) To estimate the effect of radiation upon heat transfer coefficient
The radiation heat exchange from a body when totally immersed in the bed is given by:
For a first estimate assuming a body at 80°C and a bed at 20°C with all surfaces having an emmisivity of 0.8 then
iv) To estimate the combined effects upon heat transfer coefficient
Having evaluated all the components of the heat transfer they may be combined by using the bubble fraction to express the relative behaviour in the two possible states i.e. when a bubble is at the surface or when emulsion is at the surface. Whilst a bubble is present the heat transfer will be both convection and radiation; and whilst emulsion is at the surface we have radiation and conduction to the packet surface film with the consequent conduction away from the surface film into the bulk of the packet. For the correlations being used the surface film is assumed to have a thickness of and the combined heat transfer rates can be expressed as
Then combining these equations gives
Yagi and Kunii observed that additional gas mixing takes place at the wall and introduced the additional term (where the constant ) into this equation for wall heat transfer when gas velocities exceed U_{mf} . In the bed under consideration this additional term has negligible effect but is included for completeness and the use of these equations in other conditions.
Thus
Because all results show variations of heat transfer rate dependent upon both the bed bulk temperature and the surface temperature, it is necessary, in order to obtain accurate predictions, that the temperature at which heat transfer is actually taking place between the hot surface and the impinging particles must be estimated before the predictive correlations are evaluated. The properties of the fluidising gas being of particular relevance and Gelperin suggests evaluating the effective film temperature T_{f }from:
However he defines the thermal resistances and as dependent upon the temperature in the film which is itself the unkown and required variable. From a practical engineering perspective such complex trial and error solutions can not be justified because they offer great complexity and little or no benefit over the practical approach adopted in most fluid dynamics solutions of
Both my tests and other literature suggests these equations behave quite well at ambient temperatures. At higher temperatures, whether in the bed or on the surface from which heat is transferred, they are not so good.
The effect of elevated temperatures.
It was previously mentioned that both my tests and other literature suggests these equations behave quite well at ambient temperatures. The corelations described in previous pages if evaluated across a larger range of temperature suggest heat tranfer rates will be as shown below (for alumina particles 180 micron in air)
The question arises as to how best modify any model to better predict a temperature effect. The fall in density, rise in conductivity and viscosity of the air at higher temperatures would seem to be the best correction factor to apply but how this is done is best considered in the light of other design criteria.
Mass Transfer
Mass transfer can take place in two states from a bed of solid particles. The particle material itself can sublime or evaporation of a liquid from the particle can take place. This later case is frequently encountered in drying applications.
The effect of drying.
The bed exhibits three clear states of saturation, drying and dry. Saturation occurs because the gas flow has limited ability to carry any vapour i.e. reaches saturation. Vapour driven off particles from a heater in excess of this will only recondense on other particles. Beware wasted heating in such beds.
Once the gas is able to carry whatever vapour is available the bed begins to dry. Any latent heat of hydration must be overcome at this stage and often more heat is required than was expected.
In such cases the mass of the particles (density) is continually reducing but the size remains constant. Clearly changes are taking place to the gas properties as they carry less and less vapour.
Sublimation
Beds of particles which sublime (change from solid to vapour without passing through a liquid phase) can lead to very complex bed behaviour. One clear possibility being that sublimation takes place near a hot surface and resolidifies of other cooler particles in the bed bulk. Such behaviour may or may not cause particle agglomeration and eventually fluidisation failure.