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Characterisation and control of bubbling behaviour in gas–solid fluidised beds
ARTICLE IN PRESS Control Engineering Practice 17 (2009) 67– 79
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Characterisation and control of bubbling behaviour in gas– solid fluidised beds C.N. Lim, M.A. Gilbertson , A.J.L. Harrison Department of Mechanical Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK
a r t i c l e in fo
abstract
Article history: Received 9 August 2005 Accepted 21 May 2008 Available online 24 June 2008
Bubbling fluidised beds are often exploited for the good mixing of phases they promote, which is required in many chemical processes. However, the bubbles cause heterogeneity in the fluidised system, and their dynamics must be understood for good process control. In this paper, the dynamic behaviour of the bubbles in a planar gas–solid fluidised bed is described analytically and experimentally validated. The bed resembles a temporary buffer of gas for which the bubble residence time in the bed is important. Subsequently, a closed-loop controller was developed using visual feedback to regulate the process. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Dynamic modelling Process control Chemical process System identification PID control
1. Introduction Fluidised beds are used in a variety of industrial processes, and typically comprise a column of solid particles resting on a porous surface. Gas is introduced through the floor at a flow rate sufficient to support the weight of the powder. This nearly eliminates the friction between particles, allowing them to behave in a similar manner to a fluid. Any industrial process that requires good contacting or effective mass and heat transfer to obtain high degree of efficiency is well-catered for by the attributes of the fluidisation process. Industrial applications of fluidised beds exploit the high particle surface area that exists within a relatively small volume of bed for operations such as drying, heat transfer, combustion and chemical reaction between the gas and particles. One of their principal virtues is the degree of mixing between the particles and gas. When the gas flow speed exceeds that necessary to fully support the particle (the velocity of minimum fluidisation, Umf), particlefree voids or bubbles form. These enhance the mixing of particles throughout the bed so that their properties tend towards uniformity (Eames & Gilbertson, 2005); however, the gas within these bubbles does not serve its primary purpose of reacting with the solids. The bubbles also introduce heterogeneity to the process, which reduces the efficiency of contact between solid and gas, and, as they usually travel faster through the bed than the surrounding gas in the emulsion phase, the residence time
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E-mail address: [email protected] (M.A. Gilbertson). 0967-0661/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2008.05.007
associated with the gas in the bubbles is reduced (Kaart, Schouten, & Bleek, 1999). There is little exchange of the gas from within a bubble and outside of it (Rowe, 1964); furthermore, the speed of a bubble increases with size, reducing the residence time of the gas. The result is that, in a bed with large fast-moving bubbles, a large proportion of the gas introduced into the bed does not adequately contact the particles. Therefore, a general definition of good fluidisation quality is that there are large numbers of ‘smallish’ bubbles homogeneously distributed throughout the bed. Whilst the aim of monitoring and controlling the process is to achieve high-quality fluidisation, this can be difficult to achieve practically for a number of reasons. In particular, small bubbles tend to coalesce as they move up the bed to form less desirable large ones, which move quicker up the bed further diminishing particle–gas contact. Harrison, Davidson, and de Kock (1961) pointed out that the quality of fluidisation is also affected by several factors other than gas velocity, such as the solid and gas density ratio, solid particle size, operating pressure and temperature, and fluidising gas viscosity. Depending on these factors, a fluidised bed may be in any one of a complete range of fluidisation regimes from smooth to slugging. By controlling the choice of solid and fluidising medium characteristics, the fluidisation mode and hence the quality, depending on the type of operation, could also be controlled. However, often this is not possible when the operating parameters have already been determined and the quality of fluidisation begins to vary during the process operation. Then, a dynamic means of control is required—e.g. by controlling the feed rate of solids or gas. The ability to control the state of the bed becomes more crucial when factors inherent in the bed change, e.g. the gas properties
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may vary with time due to humidity or viscosity changes; particles may agglomerate or fragment, or new particles with different properties may be introduced to the bed. These will then affect bubble formation and behaviour. Any controller must be capable of detecting and reacting appropriately with changing process dynamics to continuously deliver the desired performance. Furthermore, time-series measurements of bubbling fluidised beds are generally ‘noisy’—pressures, bed heights, bubble sizes and distributions, etc. exhibit a significant level of fluctuation. Such systems are known to be difficult to control because, if not properly accounted for in the controller design, the closed-loop nature of typical controller implementation means that noise can swamp the error feedback loop. This can result in control signal saturation and failure of the controller. In addition to noise, the behaviour of the particles and bubbles in the bed varies non-linearly as the flow of gas into a bed is increased from zero. The onset of fluidisation at Umf is particularly significant: at this flow rate the bed experiences a discontinuous state change between being a fixed bed in which friction dominates, into a suspended mobile bulk of particles and, in certain cases, the immediate onset of bubbling. Across the whole regime of fluidisation, the bed undergoes a series of dynamics transitions with increase of flow supply. Typically, there is also some form of delay or lag whereby the state of the bed does not immediately respond to changes applied to the gas supply. All the above-mentioned factors make fluidised beds difficult to monitor and control. Many techniques to monitor and control processes have been devised and applied industrially. Numerous methods for controlling the bubbling process have been suggested; from Lee, Chong, and Leung (1986), Chong, O’Dea, White, Lee, and Leung (1987) developed a control technique employing pressure measurement in a tall bed to govern the control of amount of excess gas expelled by bleed valves installed at consecutive heights in the bed. The technique proved to be successful and practical. Korte, Schouten, and van den Bleek (2001) suggested an alternative method of controlling the nucleation of bubbles in the bed such that coalescence was controlled and suppressed to an acceptable amount to maintain desired bubble population and distribution. In addition, statistical control methods have been used to control processes that rely on fluidised beds (e.g. Hilgert, Harmand, Steyer, & Vila, 2000; Simoglou et al., 2000). However, these developments have not fully addressed the causal relationship
between the control and monitored parameters, i.e. the process dynamics are not fully understood. This paper describes an attempt to tackle this issue by identifying and interpreting the dynamics of bubbles in a planar fluidised bed. From this, controllers are designed and implemented. The structure of the paper is as follows: In Section 2, the experimental arrangement is described. The steady-state and transient responses of the bed to a step change in gas flow rate are examined. For dynamical modelling purposes, the bed can be viewed as a temporary storage for gas. In Section 3, the development of this idea leads to the derivation of transfer functions relating the gas flow rate contained in bubbles to the bubble void fraction (BVF), this being the fraction of the bed’s volume that is occupied by bubbles. From these observations, it is possible to control the BVF using closed-loop linear control strategies, implemented and detailed in Section 4.
2. Experimental observations of a 2-D fluidised bed This paper describes the modelling and control of the fluidisation quality of a bubbling bed. One of the difficulties with practical fluidised beds is that they are opaque and so it is difficult to measure the number and distribution of bubbles within them. To allow unambiguous measurement, an experimental Perspexwalled two-dimensional bed and a real-time image analysis technique were used to directly measure the condition of the bubbling in the bed. This allowed the explicit observation of the behaviour of bubbles and the bed fluidisation quality to be determined. The quality of the bed was interpreted from the number and distribution of bubbles. The relationship between them and the gas flow rate was subsequently explored experimentally and theoretically. Fig. 1 shows a schematic diagram of the experimental fluidised bed and monitoring equipment. The bed was b ¼ 500 mm wide, h ¼ 1000 mm high, l ¼ 13 mm thick and so was effectively twodimensional. The bubbles were viewed by illuminating the Perspex walls from behind. The particles were Ballotini glass beads of bulk density 1500 kg/m3, approximately 80% spherical, and with a diameter range 106–212 mm. These filled the bed to a height of approximately 690 mm in the absence of gas flow. Bubbles were seen to form in the bed when incipiently fluidised (UmfEUmb), with mild expansion (3–5% of ungassed bed height) of the bed. Compressed air was used to fluidise the bed, and its
Planar fluidised bed
Feed back loop Image analysis
ControlLab
Computer
Frame grabber
Camera Monitor
D/A converter
Forward control loop
Actuation system
Air supply Fig. 1. Experimental fluidised bed and monitoring equipment.
Fluorescent light tubes
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flow rate qin was set by a servomotor-controlled valve and introduced uniformly into the base of the bed via a porous plate distributor. The distributor gave consistent uniform flow, with formation of small bubbles evenly distributed symmetrically across it. No channelling or concentration of flow nucleation sites on the plate was observed during any bubbling conditions and the windbox pressure was stable and remained constant throughout the cross-section. The valve position was measured via a potentiometer and AD/DA card in a computer. An image analysis system was developed, comprising a CCD array camera, frame grabber, software and the computer. The software used for these purposes was developed in-house at the University of Bristol. Bubbles in the bed were detected by the transparent areas they created on the flat cross-sectional plane of the planar bed through which illumination placed at the back of the bed passed. Fig. 2 shows a sequence of typical planar bed bubbling images captured by the imaging system. By analysing long sequences of these, information such as the proportion of bubbling phase, bubble size, spatial and temporal population, and other statistical measures that relate to the state of the bed were obtained, along with the uncontrolled bed dynamics. The system was able to identify the positions and sizes of the bubbles to be identified at up to 25 frames/s. This rate was rapid enough to allow real-time control of the bed.
2.1. Steady-state relationships to flow rate A number of steady-state or time-averaged measurements were obtained from the experimental fluidised bed, and in each case the principal input to the plant was the gas flow rate qin. The measurements were all taken from a flow rate datum based on the value of Umf, which corresponds to the onset of fluidisation, and bubbles begin to form at the distributor. Umf is thus seen as a critical value at which a discontinuity in system dynamics occurs: from fixed bed behaviour, when fluidised, the bulk of the particles undergo a drastic change in their hydrodynamical behaviour as the interaction between the gas and solid phase is completely altered. Umf was found to be approximately 103 mm/s of superficial gas flow. All measurements and analyses described in this paper were carried out with flows measured relative to the Umf ( ¼ Umb) datum. To characterise the bubbling process effectively, and capture the dynamical behaviour of the bed, a suitable parameter had to be determined. The parameter should explicitly and effectively correspond to the state of the process it was measured at. Relatively simple and fast to obtain, it must be consistent over a range of process conditions. According to Brown and Brue (2001a, 2001b), the minimum sampling frequency required to appropriately investigate the condition of the fluidisation process
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should be at least 20 Hz in order to capture all the dynamical features and to avoid anti-aliasing of fast dynamics. They showed from their experiments (Brown & Brue, 2001a) with pressure measurements that there are no differences in high-frequency spectral content obtained with the sampling rates between 20 and 100 Hz. Makkawi and Wright (2002) used electrical capacitance tomography to demonstrate that the optimum recording span for reliable analysis of the solid–gas hydrodynamics ranges from 20 to 100 s, while the acquisition rate should range between 20 and 100 Hz. The present image analysis system has a limited acquisition rate of 25 Hz, thus requiring that any measurement made to investigate the dynamics of the fluidised system should span an adequate period. Periods of 20–60 min periods were used, which are considerably longer than that recommended by Makkawi and Wright (2002). However, this enabled all the dynamics of the process to be captured, including very slow ones, and to generate reliable results despite the significant noise levels previously mentioned. The higher acquisition rate in Makkawi and Wright’s work was necessary for measurement of solid fraction in each slice of bed cross-section over a chosen height section (to produce a three-dimensional map), whereas a lower rate is acceptable for acquisition of BVF measurement on the two-dimensional planar bed applied here. Fig. 3(a) shows a typical steady-state measurement of BVF for a time span of 150 s acquired at the rate of 25 Hz. The apparent noisiness of BVF occurs principally because, whilst the gas enters the bed at an essentially constant rate, it manifests itself in the bed in a more random manner as bubbles form at the bottom and the gas leaves at the top as large discrete bubbles. When a large bubble leaves the bed, there is a corresponding significant drop in BVF. Fig. 3(b) shows the relationship between BVF and gas flow rate qin. There are two obvious regimes: below the minimum fluidisation flow rate Umf no bubbles are formed; at rates above Umf the BVF increases non-linearly with flow rate. The nonlinearity occurs because the average bubble size increases with flow rate owing to increased likelihood of coalescence, and larger bubbles move more rapidly than small ones; thus, the average residence time of bubbles within the bed decreases with increasing flow rate. Values shown amidst the graph are the means of BVF at each flow condition. The BVF measurement resembles a stair-like graph corresponding to step increase in flow. The changes in the time series with flow rate indicate that the measure of BVF was effective and robust for a range of bubbling condition, despite the presence of fluctuations in the signal. The 95% confidence limits are also shown on the graph for approximately 2.5 s of BVF time series for different bed conditions, all acquired at 10 Hz rate
Fig. 2. Sequence of snapshots (intervals 40 ms) showing a typical bubbling bed in a planar bed as captured by the imaging system.
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Fig. 3. (a) Variation in mean BVF for a constant superficial flow rate of 12.8 mm/s above Umf, (b) experimental BVF against superficial gas flow rate above Umf.
producing approximately 25 data points. However, the reliability of the measurement gradually decreases because the BVF standard deviation increases with flow rate. This is due to the growing fluctuation magnitude caused by the presence of large bubbles and their departure from the bed. This fluctuation is several orders of magnitude greater than any other fluctuations in the BVF measurement, such as those caused by light source fluctuation and hardware electronic noise. Small variations in flow rate (say 72.6 mm/s superficial flow) during bubbling were experimentally observed not to cause noticeable change to the state of the bed and the BVF. Fig. 4(a) shows the variation in the average temporal population of bubbles in the bed with respect to flow supply. Initially, above Umf there was an increase in bubble population with increase in gas flow. The bubble population peaked at approximately 52 mm/s of superficial gas flow and then declined because of the greater coalescence in the freely bubbling bed, creating
larger, faster bubbles. These bubbles leave the bed quickly, occasionally exhausting the bed of bubbles by sweeping through the bed and coalescing with more bubbles in their path, creating a few large ones. This phenomenon also increases the standard deviation in the measure of bubble population. Fig. 4(b) shows the size distribution of bubbles with flow rate. It should be noted that bubbles of diameter less than the bed thickness are unlikely to be detected by the imaging system; however, these do not contribute significantly to the overall BVF measurement. The average bubble size increases approximately linearly with flow. At constant flow rate, the number and mean size of bubbles are subject to large variability, and therefore would be difficult to control by varying the flow rate. However, the BVF, which is essentially the sum of the areas of each bubble, has a monotonic relationship with flow rate and is therefore more readily controllable. Moreover, because of the direct link between the
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and there is no coalescence), then
70
Bubble population / [ ]
65 60
(2)
(3)
0
55
T
qin ðtÞ qin ðt TÞ dt
The Laplace transform that describes this is
50
VðsÞ ¼
45
1 Q ðsÞ½1 esT s in
(4)
Also, the BVF is
40
BVFðtÞ ¼ Mean value Best fit (cubic) Standard deviation error bar
35 30 10
20
30
40
50
60
70
80
90
100 110
Flow rate above Umf / [mm/s]
Standard deviation error bar Mean value
160 140 120 100 80 60 40 10
20
30
40
50
60
70
80
90
100 110
Flow rate above Umf / [mm/s] Fig. 4. (a) Experimental variation of number of bubbles with gas flow rate, (b) experimental variation in mean bubble diameter with superficial gas flow above Umf.
state of the bed and gas flow rate, the BVF is a useful parameter to control as it is a reasonable indicator of the overall state of the bed.
3. Modelling the dynamic response of the bed 3.1. A simple model of a bubbling fluidised bed From the point of view of its dynamics, a bubbling bed can be considered as a temporary store for the gas supplied. Bubbles can be characterised as packets of excess gas which reside for a given amount of time determined by their physical properties before leaving at the freeboard. The volume v(t) of the bubbles in the bed at time t is given by Z t vðtÞ ¼ qin ðtÞ qout ðtÞ dt (1) 0
where qin(t) and qout(t) are, respectively, the flows (in excess of Umf) into and out of the bed at time t. If all the gas bubbles take T seconds to pass through the bed (i.e. they all have the same size
vðtÞ ¼ KvðtÞ bhl
(5)
where b, h and l are the bed’s width, height and thickness, while K is a proportionality constant relating the amount of gas supplied to the corresponding area of bubble void generated in the bed. Thus, the bed’s transfer function relating BVF to gas flow rate into the bed may also be expressed as Gb1 ðsÞ ¼
180
Average bubble size / [mm2]
qout ðtÞ ¼ qin ðt TÞ whence Z vðtÞ ¼
BVFðsÞ K ¼ ½1 esT Q in ðsÞ s
(6)
This simple model of the bed Gb1(s) is then a ‘zero-order hold’ transfer function with low frequency gain KT, i.e. in this case, the bed’s steady-state value of BVF. When there is a step change in flow into the bed, bubbles are generated and the BVF ramps up at a constant rate for T seconds, where T corresponds to the bubble residence time in the bed. Thereafter, as more bubbles are introduced into the bed, bubbles simultaneously exit at the top in equal quantity and the BVF is constant as long as the supply of bubbles (i.e. the gas supply) does not change. When the gas supply cuts off, the BVF ramps down at a similar rate. The Bode plot corresponding to Eq. (6) is shown in Fig. 5. The transfer function is approximately that of a system of relative degree one (i.e. first-order, or second-order denominator with first-order numerator) combined with a delay term. This gives it a decay rate of 20 dB/decade in the plot after the break frequency. These salient features are typical of a transfer function that describes the integrating effect of the bubbling process on the BVF measurement. It terminates at the break frequency with a flat DC level, due to the limit of the minimum size of bubble present in the bed (Lim, 2004). Additionally, the gain is effectively zero at integer multiples of 1/T. This occurs physically because at such frequencies the transport delay T ensures that the flow rates in and out of the bed are equal. In other words, these intermittent ‘dips’ correspond to the residence times of bubbles generated in the bed and the accompanying harmonic frequencies are anti-resonance frequencies at which the introduction of bubbles is just balanced by the bubbles leaving at the freeboard. At these frequencies, the change in BVF is small, i.e. the transfer function has low energy. The decaying trend of the phase part of the diagram is characteristic of a dynamical system containing elements of time delay. The model used to obtain Eq. (6) was kept simple, resulting in unrealistic artefacts such as sharp changes in gradients by assuming that bubbles are created with similar sizes and their size remained constant whilst in the bed. In this way, the computation of BVF was discontinuous at the zero–nonzero transition and increased in a linear ramp fashion. 3.2. A realistic model of a bubbling fluidised bed The above analysis assumes that all the bubbles have equal residence time T, and therefore are of the same size; however, in
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Gain Magnitude / [dB]
-60 -80
-20 dB/decade
-100 -120 -140 10-2
10-1
100
101
100
101
Frequency / [Hz]
Phase of signal / [°]
0 decay -50 -100 -150 -200 10-2
10-1 Frequency / [Hz]
Fig. 5. Bode plots of the simple model of the bed, Gb1 as defined by (7).
practice, larger bubbles travel more rapidly than small ones. When a bubble is introduced into the bed, isolated from any effects and interactions, it rises steadily, spending time T in the bed. In a bubbling bed where the single bubbles are free to interact and coalesce with other bubbles, their residence times could be significantly reduced. There is a distribution of possible residence time associated with a bubble of a specific size, ranging from the maximum amount of time for an isolated bubble to a minimum duration associated with a bubble that has coalesced with other bubbles, so forming a larger, faster bubble. A simulation replicating the experimental bed was developed based on the bubble interaction model of Clift and Grace (1970, 1971). The simulation describes an incipiently fluidised bed where, upon introduction of flow, bubbles were created at the bottom in an evenly distributed fashion, both spatially and in size range. In addition to coalescence, bubbles expand as they rise up the bed based on the linear relationship between equivalent diameter and height in the bed as proposed by Kunii, Yoshida, and Hiraki (1967). Full details of the simulation can be found in Lim (2004). An example is shown in Fig. 6 for the bubble residence time distribution (for 100,000 bubbles) in the simulated normally bubbling bed fluidised at 25.6 mm/s superficial gas flow above Umf with a superimposed uniformly random flow variation of 720.5 mm/s. The distribution gives an indication of the probability that a bubble of a specific size will spend a given amount of time in the bed. The distribution in Fig. 6 shows that, while most bubbles spent about 2.2 s in the bed, the approximately normally distributed profile is flanked by small bubbles which will have risen without much interaction with others, and by bubbles that have experienced large amount of coalescence and risen quickly
Fig. 6. The histogram of bubble residence time for a total of 100,000 bubbles generated in the simulated bubbling bed fluidised at 25.6 mm/s superficial gas flow above Umf with a uniformly random variation of flow into plenum of 720.5 mm/s.
through the bed. The profile of the distribution changes for different bubbling conditions depending on the characteristics of bubble generation and the consequent interaction and coalescence behaviour of bubbles. A modified transfer function Gb2 accommodates this range of values of T:
Gb2 ðsÞ ¼
4:3 X T¼1:1
fðTÞ
KðTÞ ½1 esT s
(7)
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3.4. Model benchmarking with experimental equivalent
1.4 Step response / [K]
1.2 1 0.8
Gb2 Gb3
0.6 0.4 0.2 0 0
1
2
3
4
5
Time / [s] Fig. 7. Comparison of the step response between Gb2, which is the realistic model of the bubbling fluidised bed, and Gb3, shown in (10) in Section 3.3, which is the equivalent of model Gb2 in the form of standard transfer function.
where f(T) is the probability density function of bubble residence time obtained from the histogram shown in Fig. 6. Also, assuming the simplified case where K is unity, the corresponding step response of Gb2 is shown in Fig. 7. The effect of using a range of values of T is to smooth the transition between the initial rise and steady-state values. A summation operator was used in (7) instead of an integral operator to emphasise the fact that Gb2 is a composite model of individual basic equations in (6), each for a specific residence time T. 3.3. Model form tailoring for controller design purposes Having examined the bed’s dynamic response, a second objective was to implement closed-loop control of the BVF. Whilst Gb2 is believed to represent the dynamics of the bed tolerably accurately, it is not well suited to controller design as the exponential term in (7) has infinitely many roots and complicates the design procedure. For this reason, a further transfer function Gb3 was sought: this was intended to have a similar step response to Gb2, but its numerator and denominator are polynomials in ‘s’, which facilitates conventional controller design. Gb3 was derived so as to have similar Bode plots to Gb2 as well. For this reason, it was chosen to have the general form: Gb3 ðsÞ ¼
lo2n Tðs þ 1=TÞ þ 2zon s þ o2n
s2
(8)
where dynamic terms l ¼ low frequency gain, T ¼ time constant, z ¼ damping coefficient and on ¼ undamped natural frequency. The best curve-fitting values of the parameters l, T, z and on were found using MATLAB’s fminsearch algorithm: l ¼ 1.00 K; K ¼ 1; T ¼ 0.155 s; z ¼ 0.82; on ¼ 1.60 rad/s, i.e. Gb3 ðsÞ ¼
73
0:40Kðs þ 6:45Þ s2 þ 2:63s þ 2:56
(9)
As this is a curve-fitting procedure, the variables do not directly have any physical and/or dynamic implications. These values ensure that the step response of Gb3 is very similar to that of Gb2, as shown in Fig. 7, and their Bode plots (Fig. 8) are in good agreement for frequencies up to around 0.4 Hz. At higher frequencies there is a marked discrepancy in phase; however, the gain is small at such frequencies compared to the ‘noise’ levels present in BVF, and therefore this discrepancy will have little effect on the overall dynamics of the system. A delay term can be included into (8) and (9) to improve the matching in the phase plot; however, it may mismatch the amplitude plot and will make controller design undesirably complicated. An alternative would be to use a non-minimum phase term, but in the long run this can cause instability.
The models Gb2 and Gb3 were validated by means of experimental comparison at similar process conditions considered in the model development. The experimental planar bed was subjected to the square wave variations in flow between 5 and 45 mm/s superficial flow above Umf and the BVF was measured (Fig. 9). This range of variation produced bubbles with similar distribution of residency in the bed used to construct Gb2 and Gb3 in this case study (refer to Fig. 6). Square wave or step variation has proved useful to investigate the dynamical behaviour of both models and to assess their time domain congruency with the experimental bed. MATLAB’s output error (OE) algorithm was used to estimate from the experimental data, in the time domain, the best-fit curve of similar form to the transfer function Gb3(s) expressed in (10), giving Gb4 ðsÞ ¼
0:0623ðs þ 5:97Þ s2 þ 2:40s þ 2:44
(10)
Gb4 is plotted in Fig. 9 overlaying the experimental curve to indicate a good fit performed by the OE algorithm. There is also good agreement of the poles and zeros of Gb4(s) with those of Gb3(s) when comparing the coefficients of (11) and (12). The constant K in (11) was set to 0.15, corresponding to the ratio of BVF to amount of flow for this case. This analysis shows that the models Gb2 and Gb3 can be successfully used to model the bubbling bed. In general, this analysis applied well to a wide range of BVF demands. The bed dynamics varied with amount of flow supply given, and this in turn resulted in different bubble residence time distribution and other dynamics characteristics. The system identifications carried out on the experimental bed with several ranges of flow variation demands are summarised in Fig. 10. Correspondingly, the numerical values in (11) and (12) varied according to different flow change conditions. Fig. 10 shows the critical dynamics parameter considered in this analysis, which is the bed’s undamped natural frequency, on. In this context, on indicates the frequency below which the BVF responds significantly to a change in gas flow rate—in this case to step changes. on is affected by the magnitude of flow variation as well as the initial condition of the bed (the extent of bubbling prior to change). Low frequency gain relates to the ratio of BVF to the amount of flow and the damping ratio describes the extent of BVF oscillation before settling to steady state. A distinctive trend of variation can be seen in Fig. 10. The first set of tests (1–4) exhibits increasing on, which corresponds to initial bubbling conditions in the bed. The more gas laden (and therefore bubbling) the bed is prior to flow changes, the faster the bed will respond, hence the larger value of on. Also, on is determined by the magnitude of flow change, as demonstrated by comparing Tests 1–4 with Tests 5–8. The larger the change is, the larger the value of on. For example, Test 5 has higher value of on compared to Test 1. This is because larger increase in flow causes a higher surge of gas and the creation of large bubbles that rise quickly up the bed. Equally, larger reduction in flow causes quicker collapse of bubbles. On both accounts, the effect on the BVF is fast compared to when the change is smaller.
4. Control of the bubbling process via the BVF measurement This section explores the ability to control the bed, demonstrating the extent to which the bubbling bed behaviour is understood. It also demonstrates a proposed method for controlling the process to achieve and maintain good fluidisation quality.
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10 Gb2 Gb3
Gain / K [dB]
0 -10 -20 -30 -40 -50 10-2
10-1
101
100 Frequency / [Hz]
0
Phase / [°]
-200
Gb2 Gb3
-400 -600 -800 -1000 10-2
10-1
101
100 Frequency / [Hz]
Fig. 8. Comparison of Bode plots of Gb2 and Gb3. Gb2 is the realistic model of the bubbling fluidised bed and Gb3 is the equivalent of Gb2 in the form of standard transfer function.
Bubble Void Fraction, BVF / [ ]
0.14 0.12 0.10 0.08 0.06 0.04 0.02 Experimental
Response of Gb4
0 50
100
150
200
Time / [s] Fig. 9. The response of the bed BVF and modelled by Gb4(s) to square wave flow rate variation (between 5 and 45 mm/s superficial flow above Umf).
The planar bed is not often used for practical purposes; however, it is the simplest system to study and allows the distribution of bubbles in the bed to be measured directly. Practical measurements of the state of any fluidised bed are dominated by the distribution of bubbles, and so the methods developed in the planar bed should be applicable to others. The ideas and knowledge derived from this study on what is required to properly control such a system will lead to better efforts in applying similar control to industrial beds.
4.1. Proportional+Integral+Derivative (PID) control The transfer function Gb4 in (10) was used for the design of the control of the BVF in the experimental bed. Given its widespread
use and practicality, a PID controller was used. It is a standard linear strategy that provides a control action that proportionally takes into account the direct and integral to the errors between the desired and measured output, as well as the rate at which the measured output changes. Fig. 11 shows the open-loop behaviour of the bubbling fluidised bed in comparison to an ideal response demand. It can be seen that the closed-loop control scheme must compensate for slightly sluggish response to changes in demand, steady-state errors, and suppress undesirable transient and steady-state oscillations of the BVF. The BVF measurement is used as the feedback signal, while the valve position admitting the gas into the bed is the control signal: its demand is determined by the error between the desired and fed-back measure of BVF. The large fluctuations in the BVF measurement could potentially saturate the control signal and
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2Umf ~77 -103 ~51 -103 ~51 -77 ~25 -77
~Umf -103
Superficial flow supply / [mm/s]
Natural frequency, n / [rad/s]
3.0
~25 -51
2.5
~Umf -51
2.0
~Umf -25 Umf
1.5
1.0
0.5
0 0
1
2
3
4
5
6
7
8
Test / [ ] Fig. 10. Composite graph showing the bed undamped natural frequency; on obtained from various system identification tests is indicated in the inset.
Bubble Void Fraction, BVF / [ ]
Desired open loop BVF Output BVF
Different fluctuation magnitudes
0.18
Time delay
0.14 Steep slope in rise
0.1
Sluggish fall in value
0.06 Undershoot in response
0.02 40
50
60
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90
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Time / [s] Fig. 11. Open loop errors between desired and measured BVF.
render the controller efforts ineffective, apart from also inducing component wear. A simple first-order low pass filter was therefore placed into the feedback loop to mitigate this effect. The low pass filter has been designed to have a first-order decay (20 dB/ decade) above the cut-off frequency of approximately 0.15 Hz (0.95 rad/s). This was sufficient to filter out undesirable excessive BVF measurement fluctuations (noise) without adversely affecting the bed response within the controller’s operating bandwidth. This filter also removed some of the faster dynamics of the bed, but this was necessary here to facilitate better control performance.
As this is an on-line and real-time control implementation, instantaneous filtered BVF values were directly measured from the bed and used to infer the state of the bed. A long record of BVF data was not necessary, in contrast with that required to generate graphs (Bode plots) to interpret the dynamics of the bed and for system identification. The implementation of the PID scheme on the fluidised bed is shown in Fig. 12. The PID controller transfer function can be expressed as 1 kp T d 2 s 1 Gc ðsÞ ¼ kp 1 þ þ Tds ¼ s þ þ T is T d T iT d s
(11)
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where kp is the proportional gain and Ti and Td are, respectively, the time constant associated with the integral and derivative gains of the controller. Revisiting the example given in Section 3, the closed-loop characteristic equation (CLCE) for Gb4 in (10), including the controller and filter dynamics, is then given by
By choosing 1/Td ¼ 2.4 and 1/TiTd ¼ 2.44, the second-order numerator term cancels the second-order denominator term in (14), reducing the CLCE to
0:0623ðs þ 5:97Þkp T d ðs2 þ ðs=T d Þ þ ð1=T i T d ÞÞð0:95=s þ 0:95Þ ¼0 sðs2 þ 2:40s þ 2:44Þ (12)
where gain kp simply dictates the closed-loop settling time of the bed. This can be rewritten as
1þ
rBVF +
eBVF
u
PID controller
yBVF
Fluidised bed (Plant)
f Filter
Value control signal / [V]
Bubble Void Fraction, BVF / [V]
Fig. 12. Block diagram of the fluidised bed closed-loop control implementation using PID controller scheme. rBVF, eBVF, u, yBVF, and f are, respectively, the demand, error, control, output and feedback signals in the closed-loop layout.
1 þ 0:026kp
0:95ðs þ 5:97Þ ¼0 sðs þ 0:95Þ
(13)
s2 þ ð0:95 þ 0:025kp Þs þ 0:15kp ¼ 0
(14)
Thus, by selecting kp in the range 1.7–3.7, a critically or slightly under-damped closed-loop response can be achieved. This ensures that there are no substantial overshoots in the BVF response. However, theoretically derived gains generally require subsequent fine tuning during experimental implementation. Additionally, there may be a slight lag between when the flow is supplied and the nucleation of visible bubbles. Accordingly, gains Ti and Td were reduced slightly below the initially designed values to moderate the integrating and derivative actions. The benefit of this is two-fold, since the presence of noise in the system also typically requires the reduction of the gains for a more desirable
0.2 BVF demand Unfiltered BVF Filtered BVF
0.15
0.1
0.05
0 600
620
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660
680
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660
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700 720 Time / [s]
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760
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0.02 0.01 0 -0.01 -0.02 -0.03 600
700
720
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Time / [s] 2 PID control demand Valve position / [V]
1.5 1 0.5 0 600
620
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680
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780
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Time / [s] Fig. 13. (a) The reference and the output signal from the experimental bed resulting from close-loop control for a demand of 0.02–0.14 BVF at 0.01 Hz step excitation using designed gains (kp ¼ 3.26, Ti ¼ 0.75, and Td ¼ 0.33). (b) The corresponding control effort exerted on the valve. (c) PID controller demand on valve position and the resulting measurement of the valve position.
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Bubble Void Fraction, BVF / [V]
control, in particular the derivative action of the controller. Online tuning for continuous assessment of controller performance was adequate to achieve general process control optimisation. In addition, a zero mean BVF demand was not practical due to noise: the minimum BVF value is clearly zero and any noise at that level will be positive only. Thus, at very low demanded values of BVF there will be a mean positive error. The rule-of-thumb is that a small non-zero BVF demand (order of 0.01) should be used for a no-bubbling demand in this bed. In an example, a closed-loop settling time, ts, of 10 s was sought after, giving TE2.5 s, hence kp ¼ 2.77. The corresponding first-order and closed-loop responses to step changes in BVF from
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0.02 to 0.14 are shown in Fig. 13. A sampling interval of 0.04 s was used. The controlled response is seen to be satisfactory. The obvious apparent drawback of a linear controller is its inability to respond to changes in system dynamics and so to maintain optimum closed-loop response. Despite this, it was applied to the bed by making a linear approximation to the non-linear bed dynamics near to the desired operating point. This non-linearity affected only slightly the closed-loop control performance features such as the settling time and steady-state value. This is shown in Fig. 14, wherein (a) a step variation in demand around high BVF bands resulted in a response that was slightly oscillatory, caused by the presence of large bubbles. The
0.24 BVF demand Unfiltered BVF Filtered BVF
0.2 0.16 0.12 0.08 0.04 600
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Bubble Void Fraction, BVF / [V]
Time / [s]
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Time / [s]
Bubble Void Fraction, BVF / [V]
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BVF demand Unfiltered BVF Filtered BVF
0.2 0.15 0.1 0.05 0 600
620
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Time / [s] Fig. 14. Implementation of control using fixed-gain policy for the PID controller scheme with gain values (kp ¼ 2.65, Ti ¼ 0.60, and Td ¼ 0.26) used for different ranges of BVF between 0 and 0.2. It can be seen that the approach is successful, but there are limitations as well. (a) BVF control result for step demand between 0.11 and 0.16 BVF, (b) BVF control result for step demand between 0.02 and 0.14 BVF and (c) BVF control result for step demand when there is too short a settling time and overshoots result.
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0.3 BVF demand BVF raw BVF
0.25 Demand = 0.2
0.2
Demand = 0.18
Demand = 0.16
0.15
Demand = 0.14
0.1 0.05
Demand = 0.02
0 0
100
Demand = 0.15
Demand = 0.08
Demand = 0.04 Demand = 0.02
200
300
Demand = 0.06
400
Demand ~ 0.00
500
600
Time / [s] Fig. 15. The time series of the closed-loop control of the process with PID controller conducted over a range of BVF demand.
response was also sluggish, with typical ts of about 20 s as the controller struggled to converge to the desired state using a persistently more noisy BVF feedback measurement, even when filtered. In Fig. 14(b), when a larger range of demanded BVF levels was imposed, better control was exhibited. The controller also coped with demand close to Umf. The filtered BVF feedback was considerably less noisy at lower BVF, hence giving a better control performance as can be seen in the figure. When designing the controller, choosing too fast a settling time can result in an undesirable outcome. Fig. 14(c) shows the response achieved with the same gains, used for previous cases, for a demand varying between 0.045 and 0.16. At ts ¼ 5 s, while the bed responded well to step-up in demand, rapid response to step-down demand was significantly oscillatory. Fast settling time demand consequently forced the valve to be operated quickly, opening and shutting the flow rapidly. This did not pose problems during flow increase. However, when the flow supply to the bed was abruptly reduced, causing the bed suspension to collapse under its own weight, bubbles were dissolved into the defluidised emulsion phase. This phenomenon would not have been readily identifiable during the system identification process and hence, equally, not accounted for during controller design. Furthermore, as a result of closed-loop control, this occurrence caused the controller to react by reopening the valve rapidly in order to compensate and minimise the error, thus causing undesirable oscillation of BVF. Also, at low flow rates the mean bubble size was small and mean speed was low; so the mean bubble residence time was longer than at high flow rates. This slightly slowed the bed’s response, and therefore the time constant of the bed on the step-down response. Therefore, a more conservative settling time employed earlier was more favourable in achieving the desired bed response in practice. Although during system modelling the full dynamics range of the bed is of interest for control, the gross behaviour of the bed was focused on, i.e. the range of dynamics that determines the macroscopic behaviour of the bed. The fast bed dynamics contained within the BVF measurement manifest themselves as fluctuations that affect control efforts. To obtain and facilitate control, the BVF measurements were subjected to a low pass filter with break frequency at 0.15 Hz. This frequency is slow enough to largely remove the ‘noise’ component of the BVF. Despite this filter, as indicated in Fig. 14(c), the controller still achieved a settling time ts ¼ 5 s, for step-up in BVF demand. Fig. 15 shows the PID controller’s performance in controlling the process over different BVF demands. The controller was designed with fixed gains based around the dynamics of the bed at an intermediate range of bubbling, thus ensuring acceptable responses (settling time, steady-state error and stability) over different BVF demands—including close to Umf.
The studies in this paper were conducted in a planar fluidised bed, so allowing the distribution of bubbles to be seen clearly and the state of the bed measured explicitly. Commercial fluidised beds are usually cylindrical in shape. Therefore, a different means of measurement is required, where pressure measurements conventionally come into place. Pressure fluctuations in a fluidised bed can be measured non-intrusively and are directly generated by the presence of the bubbles. Therefore, their dynamics could be directly related to those described in this paper, and similar methods of analysis, monitoring and control could be applied. The work described above demonstrates the ability to control the bubbling process, provided that the elements in the control strategy were properly considered. It should be possible to implement similar control using e.g. pressure measurements rather than BVF to indicate the state of the bed (Croxford, Lim, Gilbertson, & Harrison, 2004).
5. Conclusions An instrumentation system based on real-time imaging technique has been employed to detect and measure the bubbling phase in a planar fluidised bed. The bubbling process in the bed has been characterised using the developed system and the bubbling properties of the bed have been identified. The bubble void fraction, BVF, was measured and shown to relate consistently to the amount of gas supply above quiescent fluidisation. Simple yet representative models in the form of transfer functions have been subsequently developed to describe the dynamic behaviour of bubbles as well as the response of the bed to excitation in flow supply relevant in efforts to design a suitable controller for the process. Measurement of BVF was then used as the feedback of closedloop control implementation of the bubbling process via a fixedgain PID controller. This was able to control the BVF of the bed to follow various bubbling state demands. Successful control implementation using this technique demonstrates that our understanding of the dynamics of the bubbling process is adequate, and that the bed’s dynamics can be confidently approximated to be linear within this fluidisation regime. By extension, the dynamic content of pressure measurements, which similarly relates to the fluidisation quality of the bed, may be used conveniently instead of the BVF as the feedback signal in industrial beds.
Acknowledgements The authors would like to extend their gratitude to the UK government for the Overseas Research Studentship (ORS) scheme
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funding the scholarship of the project, and to British Nuclear Fuels plc (BNFL) for research equipment support. References Brown, R. C., & Brue, E. (2001a). Resolving dynamical features of fluidized beds from pressure fluctuations. Powder Technology, 119, 68–80. Brown, R. C., & Brue, E. (2001b). Use of pressure fluctuations to validate hydrodynamic similitude in fluidized media: Bubbling beds. Powder Technology, 119, 117–127. Chong, Y. O., O’Dea, D. P., White, E. T., Lee, P. L., & Leung, L. S. (1987). Control of the quality of fluidization in a tall bed using the variance of pressure fluctuations. Powder Technology, 53, 237–246. Clift, R., & Grace, J. R. (1970). Bubble interaction in fluidized beds. Chemical Engineering Progress Symposium Series, 66(105), 14–27. Clift, R., & Grace, J. R. (1971). Coalescence of bubbles in fluidized beds. AIChE Symposium Series, 67(116), 23–33. Croxford, A. J., Lim, C. N., Gilbertson, M. A., & Harrison, A. J. L. (2004). Dynamics and control of fluidised beds. Fluidization XI, U.E.F., 475–482. Eames, I., & Gilbertson, M. A. (2005). Mixing and drift in gas-fluidised beds. Powder Technology, 154(2–3), 195–203. Harrison, D., Davidson, J. F., & de Kock, J. W. (1961). On the nature of aggregative and particulate fluidisation. Transactions of the Institution of Chemical Engineers, 39, 202–211.
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Hilgert, N., Harmand, J., Steyer, J.-P., & Vila, J.-P. (2000). Nonparametric identification and adaptive control of an anaerobic fluidized bed digester. Control Engineering Practice, 8, 367–376. Kaart, S., Schouten, J. C., & Bleek, C. M. (1999). Improving conversion and selectivity of catalytic reactions in bubbling gas-fluidized bed reactors by control of the nonlinear bubble dynamics. Catalysis Today, 48, 185–194. Korte, R. J., Schouten, J. C., & van den Bleek, C. M. (2001). Controlling bubble coalescence in a fluidized bed model using bubble injection. AIChE Journal, 47(4), 851–861. Kunii, D., Yoshida, K., & Hiraki, I. (1967). The behaviour of freely bubbling fluidized beds. In Proceedings to the international symposium on fluidization (pp. 243–256). Lee, P. L., Chong, Y. O., & Leung, L. S. (1986). Experimental investigation of the quality control of fluidized beds. Powder Technology, 46, 77–80. Lim, C. N. (2004). The dynamics and control of bubbling fluidised beds. Ph.D. dissertation (p. 267), University of Bristol, UK. Makkawi, Y. T., & Wright, P. C. (2002). Optimization of data acquisition rate and experiment span for reliable ECT measurement in fluidized beds. Measurement Science and Technology, 13(12), 1831–1841. Rowe, P. N., Partridge, B. A., & Lyall, E. (1964). Cloud formation around bubbles in gas fluidized beds. Chemical Engineering Science, 19, 973–985. Simoglou, A., Martin, E. B., & Morris, A. J. (2000). Multivariate statistical process control of an industrial fluidised-bed reactor. Control Engineering Practice, 8, 893–909.
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Characterisation and control of bubbling behaviour in gas– solid fluidised beds C.N. Lim, M.A. Gilbertson , A.J.L. Harrison Department of Mechanical Engineering, University of Bristol, Queen’s Building, University Walk, Bristol BS8 1TR, UK
a r t i c l e in fo
abstract
Article history: Received 9 August 2005 Accepted 21 May 2008 Available online 24 June 2008
Bubbling fluidised beds are often exploited for the good mixing of phases they promote, which is required in many chemical processes. However, the bubbles cause heterogeneity in the fluidised system, and their dynamics must be understood for good process control. In this paper, the dynamic behaviour of the bubbles in a planar gas–solid fluidised bed is described analytically and experimentally validated. The bed resembles a temporary buffer of gas for which the bubble residence time in the bed is important. Subsequently, a closed-loop controller was developed using visual feedback to regulate the process. & 2008 Elsevier Ltd. All rights reserved.
Keywords: Dynamic modelling Process control Chemical process System identification PID control
1. Introduction Fluidised beds are used in a variety of industrial processes, and typically comprise a column of solid particles resting on a porous surface. Gas is introduced through the floor at a flow rate sufficient to support the weight of the powder. This nearly eliminates the friction between particles, allowing them to behave in a similar manner to a fluid. Any industrial process that requires good contacting or effective mass and heat transfer to obtain high degree of efficiency is well-catered for by the attributes of the fluidisation process. Industrial applications of fluidised beds exploit the high particle surface area that exists within a relatively small volume of bed for operations such as drying, heat transfer, combustion and chemical reaction between the gas and particles. One of their principal virtues is the degree of mixing between the particles and gas. When the gas flow speed exceeds that necessary to fully support the particle (the velocity of minimum fluidisation, Umf), particlefree voids or bubbles form. These enhance the mixing of particles throughout the bed so that their properties tend towards uniformity (Eames & Gilbertson, 2005); however, the gas within these bubbles does not serve its primary purpose of reacting with the solids. The bubbles also introduce heterogeneity to the process, which reduces the efficiency of contact between solid and gas, and, as they usually travel faster through the bed than the surrounding gas in the emulsion phase, the residence time
Corresponding author. Tel.: +44 117 9739732; fax: +44 117 9294423.
E-mail address: [email protected] (M.A. Gilbertson). 0967-0661/$ - see front matter & 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2008.05.007
associated with the gas in the bubbles is reduced (Kaart, Schouten, & Bleek, 1999). There is little exchange of the gas from within a bubble and outside of it (Rowe, 1964); furthermore, the speed of a bubble increases with size, reducing the residence time of the gas. The result is that, in a bed with large fast-moving bubbles, a large proportion of the gas introduced into the bed does not adequately contact the particles. Therefore, a general definition of good fluidisation quality is that there are large numbers of ‘smallish’ bubbles homogeneously distributed throughout the bed. Whilst the aim of monitoring and controlling the process is to achieve high-quality fluidisation, this can be difficult to achieve practically for a number of reasons. In particular, small bubbles tend to coalesce as they move up the bed to form less desirable large ones, which move quicker up the bed further diminishing particle–gas contact. Harrison, Davidson, and de Kock (1961) pointed out that the quality of fluidisation is also affected by several factors other than gas velocity, such as the solid and gas density ratio, solid particle size, operating pressure and temperature, and fluidising gas viscosity. Depending on these factors, a fluidised bed may be in any one of a complete range of fluidisation regimes from smooth to slugging. By controlling the choice of solid and fluidising medium characteristics, the fluidisation mode and hence the quality, depending on the type of operation, could also be controlled. However, often this is not possible when the operating parameters have already been determined and the quality of fluidisation begins to vary during the process operation. Then, a dynamic means of control is required—e.g. by controlling the feed rate of solids or gas. The ability to control the state of the bed becomes more crucial when factors inherent in the bed change, e.g. the gas properties
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may vary with time due to humidity or viscosity changes; particles may agglomerate or fragment, or new particles with different properties may be introduced to the bed. These will then affect bubble formation and behaviour. Any controller must be capable of detecting and reacting appropriately with changing process dynamics to continuously deliver the desired performance. Furthermore, time-series measurements of bubbling fluidised beds are generally ‘noisy’—pressures, bed heights, bubble sizes and distributions, etc. exhibit a significant level of fluctuation. Such systems are known to be difficult to control because, if not properly accounted for in the controller design, the closed-loop nature of typical controller implementation means that noise can swamp the error feedback loop. This can result in control signal saturation and failure of the controller. In addition to noise, the behaviour of the particles and bubbles in the bed varies non-linearly as the flow of gas into a bed is increased from zero. The onset of fluidisation at Umf is particularly significant: at this flow rate the bed experiences a discontinuous state change between being a fixed bed in which friction dominates, into a suspended mobile bulk of particles and, in certain cases, the immediate onset of bubbling. Across the whole regime of fluidisation, the bed undergoes a series of dynamics transitions with increase of flow supply. Typically, there is also some form of delay or lag whereby the state of the bed does not immediately respond to changes applied to the gas supply. All the above-mentioned factors make fluidised beds difficult to monitor and control. Many techniques to monitor and control processes have been devised and applied industrially. Numerous methods for controlling the bubbling process have been suggested; from Lee, Chong, and Leung (1986), Chong, O’Dea, White, Lee, and Leung (1987) developed a control technique employing pressure measurement in a tall bed to govern the control of amount of excess gas expelled by bleed valves installed at consecutive heights in the bed. The technique proved to be successful and practical. Korte, Schouten, and van den Bleek (2001) suggested an alternative method of controlling the nucleation of bubbles in the bed such that coalescence was controlled and suppressed to an acceptable amount to maintain desired bubble population and distribution. In addition, statistical control methods have been used to control processes that rely on fluidised beds (e.g. Hilgert, Harmand, Steyer, & Vila, 2000; Simoglou et al., 2000). However, these developments have not fully addressed the causal relationship
between the control and monitored parameters, i.e. the process dynamics are not fully understood. This paper describes an attempt to tackle this issue by identifying and interpreting the dynamics of bubbles in a planar fluidised bed. From this, controllers are designed and implemented. The structure of the paper is as follows: In Section 2, the experimental arrangement is described. The steady-state and transient responses of the bed to a step change in gas flow rate are examined. For dynamical modelling purposes, the bed can be viewed as a temporary storage for gas. In Section 3, the development of this idea leads to the derivation of transfer functions relating the gas flow rate contained in bubbles to the bubble void fraction (BVF), this being the fraction of the bed’s volume that is occupied by bubbles. From these observations, it is possible to control the BVF using closed-loop linear control strategies, implemented and detailed in Section 4.
2. Experimental observations of a 2-D fluidised bed This paper describes the modelling and control of the fluidisation quality of a bubbling bed. One of the difficulties with practical fluidised beds is that they are opaque and so it is difficult to measure the number and distribution of bubbles within them. To allow unambiguous measurement, an experimental Perspexwalled two-dimensional bed and a real-time image analysis technique were used to directly measure the condition of the bubbling in the bed. This allowed the explicit observation of the behaviour of bubbles and the bed fluidisation quality to be determined. The quality of the bed was interpreted from the number and distribution of bubbles. The relationship between them and the gas flow rate was subsequently explored experimentally and theoretically. Fig. 1 shows a schematic diagram of the experimental fluidised bed and monitoring equipment. The bed was b ¼ 500 mm wide, h ¼ 1000 mm high, l ¼ 13 mm thick and so was effectively twodimensional. The bubbles were viewed by illuminating the Perspex walls from behind. The particles were Ballotini glass beads of bulk density 1500 kg/m3, approximately 80% spherical, and with a diameter range 106–212 mm. These filled the bed to a height of approximately 690 mm in the absence of gas flow. Bubbles were seen to form in the bed when incipiently fluidised (UmfEUmb), with mild expansion (3–5% of ungassed bed height) of the bed. Compressed air was used to fluidise the bed, and its
Planar fluidised bed
Feed back loop Image analysis
ControlLab
Computer
Frame grabber
Camera Monitor
D/A converter
Forward control loop
Actuation system
Air supply Fig. 1. Experimental fluidised bed and monitoring equipment.
Fluorescent light tubes
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flow rate qin was set by a servomotor-controlled valve and introduced uniformly into the base of the bed via a porous plate distributor. The distributor gave consistent uniform flow, with formation of small bubbles evenly distributed symmetrically across it. No channelling or concentration of flow nucleation sites on the plate was observed during any bubbling conditions and the windbox pressure was stable and remained constant throughout the cross-section. The valve position was measured via a potentiometer and AD/DA card in a computer. An image analysis system was developed, comprising a CCD array camera, frame grabber, software and the computer. The software used for these purposes was developed in-house at the University of Bristol. Bubbles in the bed were detected by the transparent areas they created on the flat cross-sectional plane of the planar bed through which illumination placed at the back of the bed passed. Fig. 2 shows a sequence of typical planar bed bubbling images captured by the imaging system. By analysing long sequences of these, information such as the proportion of bubbling phase, bubble size, spatial and temporal population, and other statistical measures that relate to the state of the bed were obtained, along with the uncontrolled bed dynamics. The system was able to identify the positions and sizes of the bubbles to be identified at up to 25 frames/s. This rate was rapid enough to allow real-time control of the bed.
2.1. Steady-state relationships to flow rate A number of steady-state or time-averaged measurements were obtained from the experimental fluidised bed, and in each case the principal input to the plant was the gas flow rate qin. The measurements were all taken from a flow rate datum based on the value of Umf, which corresponds to the onset of fluidisation, and bubbles begin to form at the distributor. Umf is thus seen as a critical value at which a discontinuity in system dynamics occurs: from fixed bed behaviour, when fluidised, the bulk of the particles undergo a drastic change in their hydrodynamical behaviour as the interaction between the gas and solid phase is completely altered. Umf was found to be approximately 103 mm/s of superficial gas flow. All measurements and analyses described in this paper were carried out with flows measured relative to the Umf ( ¼ Umb) datum. To characterise the bubbling process effectively, and capture the dynamical behaviour of the bed, a suitable parameter had to be determined. The parameter should explicitly and effectively correspond to the state of the process it was measured at. Relatively simple and fast to obtain, it must be consistent over a range of process conditions. According to Brown and Brue (2001a, 2001b), the minimum sampling frequency required to appropriately investigate the condition of the fluidisation process
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should be at least 20 Hz in order to capture all the dynamical features and to avoid anti-aliasing of fast dynamics. They showed from their experiments (Brown & Brue, 2001a) with pressure measurements that there are no differences in high-frequency spectral content obtained with the sampling rates between 20 and 100 Hz. Makkawi and Wright (2002) used electrical capacitance tomography to demonstrate that the optimum recording span for reliable analysis of the solid–gas hydrodynamics ranges from 20 to 100 s, while the acquisition rate should range between 20 and 100 Hz. The present image analysis system has a limited acquisition rate of 25 Hz, thus requiring that any measurement made to investigate the dynamics of the fluidised system should span an adequate period. Periods of 20–60 min periods were used, which are considerably longer than that recommended by Makkawi and Wright (2002). However, this enabled all the dynamics of the process to be captured, including very slow ones, and to generate reliable results despite the significant noise levels previously mentioned. The higher acquisition rate in Makkawi and Wright’s work was necessary for measurement of solid fraction in each slice of bed cross-section over a chosen height section (to produce a three-dimensional map), whereas a lower rate is acceptable for acquisition of BVF measurement on the two-dimensional planar bed applied here. Fig. 3(a) shows a typical steady-state measurement of BVF for a time span of 150 s acquired at the rate of 25 Hz. The apparent noisiness of BVF occurs principally because, whilst the gas enters the bed at an essentially constant rate, it manifests itself in the bed in a more random manner as bubbles form at the bottom and the gas leaves at the top as large discrete bubbles. When a large bubble leaves the bed, there is a corresponding significant drop in BVF. Fig. 3(b) shows the relationship between BVF and gas flow rate qin. There are two obvious regimes: below the minimum fluidisation flow rate Umf no bubbles are formed; at rates above Umf the BVF increases non-linearly with flow rate. The nonlinearity occurs because the average bubble size increases with flow rate owing to increased likelihood of coalescence, and larger bubbles move more rapidly than small ones; thus, the average residence time of bubbles within the bed decreases with increasing flow rate. Values shown amidst the graph are the means of BVF at each flow condition. The BVF measurement resembles a stair-like graph corresponding to step increase in flow. The changes in the time series with flow rate indicate that the measure of BVF was effective and robust for a range of bubbling condition, despite the presence of fluctuations in the signal. The 95% confidence limits are also shown on the graph for approximately 2.5 s of BVF time series for different bed conditions, all acquired at 10 Hz rate
Fig. 2. Sequence of snapshots (intervals 40 ms) showing a typical bubbling bed in a planar bed as captured by the imaging system.
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Fig. 3. (a) Variation in mean BVF for a constant superficial flow rate of 12.8 mm/s above Umf, (b) experimental BVF against superficial gas flow rate above Umf.
producing approximately 25 data points. However, the reliability of the measurement gradually decreases because the BVF standard deviation increases with flow rate. This is due to the growing fluctuation magnitude caused by the presence of large bubbles and their departure from the bed. This fluctuation is several orders of magnitude greater than any other fluctuations in the BVF measurement, such as those caused by light source fluctuation and hardware electronic noise. Small variations in flow rate (say 72.6 mm/s superficial flow) during bubbling were experimentally observed not to cause noticeable change to the state of the bed and the BVF. Fig. 4(a) shows the variation in the average temporal population of bubbles in the bed with respect to flow supply. Initially, above Umf there was an increase in bubble population with increase in gas flow. The bubble population peaked at approximately 52 mm/s of superficial gas flow and then declined because of the greater coalescence in the freely bubbling bed, creating
larger, faster bubbles. These bubbles leave the bed quickly, occasionally exhausting the bed of bubbles by sweeping through the bed and coalescing with more bubbles in their path, creating a few large ones. This phenomenon also increases the standard deviation in the measure of bubble population. Fig. 4(b) shows the size distribution of bubbles with flow rate. It should be noted that bubbles of diameter less than the bed thickness are unlikely to be detected by the imaging system; however, these do not contribute significantly to the overall BVF measurement. The average bubble size increases approximately linearly with flow. At constant flow rate, the number and mean size of bubbles are subject to large variability, and therefore would be difficult to control by varying the flow rate. However, the BVF, which is essentially the sum of the areas of each bubble, has a monotonic relationship with flow rate and is therefore more readily controllable. Moreover, because of the direct link between the
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and there is no coalescence), then
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Bubble population / [ ]
65 60
(2)
(3)
0
55
T
qin ðtÞ qin ðt TÞ dt
The Laplace transform that describes this is
50
VðsÞ ¼
45
1 Q ðsÞ½1 esT s in
(4)
Also, the BVF is
40
BVFðtÞ ¼ Mean value Best fit (cubic) Standard deviation error bar
35 30 10
20
30
40
50
60
70
80
90
100 110
Flow rate above Umf / [mm/s]
Standard deviation error bar Mean value
160 140 120 100 80 60 40 10
20
30
40
50
60
70
80
90
100 110
Flow rate above Umf / [mm/s] Fig. 4. (a) Experimental variation of number of bubbles with gas flow rate, (b) experimental variation in mean bubble diameter with superficial gas flow above Umf.
state of the bed and gas flow rate, the BVF is a useful parameter to control as it is a reasonable indicator of the overall state of the bed.
3. Modelling the dynamic response of the bed 3.1. A simple model of a bubbling fluidised bed From the point of view of its dynamics, a bubbling bed can be considered as a temporary store for the gas supplied. Bubbles can be characterised as packets of excess gas which reside for a given amount of time determined by their physical properties before leaving at the freeboard. The volume v(t) of the bubbles in the bed at time t is given by Z t vðtÞ ¼ qin ðtÞ qout ðtÞ dt (1) 0
where qin(t) and qout(t) are, respectively, the flows (in excess of Umf) into and out of the bed at time t. If all the gas bubbles take T seconds to pass through the bed (i.e. they all have the same size
vðtÞ ¼ KvðtÞ bhl
(5)
where b, h and l are the bed’s width, height and thickness, while K is a proportionality constant relating the amount of gas supplied to the corresponding area of bubble void generated in the bed. Thus, the bed’s transfer function relating BVF to gas flow rate into the bed may also be expressed as Gb1 ðsÞ ¼
180
Average bubble size / [mm2]
qout ðtÞ ¼ qin ðt TÞ whence Z vðtÞ ¼
BVFðsÞ K ¼ ½1 esT Q in ðsÞ s
(6)
This simple model of the bed Gb1(s) is then a ‘zero-order hold’ transfer function with low frequency gain KT, i.e. in this case, the bed’s steady-state value of BVF. When there is a step change in flow into the bed, bubbles are generated and the BVF ramps up at a constant rate for T seconds, where T corresponds to the bubble residence time in the bed. Thereafter, as more bubbles are introduced into the bed, bubbles simultaneously exit at the top in equal quantity and the BVF is constant as long as the supply of bubbles (i.e. the gas supply) does not change. When the gas supply cuts off, the BVF ramps down at a similar rate. The Bode plot corresponding to Eq. (6) is shown in Fig. 5. The transfer function is approximately that of a system of relative degree one (i.e. first-order, or second-order denominator with first-order numerator) combined with a delay term. This gives it a decay rate of 20 dB/decade in the plot after the break frequency. These salient features are typical of a transfer function that describes the integrating effect of the bubbling process on the BVF measurement. It terminates at the break frequency with a flat DC level, due to the limit of the minimum size of bubble present in the bed (Lim, 2004). Additionally, the gain is effectively zero at integer multiples of 1/T. This occurs physically because at such frequencies the transport delay T ensures that the flow rates in and out of the bed are equal. In other words, these intermittent ‘dips’ correspond to the residence times of bubbles generated in the bed and the accompanying harmonic frequencies are anti-resonance frequencies at which the introduction of bubbles is just balanced by the bubbles leaving at the freeboard. At these frequencies, the change in BVF is small, i.e. the transfer function has low energy. The decaying trend of the phase part of the diagram is characteristic of a dynamical system containing elements of time delay. The model used to obtain Eq. (6) was kept simple, resulting in unrealistic artefacts such as sharp changes in gradients by assuming that bubbles are created with similar sizes and their size remained constant whilst in the bed. In this way, the computation of BVF was discontinuous at the zero–nonzero transition and increased in a linear ramp fashion. 3.2. A realistic model of a bubbling fluidised bed The above analysis assumes that all the bubbles have equal residence time T, and therefore are of the same size; however, in
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Gain Magnitude / [dB]
-60 -80
-20 dB/decade
-100 -120 -140 10-2
10-1
100
101
100
101
Frequency / [Hz]
Phase of signal / [°]
0 decay -50 -100 -150 -200 10-2
10-1 Frequency / [Hz]
Fig. 5. Bode plots of the simple model of the bed, Gb1 as defined by (7).
practice, larger bubbles travel more rapidly than small ones. When a bubble is introduced into the bed, isolated from any effects and interactions, it rises steadily, spending time T in the bed. In a bubbling bed where the single bubbles are free to interact and coalesce with other bubbles, their residence times could be significantly reduced. There is a distribution of possible residence time associated with a bubble of a specific size, ranging from the maximum amount of time for an isolated bubble to a minimum duration associated with a bubble that has coalesced with other bubbles, so forming a larger, faster bubble. A simulation replicating the experimental bed was developed based on the bubble interaction model of Clift and Grace (1970, 1971). The simulation describes an incipiently fluidised bed where, upon introduction of flow, bubbles were created at the bottom in an evenly distributed fashion, both spatially and in size range. In addition to coalescence, bubbles expand as they rise up the bed based on the linear relationship between equivalent diameter and height in the bed as proposed by Kunii, Yoshida, and Hiraki (1967). Full details of the simulation can be found in Lim (2004). An example is shown in Fig. 6 for the bubble residence time distribution (for 100,000 bubbles) in the simulated normally bubbling bed fluidised at 25.6 mm/s superficial gas flow above Umf with a superimposed uniformly random flow variation of 720.5 mm/s. The distribution gives an indication of the probability that a bubble of a specific size will spend a given amount of time in the bed. The distribution in Fig. 6 shows that, while most bubbles spent about 2.2 s in the bed, the approximately normally distributed profile is flanked by small bubbles which will have risen without much interaction with others, and by bubbles that have experienced large amount of coalescence and risen quickly
Fig. 6. The histogram of bubble residence time for a total of 100,000 bubbles generated in the simulated bubbling bed fluidised at 25.6 mm/s superficial gas flow above Umf with a uniformly random variation of flow into plenum of 720.5 mm/s.
through the bed. The profile of the distribution changes for different bubbling conditions depending on the characteristics of bubble generation and the consequent interaction and coalescence behaviour of bubbles. A modified transfer function Gb2 accommodates this range of values of T:
Gb2 ðsÞ ¼
4:3 X T¼1:1
fðTÞ
KðTÞ ½1 esT s
(7)
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3.4. Model benchmarking with experimental equivalent
1.4 Step response / [K]
1.2 1 0.8
Gb2 Gb3
0.6 0.4 0.2 0 0
1
2
3
4
5
Time / [s] Fig. 7. Comparison of the step response between Gb2, which is the realistic model of the bubbling fluidised bed, and Gb3, shown in (10) in Section 3.3, which is the equivalent of model Gb2 in the form of standard transfer function.
where f(T) is the probability density function of bubble residence time obtained from the histogram shown in Fig. 6. Also, assuming the simplified case where K is unity, the corresponding step response of Gb2 is shown in Fig. 7. The effect of using a range of values of T is to smooth the transition between the initial rise and steady-state values. A summation operator was used in (7) instead of an integral operator to emphasise the fact that Gb2 is a composite model of individual basic equations in (6), each for a specific residence time T. 3.3. Model form tailoring for controller design purposes Having examined the bed’s dynamic response, a second objective was to implement closed-loop control of the BVF. Whilst Gb2 is believed to represent the dynamics of the bed tolerably accurately, it is not well suited to controller design as the exponential term in (7) has infinitely many roots and complicates the design procedure. For this reason, a further transfer function Gb3 was sought: this was intended to have a similar step response to Gb2, but its numerator and denominator are polynomials in ‘s’, which facilitates conventional controller design. Gb3 was derived so as to have similar Bode plots to Gb2 as well. For this reason, it was chosen to have the general form: Gb3 ðsÞ ¼
lo2n Tðs þ 1=TÞ þ 2zon s þ o2n
s2
(8)
where dynamic terms l ¼ low frequency gain, T ¼ time constant, z ¼ damping coefficient and on ¼ undamped natural frequency. The best curve-fitting values of the parameters l, T, z and on were found using MATLAB’s fminsearch algorithm: l ¼ 1.00 K; K ¼ 1; T ¼ 0.155 s; z ¼ 0.82; on ¼ 1.60 rad/s, i.e. Gb3 ðsÞ ¼
73
0:40Kðs þ 6:45Þ s2 þ 2:63s þ 2:56
(9)
As this is a curve-fitting procedure, the variables do not directly have any physical and/or dynamic implications. These values ensure that the step response of Gb3 is very similar to that of Gb2, as shown in Fig. 7, and their Bode plots (Fig. 8) are in good agreement for frequencies up to around 0.4 Hz. At higher frequencies there is a marked discrepancy in phase; however, the gain is small at such frequencies compared to the ‘noise’ levels present in BVF, and therefore this discrepancy will have little effect on the overall dynamics of the system. A delay term can be included into (8) and (9) to improve the matching in the phase plot; however, it may mismatch the amplitude plot and will make controller design undesirably complicated. An alternative would be to use a non-minimum phase term, but in the long run this can cause instability.
The models Gb2 and Gb3 were validated by means of experimental comparison at similar process conditions considered in the model development. The experimental planar bed was subjected to the square wave variations in flow between 5 and 45 mm/s superficial flow above Umf and the BVF was measured (Fig. 9). This range of variation produced bubbles with similar distribution of residency in the bed used to construct Gb2 and Gb3 in this case study (refer to Fig. 6). Square wave or step variation has proved useful to investigate the dynamical behaviour of both models and to assess their time domain congruency with the experimental bed. MATLAB’s output error (OE) algorithm was used to estimate from the experimental data, in the time domain, the best-fit curve of similar form to the transfer function Gb3(s) expressed in (10), giving Gb4 ðsÞ ¼
0:0623ðs þ 5:97Þ s2 þ 2:40s þ 2:44
(10)
Gb4 is plotted in Fig. 9 overlaying the experimental curve to indicate a good fit performed by the OE algorithm. There is also good agreement of the poles and zeros of Gb4(s) with those of Gb3(s) when comparing the coefficients of (11) and (12). The constant K in (11) was set to 0.15, corresponding to the ratio of BVF to amount of flow for this case. This analysis shows that the models Gb2 and Gb3 can be successfully used to model the bubbling bed. In general, this analysis applied well to a wide range of BVF demands. The bed dynamics varied with amount of flow supply given, and this in turn resulted in different bubble residence time distribution and other dynamics characteristics. The system identifications carried out on the experimental bed with several ranges of flow variation demands are summarised in Fig. 10. Correspondingly, the numerical values in (11) and (12) varied according to different flow change conditions. Fig. 10 shows the critical dynamics parameter considered in this analysis, which is the bed’s undamped natural frequency, on. In this context, on indicates the frequency below which the BVF responds significantly to a change in gas flow rate—in this case to step changes. on is affected by the magnitude of flow variation as well as the initial condition of the bed (the extent of bubbling prior to change). Low frequency gain relates to the ratio of BVF to the amount of flow and the damping ratio describes the extent of BVF oscillation before settling to steady state. A distinctive trend of variation can be seen in Fig. 10. The first set of tests (1–4) exhibits increasing on, which corresponds to initial bubbling conditions in the bed. The more gas laden (and therefore bubbling) the bed is prior to flow changes, the faster the bed will respond, hence the larger value of on. Also, on is determined by the magnitude of flow change, as demonstrated by comparing Tests 1–4 with Tests 5–8. The larger the change is, the larger the value of on. For example, Test 5 has higher value of on compared to Test 1. This is because larger increase in flow causes a higher surge of gas and the creation of large bubbles that rise quickly up the bed. Equally, larger reduction in flow causes quicker collapse of bubbles. On both accounts, the effect on the BVF is fast compared to when the change is smaller.
4. Control of the bubbling process via the BVF measurement This section explores the ability to control the bed, demonstrating the extent to which the bubbling bed behaviour is understood. It also demonstrates a proposed method for controlling the process to achieve and maintain good fluidisation quality.
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10 Gb2 Gb3
Gain / K [dB]
0 -10 -20 -30 -40 -50 10-2
10-1
101
100 Frequency / [Hz]
0
Phase / [°]
-200
Gb2 Gb3
-400 -600 -800 -1000 10-2
10-1
101
100 Frequency / [Hz]
Fig. 8. Comparison of Bode plots of Gb2 and Gb3. Gb2 is the realistic model of the bubbling fluidised bed and Gb3 is the equivalent of Gb2 in the form of standard transfer function.
Bubble Void Fraction, BVF / [ ]
0.14 0.12 0.10 0.08 0.06 0.04 0.02 Experimental
Response of Gb4
0 50
100
150
200
Time / [s] Fig. 9. The response of the bed BVF and modelled by Gb4(s) to square wave flow rate variation (between 5 and 45 mm/s superficial flow above Umf).
The planar bed is not often used for practical purposes; however, it is the simplest system to study and allows the distribution of bubbles in the bed to be measured directly. Practical measurements of the state of any fluidised bed are dominated by the distribution of bubbles, and so the methods developed in the planar bed should be applicable to others. The ideas and knowledge derived from this study on what is required to properly control such a system will lead to better efforts in applying similar control to industrial beds.
4.1. Proportional+Integral+Derivative (PID) control The transfer function Gb4 in (10) was used for the design of the control of the BVF in the experimental bed. Given its widespread
use and practicality, a PID controller was used. It is a standard linear strategy that provides a control action that proportionally takes into account the direct and integral to the errors between the desired and measured output, as well as the rate at which the measured output changes. Fig. 11 shows the open-loop behaviour of the bubbling fluidised bed in comparison to an ideal response demand. It can be seen that the closed-loop control scheme must compensate for slightly sluggish response to changes in demand, steady-state errors, and suppress undesirable transient and steady-state oscillations of the BVF. The BVF measurement is used as the feedback signal, while the valve position admitting the gas into the bed is the control signal: its demand is determined by the error between the desired and fed-back measure of BVF. The large fluctuations in the BVF measurement could potentially saturate the control signal and
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75
2Umf ~77 -103 ~51 -103 ~51 -77 ~25 -77
~Umf -103
Superficial flow supply / [mm/s]
Natural frequency, n / [rad/s]
3.0
~25 -51
2.5
~Umf -51
2.0
~Umf -25 Umf
1.5
1.0
0.5
0 0
1
2
3
4
5
6
7
8
Test / [ ] Fig. 10. Composite graph showing the bed undamped natural frequency; on obtained from various system identification tests is indicated in the inset.
Bubble Void Fraction, BVF / [ ]
Desired open loop BVF Output BVF
Different fluctuation magnitudes
0.18
Time delay
0.14 Steep slope in rise
0.1
Sluggish fall in value
0.06 Undershoot in response
0.02 40
50
60
70
80
90
100
110
120
Time / [s] Fig. 11. Open loop errors between desired and measured BVF.
render the controller efforts ineffective, apart from also inducing component wear. A simple first-order low pass filter was therefore placed into the feedback loop to mitigate this effect. The low pass filter has been designed to have a first-order decay (20 dB/ decade) above the cut-off frequency of approximately 0.15 Hz (0.95 rad/s). This was sufficient to filter out undesirable excessive BVF measurement fluctuations (noise) without adversely affecting the bed response within the controller’s operating bandwidth. This filter also removed some of the faster dynamics of the bed, but this was necessary here to facilitate better control performance.
As this is an on-line and real-time control implementation, instantaneous filtered BVF values were directly measured from the bed and used to infer the state of the bed. A long record of BVF data was not necessary, in contrast with that required to generate graphs (Bode plots) to interpret the dynamics of the bed and for system identification. The implementation of the PID scheme on the fluidised bed is shown in Fig. 12. The PID controller transfer function can be expressed as 1 kp T d 2 s 1 Gc ðsÞ ¼ kp 1 þ þ Tds ¼ s þ þ T is T d T iT d s
(11)
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where kp is the proportional gain and Ti and Td are, respectively, the time constant associated with the integral and derivative gains of the controller. Revisiting the example given in Section 3, the closed-loop characteristic equation (CLCE) for Gb4 in (10), including the controller and filter dynamics, is then given by
By choosing 1/Td ¼ 2.4 and 1/TiTd ¼ 2.44, the second-order numerator term cancels the second-order denominator term in (14), reducing the CLCE to
0:0623ðs þ 5:97Þkp T d ðs2 þ ðs=T d Þ þ ð1=T i T d ÞÞð0:95=s þ 0:95Þ ¼0 sðs2 þ 2:40s þ 2:44Þ (12)
where gain kp simply dictates the closed-loop settling time of the bed. This can be rewritten as
1þ
rBVF +
eBVF
u
PID controller
yBVF
Fluidised bed (Plant)
f Filter
Value control signal / [V]
Bubble Void Fraction, BVF / [V]
Fig. 12. Block diagram of the fluidised bed closed-loop control implementation using PID controller scheme. rBVF, eBVF, u, yBVF, and f are, respectively, the demand, error, control, output and feedback signals in the closed-loop layout.
1 þ 0:026kp
0:95ðs þ 5:97Þ ¼0 sðs þ 0:95Þ
(13)
s2 þ ð0:95 þ 0:025kp Þs þ 0:15kp ¼ 0
(14)
Thus, by selecting kp in the range 1.7–3.7, a critically or slightly under-damped closed-loop response can be achieved. This ensures that there are no substantial overshoots in the BVF response. However, theoretically derived gains generally require subsequent fine tuning during experimental implementation. Additionally, there may be a slight lag between when the flow is supplied and the nucleation of visible bubbles. Accordingly, gains Ti and Td were reduced slightly below the initially designed values to moderate the integrating and derivative actions. The benefit of this is two-fold, since the presence of noise in the system also typically requires the reduction of the gains for a more desirable
0.2 BVF demand Unfiltered BVF Filtered BVF
0.15
0.1
0.05
0 600
620
640
660
680
620
640
660
680
700 720 Time / [s]
740
760
780
800
740
780
790
800
0.02 0.01 0 -0.01 -0.02 -0.03 600
700
720
PID control demand, Value position / [V]
Time / [s] 2 PID control demand Valve position / [V]
1.5 1 0.5 0 600
620
640
660
680
700
720
740
780
790
800
Time / [s] Fig. 13. (a) The reference and the output signal from the experimental bed resulting from close-loop control for a demand of 0.02–0.14 BVF at 0.01 Hz step excitation using designed gains (kp ¼ 3.26, Ti ¼ 0.75, and Td ¼ 0.33). (b) The corresponding control effort exerted on the valve. (c) PID controller demand on valve position and the resulting measurement of the valve position.
ARTICLE IN PRESS C.N. Lim et al. / Control Engineering Practice 17 (2009) 67–79
Bubble Void Fraction, BVF / [V]
control, in particular the derivative action of the controller. Online tuning for continuous assessment of controller performance was adequate to achieve general process control optimisation. In addition, a zero mean BVF demand was not practical due to noise: the minimum BVF value is clearly zero and any noise at that level will be positive only. Thus, at very low demanded values of BVF there will be a mean positive error. The rule-of-thumb is that a small non-zero BVF demand (order of 0.01) should be used for a no-bubbling demand in this bed. In an example, a closed-loop settling time, ts, of 10 s was sought after, giving TE2.5 s, hence kp ¼ 2.77. The corresponding first-order and closed-loop responses to step changes in BVF from
77
0.02 to 0.14 are shown in Fig. 13. A sampling interval of 0.04 s was used. The controlled response is seen to be satisfactory. The obvious apparent drawback of a linear controller is its inability to respond to changes in system dynamics and so to maintain optimum closed-loop response. Despite this, it was applied to the bed by making a linear approximation to the non-linear bed dynamics near to the desired operating point. This non-linearity affected only slightly the closed-loop control performance features such as the settling time and steady-state value. This is shown in Fig. 14, wherein (a) a step variation in demand around high BVF bands resulted in a response that was slightly oscillatory, caused by the presence of large bubbles. The
0.24 BVF demand Unfiltered BVF Filtered BVF
0.2 0.16 0.12 0.08 0.04 600
620
640
660
680
700
720
740
760
780
800
Bubble Void Fraction, BVF / [V]
Time / [s]
0.2 BVF demand Unfiltered BVF Filtered BVF
0.16 0.12 0.08 0.04 0 600
620
640
660
680
700
720
740
760
780
800
Time / [s]
Bubble Void Fraction, BVF / [V]
0.25
BVF demand Unfiltered BVF Filtered BVF
0.2 0.15 0.1 0.05 0 600
620
640
660
680
700
720
740
760
780
800
Time / [s] Fig. 14. Implementation of control using fixed-gain policy for the PID controller scheme with gain values (kp ¼ 2.65, Ti ¼ 0.60, and Td ¼ 0.26) used for different ranges of BVF between 0 and 0.2. It can be seen that the approach is successful, but there are limitations as well. (a) BVF control result for step demand between 0.11 and 0.16 BVF, (b) BVF control result for step demand between 0.02 and 0.14 BVF and (c) BVF control result for step demand when there is too short a settling time and overshoots result.
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Bubble Void Fraction, BVF [V]
78
0.3 BVF demand BVF raw BVF
0.25 Demand = 0.2
0.2
Demand = 0.18
Demand = 0.16
0.15
Demand = 0.14
0.1 0.05
Demand = 0.02
0 0
100
Demand = 0.15
Demand = 0.08
Demand = 0.04 Demand = 0.02
200
300
Demand = 0.06
400
Demand ~ 0.00
500
600
Time / [s] Fig. 15. The time series of the closed-loop control of the process with PID controller conducted over a range of BVF demand.
response was also sluggish, with typical ts of about 20 s as the controller struggled to converge to the desired state using a persistently more noisy BVF feedback measurement, even when filtered. In Fig. 14(b), when a larger range of demanded BVF levels was imposed, better control was exhibited. The controller also coped with demand close to Umf. The filtered BVF feedback was considerably less noisy at lower BVF, hence giving a better control performance as can be seen in the figure. When designing the controller, choosing too fast a settling time can result in an undesirable outcome. Fig. 14(c) shows the response achieved with the same gains, used for previous cases, for a demand varying between 0.045 and 0.16. At ts ¼ 5 s, while the bed responded well to step-up in demand, rapid response to step-down demand was significantly oscillatory. Fast settling time demand consequently forced the valve to be operated quickly, opening and shutting the flow rapidly. This did not pose problems during flow increase. However, when the flow supply to the bed was abruptly reduced, causing the bed suspension to collapse under its own weight, bubbles were dissolved into the defluidised emulsion phase. This phenomenon would not have been readily identifiable during the system identification process and hence, equally, not accounted for during controller design. Furthermore, as a result of closed-loop control, this occurrence caused the controller to react by reopening the valve rapidly in order to compensate and minimise the error, thus causing undesirable oscillation of BVF. Also, at low flow rates the mean bubble size was small and mean speed was low; so the mean bubble residence time was longer than at high flow rates. This slightly slowed the bed’s response, and therefore the time constant of the bed on the step-down response. Therefore, a more conservative settling time employed earlier was more favourable in achieving the desired bed response in practice. Although during system modelling the full dynamics range of the bed is of interest for control, the gross behaviour of the bed was focused on, i.e. the range of dynamics that determines the macroscopic behaviour of the bed. The fast bed dynamics contained within the BVF measurement manifest themselves as fluctuations that affect control efforts. To obtain and facilitate control, the BVF measurements were subjected to a low pass filter with break frequency at 0.15 Hz. This frequency is slow enough to largely remove the ‘noise’ component of the BVF. Despite this filter, as indicated in Fig. 14(c), the controller still achieved a settling time ts ¼ 5 s, for step-up in BVF demand. Fig. 15 shows the PID controller’s performance in controlling the process over different BVF demands. The controller was designed with fixed gains based around the dynamics of the bed at an intermediate range of bubbling, thus ensuring acceptable responses (settling time, steady-state error and stability) over different BVF demands—including close to Umf.
The studies in this paper were conducted in a planar fluidised bed, so allowing the distribution of bubbles to be seen clearly and the state of the bed measured explicitly. Commercial fluidised beds are usually cylindrical in shape. Therefore, a different means of measurement is required, where pressure measurements conventionally come into place. Pressure fluctuations in a fluidised bed can be measured non-intrusively and are directly generated by the presence of the bubbles. Therefore, their dynamics could be directly related to those described in this paper, and similar methods of analysis, monitoring and control could be applied. The work described above demonstrates the ability to control the bubbling process, provided that the elements in the control strategy were properly considered. It should be possible to implement similar control using e.g. pressure measurements rather than BVF to indicate the state of the bed (Croxford, Lim, Gilbertson, & Harrison, 2004).
5. Conclusions An instrumentation system based on real-time imaging technique has been employed to detect and measure the bubbling phase in a planar fluidised bed. The bubbling process in the bed has been characterised using the developed system and the bubbling properties of the bed have been identified. The bubble void fraction, BVF, was measured and shown to relate consistently to the amount of gas supply above quiescent fluidisation. Simple yet representative models in the form of transfer functions have been subsequently developed to describe the dynamic behaviour of bubbles as well as the response of the bed to excitation in flow supply relevant in efforts to design a suitable controller for the process. Measurement of BVF was then used as the feedback of closedloop control implementation of the bubbling process via a fixedgain PID controller. This was able to control the BVF of the bed to follow various bubbling state demands. Successful control implementation using this technique demonstrates that our understanding of the dynamics of the bubbling process is adequate, and that the bed’s dynamics can be confidently approximated to be linear within this fluidisation regime. By extension, the dynamic content of pressure measurements, which similarly relates to the fluidisation quality of the bed, may be used conveniently instead of the BVF as the feedback signal in industrial beds.
Acknowledgements The authors would like to extend their gratitude to the UK government for the Overseas Research Studentship (ORS) scheme
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