Solids Residence Time Distribution in Gas-Flowing Solids-Fixed Bed

A two dimensional steady-state model of the gas–solid–solid reactor. Carlos Gregorio Dallos , Viatcheslav Kafarov , Rubens Maciel Filho. Chemical ...
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Ind. Eng. Chem. Res. 2005, 44, 6509-6517

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Solids Residence Time Distribution in Gas-Flowing Solids-Fixed Bed Contactors Nikola M. Nikacˇ evic´ and Aleksandar P. Dudukovic´ * Department of Chemical Engineering, Faculty of Technology and Metallurgy, University of Belgrade, 11000 Belgrade, Karnegijeva 4, Serbia and Montenegro

Flowing solids residence time distribution in gas-flowing solids-fixed bed contactors was investigated using a tracer technique with a step signal as input. Particles of different color (all other properties being the same) were used as tracers, and the residence time distribution (RTD) was obtained by the color analysis of the outlet. In additional experiments, on the basis of the determination of the tracer fraction in static holdup, the existence of exchange between static and dynamic holdup was proven. The fraction of static particles is not subjected to exchange and represents the dead part of the flowing solids in static holdup. A mathematical model which takes into account both axial mixing and exchange between static and dynamic holdup was developed. The model, based on nonideally mixed units in series, gives reasonable agreement with experimental results. Introduction In gas-flowing solids-fixed bed contactors, fine solids particles are introduced at the top of the packed bed column while gas can be introduced either at the bottom for countercurrent contacting or at the top when cocurrent operation is desired. This type of contactor can operate as either a two-phase or three-phase system. In the first case, gas and flowing solids are active phases, while the packing enables better contacting between the flowing phases. In this case, there are no limitations on the geometry and design of packing elements. In the second case, the packing elements behave as a third phase (e.g., catalyst pellets in heterogeneous catalytic reactors with flowing solids as the adsorbent for in situ separation). In this case, the geometry of the packing elements is limited by the process requirements (i.e., by the shape of catalyst particles). The gas-flowing solids-fixed bed principle was patented in 1948.1 The first industrial realization for heat recovery occurred in 1965 (Compagnie de Saint Gobain).2 Several advantages of this type of column were reported:3-39 low pressure drop, high heat and mass transfer rates, and low axial mixing in both phases. The first studies of the countercurrent flow of gas and fine solids particles inside the packed bed were carried out by Kaveckii and Plankovskii.3,4 In later studies, researchers paid considerable attention to the fluid dynamics of such systems3-28 as well as to their heat and mass transfer rates.6-8,28-34 Westerterp and colleagues36-39 proposed the use of fine solids as a regenerative adsorbent, which flows through the bed of catalyst in the case of methanol synthesis. Similar approaches can be found in the work of Verver and Van Swaaij,7,34 who investigated the catalytic oxidation of hydrogen sulfide, as well as in the work of Kiel et al.,11,35 who studied the regenerative desulfurization of flue gases. * To whom correspondence should be addressed. Tel.: 38111-3370-472. E-mail: [email protected].

Similar to the cases of countercurrent gas-liquid systems, three flow regimes were observed in previous studies with flowing solids: preloading, loading, and flooding. Preloading and loading regimes differ in the concentration of flowing solids and in the dependence of pressure drop on the gas flow rate. When the terminal velocity of the gas approaches the velocity of the flowing solids, a sudden increase in pressure drop occurs, together with the accumulation of solids at the top of the bed. These are the characteristics of flooding. Flowing solids holdup represents the volume of flowing solids per unit bed volume, and it is usually divided into two parts: dynamic and static holdup. The latter represents the fraction of flowing solids particles that rest on the packing elements. After the flowing solids and gas inlets are shut down, these particles that constitute static solids holdup remain in the bed. Dynamic holdup represents the fraction of flowing solids that is suspended in the gas stream between the packing elements. These fine solids particles flow out after the inlets of the gas and flowing solids are closed. It is well-known that plug flow is a desirable flow pattern in countercurrent separations. Real equipment always deviates from this ideal, because of the occurrence of axial (and radial) mixing. The result of this deviation is the reduction of the driving force for heat and mass transfer. Consequently, the residence time distribution (RTD) of both flowing phases in gas-flowing solids-fixed bed contactors plays an important role in the design of the equipment. Roes and Van Swaaij27,28 used the tracer technique to determine the residence time distribution and the axial mixing in both flowing phases. The axial dispersion model was used to describe the degree of axial mixing. In countercurrent gas-flowing solids systems, it was found that the gas Bodenstein number (Bog ) ugds/Dg) increases with the increase of both the solids and gas flow rates (the axial dispersion, Dg, decreases). For the operating conditions of practical significance, layers of 2-5 Pall rings height are equivalent to a single ideal mixing unit27,28 for the gas phase.

10.1021/ie050018o CCC: $30.25 © 2005 American Chemical Society Published on Web 05/03/2005

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Roes and Van Swaaij27,28 found that the Bodenstein number for the flowing solids phase (Bos ) Usds/Dsβ) was almost independent of the gas flow rate and that it increases with solids flow rate. The increase of axial dispersion in the flowing solids phase was registered near the flooding point. At the operating conditions of interest in industrial practice, 5-15 Pall ring layers corresponds to a perfect mixing unit27,28 for flowing solids. It should be noted that, in the work of Roes and Van Swaaij,27,28 two differently defined flowing solids holdups were used. Besides the total solids holdup, the corrected one was used, which excluded that fraction of flowing solids which was named by the authors as the “permanent fraction of static holdup”. It was assumed that this fraction behaved like a part of the large stagnant packing of the bed. The corrected solids holdup was in much better agreement with the results of the tracer analysis. In our previous work,40 it was shown that there exists a permanent exchange between the bulk of flowing solids (dynamic holdup) and the part of the flowing particles resting on the packing elements (static holdup). However, a fraction of the static particles that is not taking part in the exchange represents a “dead” portion of static holdup. This “dead zone of static holdup” probably describes the same particles as the “permanent fraction of static holdup” defined by Roes and Van Swaaij,27,28 but they might not be identical. The objective of this study is to investigate the residence time distribution of the flowing particles in gas-flowing solids-fixed bed contactors. We believe that the residence time of the flowing particles combines the residence time of suspended, fast moving particles and the residence time of particles which spend part of the time resting on the packing elements. Hence, we believe that this exchange between the flowing solids and the solids temporarily at rest on the packing of the bed contributes most to the tails of the RTD curves and to the “axial dispersion” observed by Roes and Van Swaaij.27,28 We propose in this manuscript to analyze the response of the system to a step change of flowing solids, based on the physical picture we just presented, and to develop a model which can describe the overall flow pattern in countercurrent gas-flowing solids-fixed bed contactors. Modeling The flow of fine solids through a fixed bed deviates from plug flow for two reasons: axial mixing and the presence of static solids holdup, which exchanges solids with the moving solids particles. A plausible phenomenological model is presented in Figure 1. The main stream of flowing solids (dynamic holdup) is represented with a tanks-in-series model. To take into account the time delay which comes from settled particles (static holdup), a parallel plug flow region is added to each of the mixing units. The plug flow holding time represents the mean residence time of particles while settled on the packing. It is assumed that the exchange between the static and dynamic portions of the solids occurs between the two consecutive tanks in series. The portion of static holdup, which is not taking part in the exchange, is represented with dead volume. At stationary conditions, the dead zone becomes a part of the fixed bed and the model equations can be

Figure 1. Schematic for the solids flow model.

derived from the material balance for one nonideal mixing unit (i)

Vidyn

dxidyn(t) ) νdynxi-1(t) - νdynxidyn(t) dt

(1)

xist(t) ) xi-1(t-τist)

(2)

νdynxidyn(t) + vstxist(t) ) νxi(t)

(3)

where xidyn is the volumetric fraction of tracer particles flowing through the ideally mixed unit i, and xist is the volumetric fraction of tracer particles at the exit of the plug flow unit adjacent to the well-mixed unit i. The reader should note that, to avoid double subscripts, we opted to indicate the unit number by a superscript. Due to continuity, the volumetric flow rate of the particles clearly is v ) vst + vdyn. The model can be readily solved in the Laplace domain (xj(s) ) L{x(t)}):

xidyn(s) x

i-1

)

(s)

xist(s)

1 τidyns

+1

τsti

) e- s

(4)

(5)

xi-1(s) For simplicity, and in the absence of data or adequate theory, we assume that the flow split is constant, vdyn/v ) ydyn and vstat/v ) yst, so that each unit has the same holdup

τidyn )

Vidyn Vcβdyn/N τdyn ) ) νdyn νdyn N

(6)

where N is the number of mixing units and βdyn is the overall solids dynamic holdup, which can either be obtained experimentally or be predicted by the proposed

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models.21,25,26 The time delay of the active portion of the static holdup for one mixing unit is

τist

i Vst,a Vc(βst - βdead)/N τst ) ) ) νst νst N

(7)

where βst and βdead are the static holdup and the volumetric fraction of the stagnant (dead) portion of the static holdup, respectively. In view of eqs 3-7, one can show that the exit flowing solids tracer concentration from N units in the Laplace domain is

[

]

τst N 1 ydyn + yst e- N s 0 τdyn x (s) ) x (s) s+1 N N

{

()

xN t ) L-1 xN(s)

}

(8)

(9)

For an impulse input, xjo(s) ) 1, eq 8 produces the transfer function for the system and eq 9 gives the E curve for the flowing solids. For a unit step input, as used in the current study, xjo(s) ) 1/s and eq 9 produces a step response at the outflow, i.e., the F curve N-1

F) e

∑ k)0

() [

N N-k k y y 1k dyn st

-(Nθdyn-kθR)

]

N-k-1

(Nθdyn - kθR)m

m)0

m!

∑

Η(Nθdyn - kθR) +

yN stΗ(Nθdyn - NθR) (10)

Figure 2. Experimental apparatus: 1 ) gas inlet; 2 ) rotameter; 3 ) column with fixed bed; 4 ) supporting grid; 5 ) flowing solids valve; 6 ) container; 7 ) flowing solids inlets with dosage system; 8 ) sliding plate with orifice (the “switch” for the step signal); 9 ) narrow sample tube; 10 ) exit valve; 11 ) gas valve.

The space time for mixed tanks in series due to continuity (v ) vdyn + vst) is

where

θdyn )

t τdyn

(11)

In eq 10, Η stands for the Heaviside function and θR ) τst/τdyn. One should note that the response obtained from eqs 8-10 represents the weighted sum of the responses of N tanks in series, N - 1 tanks in series with time lag τst/N, N - 2 tanks in series with time lag 2τst/N, etc., and, finally, the plug flow response with time lag τst! The above model contains a number of parameters: ydyn ) vdyn/v, ystat ) vst/v, τst, τdyn, and N. We try to provide an estimate of as many of them as possible before we compare or fit the model-calculated RTD to data. The volumetric flow through the static time delay zone, νst, due to the exchange of particles, can be found from the rate of exchange between the dynamic and static holdup, as defined in our previous paper.40 This volumetric exchange flow rate is given by

νst ) fVc

(12)

where f is the frequency of exchange between static and dynamic holdup.40 Consequently, the time delay for particles in the active portion of static holdup is

τst )

Vc(βst - βdead) βst - βdead ) fVc f

(13)

τdyn )

Vcβdyn ν - vst

(14)

The total volumetric flow of particles can be found from

ν)

S A Fs c

(15)

where S is the mass flux of flowing solids, Fs is the solids density, and Ac is the cross-sectional area of the packed column. Experimental Work In these experimental investigations, a step input of tracer at the inlet of the flowing solids was used. The sketch of the apparatus is presented in Figure 2. A 58.8 mm inner diameter glass column was used. It was packed with 16 mm glass spheres, supported by a grid. The void fraction was 0.512, and it was determined experimentally. Flowing solids particles of two different colors were used in the step change experiments. All the properties (density, mean particle diameter, particle diameter distribution, and sphericity) were the same for the different colored particles. Two flowing solids dosage systems were used for each powder separately. A constant solids flow during the experiments was achieved by the use of two funnels, one above the other, as shown in Figure 2, while the flow rate was adjusted by using

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Table 1. Flowing Solids (FS) Properties and Operating Conditions FS material FS colors FS density, Fs (kg/m3) FS mean diameter, ds (µm) FS mass flux, S (kg/(m2 s)) superficial gas velocity, Ug (m/s) fixed bed height (cm)

I

II

glass black/yellow 2500 700 0.69 0.122 67

alumina blue/white 3970 291 1.26 0.0891 78

the plates with different orifice diameters, which were previously calibrated. The shift of the plate with the orifice (8 in Figure 2) closes one dosage system while opening the other one, which corresponds to the step input. A glass sample tube (8 mm i.d.) with valves at each end was positioned at the bottom of the apparatus for analysis of the samples (9 in Figure 2). Solids accumulated in the sample tube after closing the bottom valve (10 in Figure 2) represent the “frozen” response to the step signal. The analysis of the vertical color profile leads to the RTD curve. The sample tube was calibrated to relate the level of solids in it with the time elapsed after the step input. The gas flow rate was adjusted with precision valves and determined by using a calibrated rotameter. Two sets of experiments were carried out using different materials for the flowing solids phase: glass and alumina. The characteristics of the flowing solids materials and the operating conditions used are presented in Table 1. The same measurement technique was applied in two types of experiments, one for the RTD measurements and the other for the determination of the exchange rate between the dynamic and static fractions of flowing solids. The procedure for the residence time distribution experiments was as follows. After steady state (3 min) was reached, the step signal of flowing solids was introduced at the top of the sample tube and simultaneously the valve (10 on Figure 2) on the bottom of the sample tube was closed. After the tube was completely filled with flowing solids, the inlets for the flowing solids and for the gas and the valve on the top of the tube were simultaneously closed. The sample tube (containing the frozen response to the step signal) was photographed with a digital camera in order to analyze the step change response. The camera (Kodak DC 260) was used for the experiments with glass as the flowing solids material. Another camera (Nikon Coolpix 4300) was used for alumina as the flowing solids phase. All of the camera settings were the same for one set of experiments, and artificial lighting (reflector) was employed in order to accomplish constant conditions. Digital photographs were analyzed by Sigma Scan 5 software. Previously prepared calibration curves were used to relate the relative scale of the color intensity with the fraction of the tracer. In summary, step response curves were obtained that relate the fraction of the tracer (digital photo analysiscalibration curve) with the corresponding time (via sample tube height calibration).

above the valve represent the dynamic holdup.4-10,12 The dynamic solids holdup was calculated using the following equation

βdyn )

mdyn FsVc

(16)

where mdyn is the mass of solids in the container above the tube valve. Experiments for the determination of the rate of exchange between the dynamic and static holdup were carried out in a similar manner, but with analysis of the tracer fraction in static holdup as a function of time. At different times after the step signal, the inlets of the gas and flowing solids were closed simultaneously. The content of the column was discharged and sieved in order to separate the stagnant flowing particles from the packing elements. After the flowing solids that remained in the bed were weighed, the static holdup was calculated according to the equation

βst )

mst Fs V c

(17)

The discharged particles were well-mixed, placed in the sample tube, and photographed in the same manner and at the same conditions as in the RTD experiments. The obtained results represent the mean value of the tracer fraction in static holdup at a certain time after the step input. These experiments were repeated for different time intervals, giving the fraction of tracer in the static part of the column as a function of time. Experimental results for the tracer fraction in the static holdup as a function of time are presented in Figure 3 for (a) glass particles and (b) alumina particles as the flowing solids. From the horizontal asymptote, the fraction of the dead part of static holdup can be estimated. For alumina particles it was found to be 41% and for glass particles 35%. We used a single mixed zone approximation for static solids holdup, employed in our previous paper40 and represented by eq 18 below

(

φ ) xst 1 -

) (

)

βdead βdead ) 1(1 - e-f/(βst-βdead)) βst βst

(18)

to estimate the exchange frequency f. When eq 18 is represented as a plot of ln[1 - φ(1 βdead/βst)] vs t, the slope is -f/(βst - βdead) or -1/τst (from eq 13). This provides us with a first estimate of τst. These plots are presented in Figure 4 for experiments with (a) glass and (b) alumina particles. The obtained values for the exchange frequency (f) and time delay (τst) are presented in Table 2. The lines in Figure 3 represent the calculated values of φ by eq 18, using the experimentally determined value of the exchange frequency (f). The average errors between the experimental data and calculated values are 8.1% for glass and 14.2% for alumina particles.

Estimation of Parameters

Results and Discussion

The dynamic holdup was determined during the same RTD experiments. After the simultaneous closing of the gas and flowing solids inlets and valve 5 in Figure 2, particles which drain from the column and accumulate

Now we have estimates of all the parameters required for applying our flow model for RTD prediction (eqs 1-9) except for the number of mixing units N. As shown above, other parameters were estimated from our data.

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Figure 3. Fraction of the tracer particles in static holdup at different experimental times. Comparison of the experimental results and the model (eq 18): (a) glass particles as flowing solids; (b) alumina particles as flowing solids.

The mean values of the dynamic and static holdups for all experimental runs were used in the calculations. The number of mixing units in the model, N, can now be estimated by fitting the experimental RTD data and can, in principle, be optimized via trial and error procedures. For the first guess of N, we used the RTD experimental results and the well-known expression for the F curve of N tanks in series:41

F ) 1 - e-Nθ

N-1 (Nθ)m

∑

m)0

m!

(19)

“long tail”, i.e., the exchange with static particles. However, a trial and error procedure proved that the first guess for N obtained from eq 19 was the most appropriate. For the glass particles, one mixing unit corresponds to 8 layers of packing elements, while one mixing unit corresponds to 12 packing layers for alumina particles. These results agree with the previous investigations of Roes and Van Swaaij,27,28 where the range of 5-15 layers of packing elements was determined to behave as one mixing unit (but with bed packing consisting of Pall rings, not solid spheres). The mean residence time for the flowing solids can be expressed by

where θ is dimensionless time:

θ)

t ht exp

The experimental mean residence time, htexp, is the integral from 0 to infinity of the (1 - F) curve and is found by using a simple method with a fitted curve and equal intervals of tracer fraction.42 The minimization of error between the experimental data for the early S rise of the curve and the results obtained with eq 19 gives a first guess for N. Rigorously, eq 19 cannot be used for the step response of the gas-flowing solids-fixed bed system, because it presumes ideally mixed tanks in series only. Consequently, eq 19 cannot predict the

ht )

Vc(βdyn + βst - βdead) ν

(20)

Taking into account eqs 12-14, the mean residence time becomes

ht )

νst νdyn τdyn + τst ν ν

(21)

The calculated mean residence time from eq 21 was compared with the experimentally determined mean residence time (thexp) in Table 2. The values are in good agreement with the experiments, with relative errors being 0.2% and 10% for glass and alumina flowing

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Figure 4. Determination of the rate of exchange between static and dynamic holdup: (a) for glass particles; (b) for alumina particles. Table 2. Solids Flow Model Parameters and Mean Residence Time FS material number of mixing units, N dynamic holdup, βdyn static holdup, βst dead zone, βdead (% of βst) rate of exchange, 105f (m3/(m3 s)) space time for tanks in series, τdyn (s) time delay, τst (s) mean residence time, ht (s) exp mean residence time, htexp (s) relative error, δ ) (th - htexp)/thexp (s)

I

II

glass 5 0.227 0.233 35 4.96 4.8 30.6 7.95 7.96 0.18

alumina 4 0.126 0.104 41 1.42 3.0 43.3 4.41 3.96 10.1

solids, respectively. Table 2 also lists all the required parameters for the flow model (Figure 1, eqs 1-3, 1314, and 21). The model was solved using the toolbox Simulink in MATLAB 6. The results of the RTD simulation are presented in Figure 5 and compared with experimental data. The agreement between the model fit and the experimental data is satisfactory. Clearly, the model predicts a somewhat higher degree of axial mixing. Slight waves can be noticed in the response curves as well. Both are the consequence of the fact that the flow model consists of a finite number of mixing units and the assumption that all exchange between dynamic and static particles

(mixed and plug flow units on Figure 1) occurs between consecutive tanks only. In reality, this exchange takes place at every point of the column. On the other hand, the prediction of the long tail is good. The model could be fitted to the data better by using a nonlinear regression in multiparameter space. That was not our objective. We merely wanted to show that the exchange between flowing solids and static solids is a good way of describing the observed long tails in the RTDs, based on physical grounds rather than the axial dispersion model. Different results (Figures 3-5 and Table 2) for alumina and glass particles were primarily the consequence of different flowing solids characteristics. One should note (Table 1) the differences in density and, particularly, in the mean diameter values. Furthermore, the glass particles were nearly ideal spheres, while that was not the case with the coarse alumina particles. These differences in properties were reflected in the values of dynamic and static holdup, which are in agreement with previous investigations.3-6,9-10,12-21,23-26 Accordingly, the differences found for the exchange rate, for the dead zone, and, finally, in the RTD curves were expected. The dependence of dynamic holdup on the properties of flowing solids and the operating conditions was reported earlier as both an empirical correlation21 and a fundamentally based model.26 This has not been

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Figure 5. Response of the gas-flowing solids-fixed bed contactor to the flowing solids step input. Comparison of the experimental results and the solids flow model: (a) glass particles; (b) alumina particles.

done for static holdup to date. Due to very limited sources of experimental data, it is not possible to offer an equation which would correlate exchange rate, dead volume, and backmixing with flowing solids properties at the moment.

Acknowledgment The authors are indebted to the Ministry of Science and Environmental Protection of Serbia for their financial support (Contract No. 101700). Nomenclature

Conclusions The residence time distribution in gas-flowing solidsfixed bed contactors is mainly influenced by two factors: axial velocity distribution and mixing and the exchange between flowing particles (dynamic holdup) and particles settled on the packing (static holdup). However, a fraction of the static holdup is not affected by the exchange process and can be treated as a dead portion of flowing solids. The exchange frequency and dead volume can be successfully estimated by analyzing the tracer content of the static holdup as a function of time. The flow compartmental model (Figure 1 and eqs 1-11) predicts well the effect of the exchange between the static and dynamic holdup. Axial mixing is taken into account through the number of mixing units. The number of packed bed particles corresponding to the height of a mixing unit is found to be 8-12 (for glass and alumina particles as flowing solids, respectively).

Ac ) cross-sectional area of the column, m2 Dc ) column diameter, m dp ) mean diameter of the packing elements, m ds ) mean diameter of the flowing solids, m F ) dimensionless response to a step input f ) rate of exchange between static and dynamic holdup per unit bed volume, m3 m-3 s-1 Η ) Heaviside function mdyn ) mass of dynamic holdup, kg mst ) mass of static holdup, kg N ) number of mixing units S ) mass flux of flowing solids, kg m-2 s-1 s ) Laplace operator t ) time, s ht ) mean residence time, s htexp ) experimental mean residence time, s Ug ) superficial gas velocity, m s-1 Vc ) packed bed volume, m3 Vdyn ) total volume of ideally mixed tanks in series (dynamic holdup volume), m3 Vst,a ) total volume of plug flow units (active static holdup volume), m3

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Vidyn ) single ideally mixed unit volume, m3 Vist,a ) single plug flow unit volume, m3 ν ) volumetric flow rate of the particles, m3 s-1 νdyn ) volumetric flow rate through mixed tanks, m3 s-1 νst ) volumetric flow rate through plug flow unit, m3 s-1 x ) volumetric fraction of the tracer particles xdyn ) volumetric fraction of the tracer particles in the ideally mixed unit (in dynamic holdup) xst ) volumetric fraction of the tracer particles at the exit of the plug flow unit (in active part of static holdup) xj ) volumetric fraction of the tracer particles in the Laplace domain xdyn ) volumetric fraction of the tracer particles in the ideally mixed unit (in dynamic holdup) in the Laplace domain xst ) volumetric fraction of the tracer particles in the plug flow unit (in active part of static holdup) in the Laplace domain ydyn ) fraction of dynamic solids flow rate ()vdyn/v) yst ) fraction of static solids flow rate ()vst/v) βdead ) volumetric fraction of the stagnant part of static holdup (dead zone) βdyn ) dynamic holdup βst ) static holdup δ ) relative error between the experimental and calculated values φ ) volumetric fractions of the tracer particles in static holdup θ ) dimensionless time θdyn ) dimensionless time for an ideally mixed tank ()t/ τdyn) θR ) ratio between the overall time delay and space time ()τst/τdyn) Fs ) flowing solids density, kg m-3 τdyni ) space time for a single ideally mixed unit, s τsti ) holding time for a single plug flow unit, s τdyn ) overall space time in ideally mixed tanks in series, s τst ) overall holding time in plug flow (time delay of flowing solids), s

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Received for review January 5, 2005 Revised manuscript received March 21, 2005 Accepted March 28, 2005 IE050018O