Residence Time Distributions in Staged Spouted Beds Enrique Arriola, Carlos Francisco Cruz-Fierro, Khaled Hamed Alkhaldi, Brian Patrick Reed and Goran Jovanovic* 102 Gleeson Hall, Oregon State University, Corvallis, OR 97331-2702
C
ontinuous multi-stage operation of spouted beds, with countercurrent flow of solids and fluid, has several advantages over batch operation. It makes better use of the equipment, provides a steady state environment, has better controllability, and narrows the residence time distribution (RTD) of solids, thus increasing the efficiency of the operation (El’ perin and Khokhov, 1965; Madonna et al, 1961; Malek and Walsh, 1996). In a single spouted bed, the RTD of solids is wide due to the solids back mixing. This can be a serious disadvantage if uniform treatment of particles is desired. By increasing the number of stages, the variance of the RTD of the solids decreases, approaching plug-flow behaviour. Comparing the experimentally obtained external age distribution, E(t), curves with theoretical curves for combinations of ideal vessels, or compartments that reflect the observed flow patterns and design of the bed, provide the best approach to fit the experimental data. The addition of a rectangular draft tube changes operating and design characteristics of an ordinary spouted bed, as shown by Stocker (1987), and Ji et al. (1998). The major advantage in using draft tubes is that there is no limitation on the maximum spoutable bed height. The solid circulation rate in the bed can be easily controlled while pressure drop and minimum gas flow rate are both lower compared to a bed without a draft tube. Industrial application of multi-stage spouted bed systems has always been problematic and users are still looking for a good design. Multi-stage, vertically superimposed spouting beds, have not been well described in the literature and very little is known about this design.
Experimental Design This investigation directly addresses the problem of determining the RTD of solids in a much simpler design of a continuous multi-stage spouted system with draft tubes. This design has no problems with hydrodynamic instabilities (oscillations in solids flow and hold-up), and improves operating characteristics and controllability of the bed. Also, the start-up and shut-down procedures/conditions of the column are very simple. The column does not require special devices with moving parts. In this work, a three-parameter model is introduced to represent single-stage and multi-stage spouted systems, and model simulation is compared to the RTD of solids in the systems obtained experimentally. The form of the model is generic; it reflects solid particle distribution * Author to whom correspondence may be addressed. E-mail address:
[email protected]
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A novel design of a multi-staged, continuous flow spouted bed, suitable for counter-current, crosscurrent, and fed-batch gas-solid contacting is developed. Single and three-stage conter-flow units are studied, the residence time distributions (RTD’s) of the solids therein are measured and two compartmental models are developed to fit this family of RTD’s. The first (two-parameter) model is satisfactory for the single stage units, and three-stage units at high gas flow rates. The second (three-parameter) model describes each stage as two CSTVs in parallel with a PFV in series. It provides better agreement with experimental data in the three-stage case. On a mis au point un nouveau concept de lit jaillissant polyphasique à écoulement continu, pour le contactage gaz-solide à contre-courant, à courant transversal ou à alimentation discontinue. Des unités à contre-courant monoétagé et triéatagé ont été étudiées et les distributions de temps de séjour (DTS) des solides y ont été mesurées. Deux modèles à compartiments ont été mis au point pour représenter cette famille de DTS. Le premier modèle (à 2 paramètres) s’avère satisfaisant pour les unités monoétagés, et pour les unités triétagés à des débits de gaz élevés. Le second modèle (à 3 paramètres) décrit chaque étage comme deux CSTV en parallèle avec un PFV en série. Celui-ci offre un meilleur accord avec les données expérimentales dans le cas triétagé. Keywords: multi-stage spouted beds, residence time distribution, spouting, gas-solid fluidization.
within each stage, types of particle flow within the bed as observed experimentally, and the air velocity distribution affected by inlet and exit regions of the systems. However, the aim here is not to obtain particular values of model parameters. This is necessarily so because of the narrow range of system parameters; compartment size and air velocity. With some exceptions, Arriola (1997), almost all previous investigation on multi-staged spouted beds has been conducted in either cylindrical or semi-cylindrical batch columns. Furthermore, none of the previous The Canadian Journal of Chemical Engineering, Volume 82, February 2004
Figure 1. Schematic of a spouted bed stage (all dimensions in cm).
Figure 2. Three-stage spouted bed experimental setup.
works in this field were done using two-dimensional, halfrectangular, continuous, multi-stage spouted beds with draft tubes, which we believe is a better design in terms of operation, control and scale-up criteria. To prove the above claims we use a solid tracer technique and RTD analysis. Even though the RTD does not carry complete information about the flow conditions of a particular bed, it is still the best tool to determine mixing phenomena occurring in the bed (Fogler, 1992; Levenspiel, 1996; Bischoff, 1966). The column design reflects special attention to selfcontrolled hold-up of the particles. By using a novel design for the solids inlet the column will self-regulate the solids feed rate to maintain a constant solids level in the bed. This design also allows for easy and complete loading or unloading of the system at startup or shutdown. See Figure 1 for a schematic of this design. Emphasis is placed on the geometry of the bed and on the simplicity of its operation. The small angled plate (solid level limiter) below the solids inlet tube in each stage acts to block the flow of solids from the inlet tube when the bed height is at the level of the plate. The bed height is therefore controlled by changing the location of the plate. This design feature allows continuous operation with constant level of solids within the bed. Figure 1 shows schematically the design features of the vessel, which is made of Plexiglas®. The multi-stage system is constructed using three identical spouted beds placed one upon the other, and connected with Plexiglas® tubes for circulation of ascending air and descending solids. Figure 2 shows the three-stage spouted bed system used in the experiments. As expected, the addition of a draft tube in each stage improves the internal re-circulation rate of solids, and should provide practically no limitation in the number of stages and in the maximum spoutable bed height. Compressed air and glass beads (particle diameter = 2.7
mm, particle density = 2.4 g/cm3) are used in all experiments. Small screens are installed in each spouted bed to avoid entrainment of particles into the air pipes, as shown in Figure 1. Pressure drop in each of the beds is measured with a U-tube water manometer (1500 Pa maximum pressure drop per stage). The solids are continuously withdrawn from the bottom vessel through a modified L-valve. This valve plays an important role in the continuous removal of solids from the system without any mechanical device. Although it has not yet been thoroughly documented, a pulsating valve is recommended for feeding air to the L-valve. Visual observations show that a more uniform flow of solids is obtained when oscillating air pressure is used. This modification of the L-valve transforms it into a spitting valve, which should be tested for further studies. The RTD curves were constructed by collecting the entire effluent from the last stage of the system into cups at regular time intervals (10 or 20 seconds) and then counting the number of tracer particles from each interval. The tracer particles were prepared by staining the glass with a red dye. The tracer particles were introduced at the top of the system in a single instantaneous bolus, thus producing an almost ideal input pulse of tracer. As the column was operated and particles collected at the bottom of the bed, a pulse of air would blow them into a sample cup.
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Mathematical Model Two models were considered when characterizing the behavior of the spouted bed system. It is important to note that the dead volume does not appear as a parameter in the model equations. At the conditions that were studied the dead volume was negligible, only becoming significant as the gas flow rate in the system is reduced to zero. The 1-CSTV model, shown in Figure 3, consists of a plug-flow
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Figure 3. Single CSTV compartmental model.
vessel, with dead space, in series with a continuous-stirred-tank vessel. The E(t) representation for a single stage is shown in Equation (1). E[1] (t ) =
t − τP 1 exp − τM τM
(1)
Figure 4. Identification and representation of multiple flow compartments in the spouted bed.
Assuming identical stages the self-convolution of Equation (1) is performed, which yields the E(t) model for three identical stages, shown in Equation (2). E[3] (t ) =
( t − τP ) 2 2τ3M
t − 3τP exp − τM
(2)
This model gave satisfactory representation of the RTD curves obtained from the single-stage experiments. However, the corresponding self-convolution to three identical stages shows a less satisfactory fit in several of the data sets presented, namely those at higher gas flow rate. The existence of two mixing regions within the spouted bed is the probable cause of this behaviour. An illustration of the two mixing regions is shown in Figure 4. The first region of moderate mixing, just below the solid input, exists because of the change of the cross-section of the vessel and experimentally observed not-flat velocity profile of solids. Whatever the gas flow rate is, solids always occupy this region. The second region of extensive mixing is located near the gas outlet and exists only at high gas flow rate, high enough to push part of the solids to that region. A modification of the first model is presented in Figure 5. The single stirred tank is replaced by two stirred tanks in parallel, of different residence times, to account for the two different mixing regions. With this modification, we expect to better represent the two distinct degrees of mixing in the spouted bed. The residence time distribution function of a single stage with this arrangement is E[1] (t ) =
t − τP 1 − x t − τP exp − exp − + for t > τ p (3) τM 1 τM 1 τM 2 τM 2 x
where x represents the fraction of the flow rate moving to the first CSTV. The mean residence time is then given as τ = τM + τP
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(4)
Figure 5. Parallel CSTV compartmental model.
where, as shown by Levenspiel (1996), from the conservation of mass one can obtain τM = xτM1 + (1 − x )τM 2
(5)
The fractional volume of the first CSTV is computed from r =
0 xτM1 τM = 01 τR τR
(6)
where τ0M1 is the residence time in the mixed vessel at the condition of zero gas flow rate, τ0R is the real total residence time in the spouted bed at zero gas flow rate. The fraction of the flow rate going through each of these tanks (x) is taken as a model parameter. However, the fractional volume of the first CSTV is computed from zero gas flow results, for both single-stage and three-stage spouted beds, and taken as a fixed value (r) for each system. Therefore, x is no longer an independent paramenter, since it is determined by Equation (6), and the only model parameters are τM1, τM2, and τP. To obtain the self-convolution of this RTD function, necessary for analyzing the three-stage experimental data, it is convenient to transform it into Laplace domain. Because
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convolution in time domain is equivalent to multiplication in Laplace domain, there is no mathematical difference if the convoluted functions are multiplied in any particular order, as long as these functions represent elements that are connected in series. This allows us to perform the convolution of the parallel CSTVs of the three stages and then convolute the result with the three plug flow vessels. This last convolution is nothing more than a time delay of 3τP. The RTD function of the two CSTVs in parallel is E (t ) =
x t 1− x t exp − exp − + τM 1 τ τ τ M1 M2 M2
(7)
1− x αs + 1 x + = τM1s + 1 τM2s + 1 ( τM1s + 1)( τM2s + 1)
(8)
where α = τM2x + (1 − x )τM1 For the 3 stages E˜ 3 (s ) =
α3s3 + 3α2s2 + 3αs + 1 3
(9)
3
(τM1s + 1) (τM2s + 1)
In order to transform Equation (9) back to the time domain, partial fraction decompositions is necessary to find the coefficients a through f satisfying α3s3 + 3α 2s 2 + 3αs + 1
(τM1s + 1)3 (τM 2s + 1)3
a b = + τM1s + 1 ( τ s + 1)2 M1 +
+
c
(τM1s + 1)
3
e
(τM 2s + 1)
2
+
+
d τM 2s + 1
(10)
f
(τM 2s + 1)3
After carrying on the addition of the fractions in the right hand side, it is possible to obtain a system of equations with a through f as unknowns. The coefficients of equal powers of s in the numerator of each side of Equation (10) are equated, obtaining the system of linear Equations (11) to (16). (11)
τM12τM23a + τM13τM22d = 0
(3τ
)
2 2 3 3 M1 τM 2 + 2τM1τM2 a + τM1τM2 b
(
(12)
)
+ 2τM13τM2 + 3τM12τM22 d + τM13τM2e = 0
(3τ
) (
)
2 2 3 2 3 M1 τM 2 + 6 τM1τM2 + τM2 a + 3τM1τM2 + τM 2 b
(
)
+ τM23c + τM13 + 6τM12τM2 + 3τM1τM22 d
(
) (
)
2 2 2 M1 + 6τM1τM 2 + 3τM 2 a + 3τM1τM2 + 3τM2 b
(
(13)
(14)
)
+ 3τM22c + 3τM12 + 6τM1τM2 + τM22 d + 3τM12e + 3τM1τM2f = 3α2
(2τ1 + 3τ2 )a + (τ1 + 3τ2 )b + 3τ2c
(15)
+ (3τ1 + 2τ2 )d + (3τ1 + τ2 )e + 3τ1f = 3α a +b +c +d +e +f =1
and its corresponding Laplace transform is E˜ (s ) =
(τ
(16)
The Equations (11 to 16) completely determine the values of the constants a though f based on the three parameters, τM1, τM2, and x (included in α). The inverse Laplace transform of the right hand side of Equation (10) is a bt ct 2 t + + exp − 2 3 τ τ M1 2τM1 M1 τM1
(18)
d et ft 2 t + + + exp − 2 3 τM 2 2τM2 τM 2 τM 2
An analytical solution, however possible, would be rather complicated. For the purposes of the analysis presented here, it is sufficient to obtain a numerical solution. The solution procedure was programmed as a set of macros in Microsoft Excel, that calculated the RTD curves for the given parameters τM1, τM2, τP, and x. Finally, introducing the time delay due to the three plug flow vessels, we get the desired RTD function of the threestage system 2 a (t − 3τP ) b (t − 3τP ) c (t − 3τP ) E[3] (t ) = + + exp − 3 2 τM 1 τM1 τM 1 2τM1 2 d (t − 3τP ) (19) e (t − 3τP ) f (t − 3τP ) + + + exp − τ τM 2 τM 22 2τM 23 M2
for t > 3τP Both single-stage and three-stage RTD functions (Equations (3) and (19)) are used to represent the experimental data. It is important to notice that the constants a, b, c, d, e, and f are not adjustable parameters of the model, but constant coefficients obtained from the partial fraction representation. The system parameters are τM1, τM2, τP, and x, subject to one constrain of the first CSTV fractional volume r. This makes the system net parameters equal to three. The values of the parameters are chosen to minimize the sum of squared residuals in each data set.
)
+ τM13 + 3τM12τM2 e + τM13f = α3
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Table 1. Calculated parameters of proposed model (single-stage). Run #
G [L/s]
W [g]
1030 1031 1032 1033 1034 1035 1036 1037 1038 1040 1041 1042 1043
3.5 3.0 3.7 3.2 0.0 3.0 3.0 3.2 3.5 3.5 3.7 3.2 3.7
1198 1200 1218 1136 977 993 1069 970 1310 1215 1211 1357 1227
S [g/s] 6.9 5.9 11.9 4.8 8.2 3.0 20.4 2.7 17.0 2.3 2.8 18.6 5.4
τR [s] 173 204 102 235 119 333 52 358 77 524 426 73 229
τM1 [s]
τM2 [s]
τP [s]
x [–]
τmodel [s]
r [–]
98 73 73 103 29 119 16 175 45 436 395 29 190
190 220 77 296 – 900 66 470 70 450 350 82 170
49 30 30 50 14 70 18 101 50 80 60 35 52
0.44 0.70 0.35 0.57 1.00 0.70 0.82 0.51 0.43 0.30 0.27 0.63 0.30
199 147 106 236 43 424 43 420 109 526 422 83 228
0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25
τM1 [s]
τM2 [s]
τP [s]
x [–]
τmodel [s]
r [–]
473 165 481 24 66 29 86 15 19 42
600 630 800 63 147 50 135 91 – 180
70 70 130 23 45 29 42 16 101 77
0.19 0.45 0.23 0.41 0.36 0.36 0.27 0.61 1.00 0.51
1938 1471 2573 210 490 213 490 180 361 561
0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15 0.15
Table 2. Calculated parameters of proposed model (three-stage). Run #
G [L/s]
W [g]
2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
3.7 3.0 3.5 3.5 3.5 3.7 3.7 3.0 0.0 3.0
3785 3068 4117 3403 3478 3370 3474 3074 2763 2962
S [g/s] 2.1 2.1 1.9 17.4 7.4 16.2 7.4 17.1 7.1 7.0
τR [s] 1794 1484 2181 195 469 208 471 180 390 421
Data Analysis and Results Tables 1 and 2 summarize the conditions of each experiment and the parameters for the fit with the model presented here. All experimental data are compared to the prediction of both the 1-CSTV and 2-CSTV single-stage and three-stage models. Two examples of the fit are shown in Figures 6 and 7. Figure 6 shows both model fits with the data from a single-stage experiment (# 1037, Table 1), while Figure 7 shows both models applied to the data from a three-stage experiment (# 2007, Table 2). For the single-stage case both models fit the single-stage data excellently. The 2-CSTV model fits the RTD data for the three-stage experiments much better than the 1-CSTV model. The fractional volume of the first CSTV is r = 0.25 for a single-stage and r = 0.15 for three-stage spouted bed. The value of τp increases for the three-stage system because of the connections between the stages of the spouted bed, thus effectively increasing the plug-flow volume of the system. This increase in τp causes the value for r to be lower in the threestage case. The proposed model provides a successful presentation of the data in all runs for both single-stage and three-stage spouted beds as illustrated in Figures 6 and 7. Even though all stages in the three-stage unit are physically identical, the parameters obtained from the three-stage unit
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differ from those corresponding to the single-stage unit. Air flows in series through the stages, with a pressure drop at each stage. Thus the air pressure in each stage is different corresponding to the pressure drop across stages. Although the air flow rate for each stage is identical, air velocity is not. Additionally, the three-stage system includes connections between stages, which are absent in the single-stage system. The connections effectively increase the total plug flow volume of the system. The values of the parameter estimates shown in Tables 1 and 2 were analyzed in order to find a correlation with the operating conditions in the spouted bed. The main explanatory variables used were the gas flow rate (G) in L/s, the solids flow rate (S) in g/s, and multi-stage (M), a categorical variable that takes the value of 0, for single-stage experiments, and 1, for three-stage experiments. A least-squares multiple linear regression was performed using these explanatory variables and their possible interaction. The response variables for which a correlation was found were the fractional flow rate (x) and the overall mixed-flow residence time [τM = xτM1 + (1 – x)τM2], normalized with the total residence time, (τM/τ). All data points were included in the regression, except those of zero gas flow rate.
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Figure 6. Measured RTD and model fit for experiment # 1037 (singlestage).
Table 3. Regression coefficients for the model 1nx = β0 + β1 × G + β2 × S + β3 × M. Coefficient β0 β1 β2 β3 β0 + β3
Estimate 2.675 –1.078 0.022 –0.236 2.439
Standard Error 0.317 0.093 0.004 0.052 0.369
95% CI (2.007, 3.344) (–1.275, –0.884) (0.013, 0.031) (–0.347, –0.126) (1.660, 3.218)
Fractional Flow Rate (x) It was found that a logarithmic transformation of the response variable x provided the best fit. The model allowing for different slopes and intercepts between the single- and three-stage experiments is ln x = β0 + β1 × G + β2 × S + β3 × M + β4 × G × M
(20)
+ β5 × S × M For single-stage experiments, M = 0, and Equation (20) reduces to ln x = β0 + β1 × G + β2 × S
(21)
and for three-stage experiments, M = 1, and Equation (20) reduces to ln x = (β0 + β3 ) + (β1 + β 4 ) × G + (β2 + β5 ) × S
(22)
There is no statistical evidence that β4 ≠ 0, therefore there is no interaction between G and M (p-value is 0.2 from extra SS F-test). Also, there is only moderate evidence that β5 ≠ 0,
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Figure 7. Measured RTD and model fit for experiment # 2007 (threestage).
indicating interaction between S and M (p-value is 0.03 from extra SS F-test). This interaction term could only account for an additional 2% of the variability in the data (R2 changes from 0.95 to 0.93 when S × M is removed from the model). Therefore it is of little practical significance as a correlating variable. Once β4 and β5 are removed, the model from Equation (20) becomes ln x = β0 + β1 × G + β2 × S + β3 × M
(23)
The value of R2 using Equation (23) is 0.92. The estimates of the parameters in Equation (23) as well as their standard errors and 95% confidence intervals are summarized in Table 3. The sensitivity of β1 (from Table 3) is such that for each increase of 0.1 L/s in the gas flow rate, there is a corresponding change of [exp(–1.078 × 0.1) – 1] × 100% = –10.2% in the value of x. Similarly, the sensitivity of β2 demonstrates that for each 1 g/s increase in the solids flow rate, there is an increase of [exp(0.002) – 1] × 100% = +2.2% in the value of x. The corresponding simplification of Equation (23) for single-stage is given by Equation (24), whereas that for three stages is given by Equation (25). ln x = β0 + β1 × G + β2 × S
(24)
ln x = (β0 + β3 ) + β1 × G + β2 × S
(25)
The experimental values of x, along with the predictions obtained from the least-squares model represented by Equations (24) and (25), are shown in Figures 8 and 9.
Overall-Mixed Flow Residence Time (tM/t) There is moderate evidence of a correlation between the normalized overall mixed-flow residence time and the solids flow rate, as suggested by Equation (26). The model is τM = γ 0 + γ1 × S τ
(26)
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Figure 8. Experimental values of x and model prediction using Equation (24) (single-stage).
Figure 9. Experimental values of x and model prediction using Equation (25) (three-stage).
The least-squares linear regression using this model has R2 = 0.71. The estimates of the parameters as well as their standard error and 95% confidence interval are summarized in Table 4. The scatter plot in Figure 10 shows the trend in experimental data along with the model prediction (Equation 26). The sensitivity of γ1 indicates that for each 1 g/s increase in the solids flow rate there is a reduction of 0.0139 in the value of τM/τ. There is no statistical evidence of influence of gas flow rate (G) or number of stages (M) (p-value is 0.7 in both cases, from an extra SS F-test)
Conclusion The following conclusions can be drawn based on the results obtained in this study: The design of a staged spouted bed column used in this study has clear operating and scale-up advantages over designs reported in literature. With the addition of a draft tube to each stage internal re-circulation of solids is improved, allowing an operation with less gas and lower pressure drop. Under these conditions there should be practically no limitation in the number of stages that could be used in a column. The internal design of the column, especially the solids entrance, allows for very simple start-up and shut-down procedures. During start-up the whole column will fill to the designed hold-up and the flow of solids will stop automatically. The column is also capable of emptying readily for shut-down. Once the flow of solids to the top stage is stopped the column will empty itself leaving very few particles in the system. This design allows for simple operation without needing any sophisticated method for loading and unloading the column. The developed model predicts the data well for the systems investigated in this study. The 2-CSTV model allows for the non-idealities of mixing in the system as well as accounting for the experimentally observed multiple mixing regions. The developed correlations reflect the physical nature of the system, giving practical value to the study performed, and providing a good guide for the design of future spouted bed systems.
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Figure 10. Experimental values of τM/τ and model prediction using Equation (26).
Table 4. Regression coefficients for the model (τM/τ) = γ0 + γ1 × S. Coefficient
Estimate
Standard Error
γ0 γ1
0.857 –0.0139
0.022 0.0021
95% CI (0.811, 0.903) (–0.0182, –0.0096)
Nomenclature a-f E(t) ~ E(s) G M r S s t W x
partial fraction coefficients external age distribution Laplace transform of external age distribution gas flow rate,(L/s) categorical variable fractional volume of the first CSTV solids flow rate, (g/s) Laplace domain variable time, (s) bed weight, (g) fractional flow rate to first CSTV
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Greek Symbols
References
Fogler H. S., “Elements of Chemical Reaction Engineering”, 2nd Edition. Prentice Hall Englewood Cliffs, NJ (1992). Ji, H., A. Tsutsumi and Y. Kunio, “Solids Circulation in a Spouted Bed with a draft tube”, J. Chem. Eng. Jpn. 31, 842–845 (1998). Levenspiel, O. “The Chemical Reactor Omnibook. Fourth Edition”, OSU Book Stores, Inc. (1993). Madonna, L.A., R. F. Lama and W. L. Brisson, “Solids-Air Jets” B. Chem. Eng. 8, 524–528 (1961). Malek, M. A. and J. H. Walsh, “The Treatment of Coal for Coking by the Spouted Bed Process”, Dept. of Mines and Tech. Surveys Mines Branch, Division Report FMP 66/54-SP Ottawa ON (1996). Stocker, R. K., “Ultrapyrolysis of Propane in a Spouted Bed Reactor With a Draft Tube” PhD Thesis, University of Calgary, AB (1987).
Arriola E., “Residence Time Distribution of Solids in Staged Spouted Beds”, Chemical Engineering. PhD Thesis, Oregon State University, Corvallis, OR (1997). Bischoff, K. B., “Mixing and Contacting in Chemical Reactors”, Ind. Eng. Chem. 58 (11), 18–32 (1966). Elperin, I. T. and B. K. Khokholov, “Rapid Drying and Roasting of Granular Materials”, Gos. Vses. Nauchn-Issled. Inst. Tsement. Prom. 18, 5–9 (1964).
Manuscript received August, 2002; revised manuscript received May 12, 2003; accepted for publication Sepember 18, 2003.
α β0-β5 γ0-γ1 τ τM τM1 τ0M1 τM2 τP τ0R
transform variable model parameter estimates model parameter estimates residence time, (s) residence time of CSTV, (s) residence time of first CSTV, (s) residence time of first CSTV at zero gas flow rate, (s) residence time of second CSTV, (s) residence time of plug flow vessel, (s) real total residence time at zero gas flow rate, (s)
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