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Ind. Eng. Chem. Res. 2001, 40, 3230-3236
Experimental Observation of Pressure and Holdup Overshoot Following a Sudden Increase of Liquid Flow Vladimı´r Staneˇ k, Petr Svoboda, and Vladimı´r Jirˇ icˇ ny´ Institute of Chemical Process Fundamentals, Czech Academy of Sciences, Rozvojova´ 135, 165 02 Prague 6, Czech Republic
This paper presents the results of an experimental investigation of the hydrodynamics of a countercurrent packed-bed column exposed to a step change in the liquid flow rate. For the first time in the literature, this paper reports observations of the overshoots of the transient curves of pressure as well as liquid holdup when the change in the liquid rate brings the hydrodynamic regime of the bed close to the flooding line. The experimental technique indicated that column flooding did not appear to occur, even temporarily. The experimental transient curves for the pressure along the packed section, the total pressure drop across the column, and the liquid holdup are interpreted in terms of a simple three-parameter transfer function. Plots of the magnitude of the overshoot and the instant of its appearance as functions of the gas-to-liquid mass flow rate ratio are presented and analyzed. On the basis of these results, the existence of a pressure overshoot is attributed to the extra energy required for redistribution of the gas and liquid flow according to the new prevailing flow regime. 1. Introduction
physical quantity H(t) and a step change takes the form
Time-variable behavior of multiphase systems receives a great deal of attention1-4 because of its theoretical as well practical aspects. Nevertheless, some features of the transients of the hydrodynamics in these systems are not fully understood. In our previous communications,5,6 we have experimentally demonstrated the existence of overshoot phenomena in countercurrent packed beds. The overshoots have been observed when the column is exposed to a sudden change in the gas velocity. In this work, we have modified our experimental setup7-9 for close monitoring of the hydrodynamic transients of gas-liquid countercurrent beds exposed to time-variable liquid velocities. The aim of this paper is to investigate the possibility of the existence of pressure and liquid holdup overshoots when the bed is exposed to a sudden increase in the liquid rate.
[
H(t) ) H(∞) - [H(∞) - H(0-)]
+ T2 (T - τ1)(T - τ2) t t exp (2) T T3
G(s) )
(τ1s + 1)(τ2s + 1) (Ts + 1)2
(1)
where s is the Laplace variable. All three parameters appearing in eq 1 have the physical dimension of time. The dimensional solution in the time domain for a * Author to whom correspondence should be addressed. Phone: +420-2-20390233. E-mail:
[email protected].
] ( )
where H(0-) designates the initial steady-state value of the quantity H(t) at time zero prior to the onset of the step change and H(∞) designates the final steadystate value. The magnitude of the overshoot of the quantity H(t) is given by
( )( ) ( )
τ2 τ1 -1 1T T τ1 τ2 exp (3) τ1 - T τ2 - T
Hovershoot ) [H(∞) - H(0-)]
2. Theory The aim of this work is to describe the transient response of the column hydrodynamics by a suitable transfer function, G(s). The transfer function should have a small number of parameters while allowing for the possibility of overshoot. An attempt to use a twoparameter function failed to provide a sufficiently accurate fit to the experimental data. Therefore, the following three-parameter transfer function was selected
T2 - τ1τ2
A maximum of the function in eq 2, and hence the maximum overshoot, appears at the time tmax given by
tmax τ1 τ2 ) + T τ1 - T τ2 - T
(4)
A condition for the existence of the overshoot is
τ2 τ1 > 1 and 1 T T (5)
The form of the presented formulas indicates that, if either τ1 or τ2 equals zero, then the model G(s) reduces to a two-parameter model that still retains the capability of predicting an overshoot. If both τ1 and τ2 are zero, the response degenerates to a one-parameter model that is unable to describe an overshoot. If either τ1 or τ2 is zero while the other equals T, the model degenerates to a simple first-order system.
10.1021/ie990659h CCC: $20.00 © 2001 American Chemical Society Published on Web 06/19/2001
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It is stressed that, at this stage, the theoretical approach employed has no ambition to link the overshoots to the physical properties of the system or the fluid flow behavior. The aim was to define an unbiased way of evaluating the magnitude and time of appearance of the overshoots from experimental time series. 3. Experimental Section The experimental setup has been described in detail in our previous communications.5-9 The column is a 0.19-m-diameter Perspex cylinder packed to 1 m in height with 0.01-m glass spheres. The arrangement of the flows of air as a gas phase and water as a liquid phase is countercurrent. The experimentally measured quantities are pressures p1-p6 at the ports located 0, 0.2, 0.4, 0.6, 0.8, and 1 m from the top level of the packed bed. Pressure p1 provides an indication of the eventual presence of the gas/liquid mixture on top of the packing during the eventual onset of flooding conditions. Pressure p6 actually represents the overall pressure drop along the column. All pressures are measured by piezoelectric pressure transducers with an error of about 0.005 m of water head. The column is suspended on a strain gauge that measures the column’s weight under operating conditions. The liquid holdup is evaluated on-line from the measured apparent weight of the column, pressure p6, and the weight of the dry column. The sensitivity of the strain gauge in detecting the apparent column weight is 0.01 kg, while the current holdups are on the order of kilograms. The function of weighing under operating conditions was tested by weighing the dry column under variable gas velocities covering the entire experimental range with a constant weight as the result. Above the top of the packed section, there is a distributor of liquid. The distributor is a large cylindrical chamber whose bottom is equipped with 0.2-m-long hollow rivets arranged in a 0.01-m square pitch. The hollow rivets actually deliver the liquid onto the packing. Throughout system operation, the entire volume of the distributor and the rivets was full of water. During the experiments, the rate of the liquid brought on the top of the packed bed was suddenly increased. The step change in the liquid rate was realized in such a way that the liquid supply line was actually split into two parallel branches. The flow rates in the main and auxiliary branches were adjusted independently by two control valves, one located in each branch. The auxiliary branch was further equipped with a remote-control magnetic valve. Upon actuation of the magnetic valve, the liquid, which had initially been passing only through the main branch, began to enter the column via both branches at a higher rate. The auxiliary liquid supply branch was permanently full of liquid, even under standby conditions. In view of the virtual incompressibility of liquid, any significant time-lag or accumulation effects in the line are highly unlikely to occur. The entire experiment was controlled by a program in the Hewlett-Packard computing data logger. The data logger measures the electric signals of the transducers to four digits at the frequency of sampling, on-line processing, and data logging of 0.08 s per channel. After the gas rate and the lower and upper levels of the liquid rates had been adjusted using the control valves, the system was left to reach steady state at the
Table 1. Superficial Gas and Liquid Mass Velocities Used in Experiments of Type I superficial mass velocity [kg/(m2 s)] liquid gas
lower
upper
0.096 0.106 0.106 0.117 0.150 0.189 0.212 0.220 0.250 0.317 0.380 0.409 0.464
0 0 0 0 0 0 0 0 0 0 0 0 0
13.69 12.71 11.74 10.76 9.78 8.80 7.82 6.85 5.87 4.89 3.91 3.42 3.42
Table 2. Superficial Gas and Liquid Mass Velocities Used in Experiments of Type II superficial mass velocity [kg/(m2 s)] liquid gas
lower
upper
0.4737 0.4413 0.2584 0.397 0.3191 0.3157 0.2126 0.1945 0.1618 0.1146 0.1191 0.0983
1.96 1.96 1.96 1.96 1.96 1.96 1.96 1.96 6.85 6.85 6.85 9.78
3.42 3.42 5.87 3.91 4.89 5.38 7.33 8.31 9.78 10.76 12.71 13.69
lower (magnetic valve in the auxiliary liquid branch shut) liquid irrigation rate. Then, the control program was initiated. Initially, the data logger recorded all six pressures for 10 s and computed and recorded the liquid holdup. Then, the magnetic valve in the auxiliary liquid branch valve was opened, and recording of all experimental data continued until the time instant set as an input in the control program. During the experiment, the data were saved to a hard disk. The study covered two types of experiments. In one type, the liquid rate prior to the step change was zero; in the other, the initial liquid rate was nonzero. The superficial gas and liquid mass velocities used in the experiments are summarized in Tables 1 and 2. The “lower” and “upper” superficial velocities are those prevailing before and after the step change in the liquid rate, respectively. The lower liquid rates in Table 1 all equal zero, indicating that these experiments studied the column “startup”. In order that the initial hydrodynamic regime be reproducibly defined, the column in each type of experiment was irrigated prior to the experiment by liquid at the rate selected as the rate of the final state. After irrigating and flushing the bed, the transient experiment was carried out. 4. Experimental Data Processing The experimentally obtained transients, induced by a sudden increase in the liquid rate, were found to exhibit overshoots. That is, the pressure profile, overall pressure drop, and liquid holdup on transition to the new steady state temporarily reached values greater
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Figure 1. Transient experimental (points) and theoretical pressure profiles at ports p1-p6 following a step change in the liquid velocity.
than those corresponding to the new steady state. The hydrodynamics of the column thus approached the new steady state from above. The overshoots were observed when the conditions of the new steady state were close to the flooding point. The experimentally obtained transients of pressure and liquid holdup were fitted by eq 2 using an Excel spreadsheet optimization routine that minimizes the sum of the squares of the deviations between the experiment and the model. The optimized parameters were T, τ1, and τ2. The necessary H(0-) and H(∞) values were taken as averages of the experimentally measured quantity (pressure or liquid holdup) prior to the initial steady state and after reaching the new steady state, respectively. It is stressed that the only optimized quantities are T, τ1, and τ2. The optimum values of T, τ1, and τ2 were used, in turn, to calculate the magnitude and time of appearance of the overshoot in the response curve using eqs 3 and 4. In case of degeneration of the model, when either τ1 or τ2 approaches T so that the overshoot is zero and the time of maximum overshoot goes to infinity, tmax was restricted to 20 s. This value in the following figures thus indicates the absence of the overshoot phenomenon. 5. Results and Discussion Pressure and Liquid Holdup Overshoot Observation. Figure 1 shows plots of typical experimental (points) transient profiles of the pressures measured at ports p1-p6. The data were obtained for a step change in the superficial liquid mass velocity from 6.85 to 9.78 kg/(m2 s) at a gas superficial mass velocity of 0.162 kg/ (m2 s). Theoretical curves were computed from eq 2 for optimum values of the parameters τ1, τ2, and T. The transfer function selected for evaluation of the experiments is seen to fit the experimental data well enough for the purposes of this paper. The flow conditions of the experiment after the step change in the liquid rate were close to the flooding line,
but the process of flooding of the column did not occur even temporarily. This is clearly seen (Figure 1) on the plotted output of the pressure at port p1, which is located at the top of the packed section. The reading of the pressure port invariably remains zero throughout the transient experiment. The experimentat results plotted in Figure 1 demonstrate the existence of the overshoot phenomenon under conditions of a sudden change in the liquid rate. To the best of our knowledge, this is the first paper that demonstrates the occurrence of the overshoot phenomenon induced by a liquid rate change in a two-phase countercurrent bed. The following paragraphs analyze this phenomenon. Magnitude of Pressure and Liquid Holdup Overshoots. Figure 2 shows plots of the magnitudes of the pressure overshoots at ports p2-p6 and the liquid holdup overshoot as functions of the gas-to-liquid superficial mass velocity ratio. The initial liquid rate in the plotted experiments was zero. The flow ratio is on a logarithmic scale. The gas/liquid mass flow ratio was evaluated using the value of the liquid rate after the imposed step change. The overshoots at ports p2 and p3 and, at low gasto-liquid rate ratio, also at port p4 are significantly larger than those at the ports located near the supporting packing grid. The overshoot thus appears to originate at the top of the packed section where the liquid enters. Except for the two lowest flow ratios, port p6, which monitors the overall pressure drop, “feels” a strongly attenuated overshoot. The profiles of the pressure overshoots at the individual ports appear to display two maxima in Figure 1. The appearance of the maximum at the low end of the gas-to-liquid flow ratios might be speculatively ascribed to the following phenomenon: At high liquid rates, the flow of liquid entrains gas, which becomes manifest by a negative “pressure drop” at zero gas velocity. At low gas-to-liquid ratios, this phenomenon might be respon-
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Figure 2. Pressure and holdup overshoots as functions of the gas/liquid mass flow ratio for a step change in the liquid rate from zero initial rate.
Figure 3. Pressure overshoot as a function of distance of the port from the top of the packed bed for a step change in the liquid rate from zero initial rate for several mass flow ratios.
sible for an increased pressure drop needed to overcome the entraining effects of the powerful liquid flow. Plotted on the secondary axis in Figure 2 is the overshoot of the liquid holdup, which is also evaluated from eq 3. The liquid holdup is evaluated as the volume of liquid per unit volume of the column. The plotted holdup overshoots are small, yet the adopted experimental method succeeded in detecting them. In the normalized form (step change scaled between zero and unity), the liquid holdup overshoots are much smaller than the normalized pressure overshoots and do not exceed approximately 5% of the difference between the steady-state holdups before and after the liquid rate step change. In contrast, the normalized pressure overshoots at port p2 reach up to 200% of the corresponding steady-state pressure difference. The liquid holdup overshoot is seen to exist over the same range of flow ratios as the pressure overshoots. However, the largest holdup overshoots appear at the lowest gas-to-liquid mass ratio and monotonically de-
crease with increasing flow ratio without exhibiting maxima analogous to those seen for the pressure overshoots. A plot similar to that shown in Figure 2 for a step change from a nonzero liquid rate leads to virtually the same conclusions as those for a zero initial liquid rate. The overshoot appears at a low mass flow ratio of about 0.005, passes through two maxima and disappears at a high flow rate ratio equal to about 0.1. Similarly, the overshoots for ports p2-p4 do not differ very much, whereas those for ports p5 and p6 are distinctly smaller. Within experimental error, the magnitudes of the overshoots for the two types of experiments are the same regardless of the liquid rate prior to the step change. Figure 3 shows plots of the pressure overshoot as a function of the distance of the measuring port from the top of the packed section for several mass flow ratios. The plotted data are those for zero initial liquid rates, but the results do not differ markedly from the experiments for nonzero initial liquid rates.
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Figure 4. Times of pressure and holdup overshoots as functions of the gas/liquid mass flow ratio for a step change in the liquid rate from a nonzero initial rate.
The plotted profiles show that the overshoot remains virtually unattenuated within approximately the upper half of the bed, whereas in the bottom section, it rapidly decays. In fact, it appears that the overshoot in the upper part of the bed actually becomes slightly amplified. Judging from these results, one could conclude that the period of formation of the pressure overshoot is slow enough that the overshoot spreads into the bed while it is still growing. Even if the tendency to overshoot is very strong, the decay of the overshoot, after its full development, is so rapid that the pressure at port p6 virtually does not experience the overshoot. We note that port p6 measures the pressure loss across the entire packed section of the bed. The only noticeable difference in the profiles for the nonzero initial liquid rate experiments is that the sharp decay of the overshoot in the lower part of the bed actually starts higher in the bed. This observation is apparently associated with the fact that, for nonzero initial liquid rates, the bed requires less extra liquid and the overshoot takes less time to build up. Hence, the buildup overlaps less with the spreading of the disturbance. Time of Appearance of Pressure and Liquid Holdup Overshoots. Figure 4 displays the times of the maximum overshoots, tmax, at individual ports as functions of the gas-to-liquid mass velocity ratio. In the previous paragraph, we saw that the overall pressure drop, p6, essentially did not experience overshoot. Absence of the overshoot formally results in large (infinite) tmax times. For plotting purposes, these values were limited to 20 s. tmax values equal 20 s thus indicate the absence of an overshoot. Figure 4 shows that the time of maximum overshoot, tmax, increases only slightly with increasing flow ratio for the pressures that exhibited largest overshoots. At low flow ratios, the overshoot takes about 3 s to develop at the top of the column where the liquid phase enters. In the sections immediately below the top (ports p3 and p4), the overshoot peaks with only a small delay compared to the peak at port p2. However, in the lower half of the flow ratio range, it takes approximately 2-3 additional seconds for the overshoot peak to move from
port p4 to port p5. The slowing of the motion of the overshoot in the lower half of the bed (ports p4- p6) becomes more conspicuous at higher flow ratios. The slower motion of the overshoot in Figure 4 coincides with the region of gradual decay of the overshoot seen in a previous figure. The secondary axis of Figure 4 indicates the time of maximum overshoot of the liquid holdup. For lower values of the gas-to-liquid flow ratio, the model did detect the overshoot of the holdup. The corresponding tmax times, being in excess of 10 s, significantly exceed those of the pressure overshoots at ports p2-p5. The tmax data for the liquid holdup approximately copy the overall hydrodynamic flow regime reflected in the pressure drop p6. The overall pressure drop overshoot thus seems to be related to the buildup of liquid holdup after the step change in the liquid rate. It should be noted that the liquid holdup, as it is defined and experimentally determined, is a quantity that integrates the properties of the entire packed bed. A similar plot for the experiments with zero initial liquid rates exhibited generally longer tmax times compared to the those for step change from nonzero liquid rates. Also, the increase in tmax with the gas/liquid rate ratio was stronger. These observations are in line with the fact that the bed now requires more extra liquid for the liquid holdup buildup, which depends on the liquid rate. Figure 5 shows plots of the times of appearance of the pressure overshoot as axial profiles along the height of the packed bed. tmax is seen to generally increase with distance from the top of the packing. However, in the top sections of the bed, the time lag of the overshoot increases only slowly. In the lower sections, the overshoot grows weaker and the corresponding tmax times become large and approach the 20-s limit as the overshoot vanishes. At high relative liquid rates (low flow ratios), it takes less time to supply the extra liquid needed for the increase of the liquid holdup. The overshoot, though larger in magnitude, still develops somewhat more quickly. The depth of penetration of the overshoot into the bed remains essentially constant. The flow redistribution and the associated pressure overshoot thus
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Figure 5. Time of pressure overshoot as a function of distance of the port from the top of the packed bed for a step change in the liquid rate from zero initial rate.
take place in the upper section of the bed with enough liquid. The lower part of the bed then receives liquid with mostly restructured flow. Interpretation of the Overshoots. At the highest flow ratios, more time is required to supply the extra liquid needed for the buildup of the liquid holdup. The overshoot, though weaker and with less flow redistribution work to be done, penetrates deeper into the bed so that it is still felt strongly at port p6 as the overall pressure drop. The observed overshoots are interpreted as an effect of the flow redistribution taking place in the bed after an increase of the liquid rate. Flow redistribution requires extra energy dissipation, which appears as a pressure overshoot. After a step change in the liquid rate is imposed, the bed begins to absorb the liquid needed for the buildup of the holdup to a level corresponding to the higher liquid rate. The overshoot mechanism, assumed to be associated with flow redistribution, now takes effect. The process of overshoot buildup then overlaps with the process of overshoot spreading (supply of additional liquid feeding the overshoot). As a consequence, the time of the overshoot in the upper section (overlapping mechanism) of the bed does not change appreciably. Once the buildup of the overshoot is complete, the overshoot rapidly decays as it travels down the bed. During the transient experiment, one could actually visually observe, through the walls of the Perspex column, the process of flow restructuring, its spreading down the bed, and its decay. 6. Conclusions The experiments conducted demonstrate the existence of the overshoot phenomenon in countercurrent packed beds exposed to sudden changes in the liquid rate. The overshoots were observed in the pressure as well as in the liquid holdup transients. The magnitudes and times of appearance of the overshoots were evaluated from experimental time series using a three-parameter transfer function from control theory without the intention of relating the function to the physics of fluid flow.
The overshoots were observed when the upper liquid rate brought the regime of the column close to the flooding point. At the same time, the experiments did not reveal the onset of flooding conditions even temporarily. The overshoots were observed when the liquid step change took place from zero as well as nonzero initial liquid rates. The magnitudes of the pressure overshoots were found to be virtually the same, regardless of whether the step change occurred from zero or nonzero initial liquid rates. Pressure overshoots appeared first at the top of the packed section, where the liquid stream enters the bed. Here, the pressure overshoots were largest. Near the bottom of the packing, the overshoot was essentially not experienced. The liquid holdup overshoots take substantially longer to develop than the pressure overshoots near the top of the bed. The existence of overshoots is explained by the redistribution of gas/liquid flow after the imposed liquid rate change. Flow redistribution is a process consuming extra energy that manifests itself in a temporary pressure overshoot. The principal part of the redistribution occurs near the top of the bed where liquid enters and the flow regime responds to the uniform liquid distribution imposed by the liquid distributor. In our previous papers,5,6 we demonstrated and analyzed the overshoots after a step increase in the gas velocity. A comparison with the present study indicates that the magnitudes of the pressure and holdup overshoots are roughly the same. A gas rate change, however, causes the overshoot to develop faster and at the opposite end of the bed. The importance of this work for chemical engineering theory is in reporting pressure and holdup overshoots as entirely new phenomena. The importance for practice lies in identifying new hydrodynamic states in countercurrent systems exposed to sudden liquid flow changes that might be, on one hand, potentially hazardous and, on the other hand, beneficial if properly utilized for improved efficiency of operation under controlled hydrodynamic transients involving liquid rate changes.
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Acknowledgment The authors gratefully acknowledge financial support of the project by the Grant Agency of the Czech Republic under Grant A4072004/00. Nomenclature G(s) ) three-parameter transfer function H(t) ) dimensional three-parameter step change function, kPa for pressure or dimensionless for liquid holdup H(0-) ) value of H(t) prior to step change, kPa for pressure or dimensionless for liquid holdup H(0+) ) value of H(t) at onset of step change, kPa for pressure or dimensionless for liquid holdup H(∞) ) final steady-state value of H(t), kPa for pressure or dimensionless for liquid holdup Hovershoot ) overshoot, kPa for pressure or dimensionless for liquid holdup pi ) pressure at port pi, where i ) 1-6, kPa s ) Laplace variable, 1/s t ) time, s tmax ) time of maximum overshoot, s T ) model parameter, s τ1, τ2 ) model parameters, s
Literature Cited (1) Billet, R.; Schultes, M. Fluid Dynamics and Mass Transfer in the Total Capacity Range of Packed Columns up to the Flooding Point. Chem. Eng. Technol. 1995, 18, 371.
(2) Helwick, J.; Dillon, P.; McCready, M. Time varying behaviour of cocurrent gas-liquid flows in packed beds. Chem. Eng. Sci. 1992, 47, 3249. (3) Lange, R.; Hanika, J.; Stradiotto, D.; Hudgins, R. R.; Silveston, P. L. Investigation of Periodically Operated Trickle Bed Reactors. Chem. Eng. Sci. 1994, 49, 5616. (4) Silveston, P. New Application of Periodic Operations. In Unsteady-State Processes in Catalysis; Matros, Yu. Sh., Ed.; VSP BV: Utrecht, The Netherlands, 1990. (5) Staneˇk, V.; Jirˇicˇny´, V. Experimental observation of pressure drop overshoot following an onset of gas flow in counter-current beds. Chem. Eng. J. 1997, 68, 207. (6) Staneˇk, V.; Jakesˇ, B.; Ondra´cˇek, J.; Jirˇicˇny´, V. Characteristics of Pressure and Holdup Overshoot Following a Sudden Increase of Gas Flow. Chem. Biochem. Eng. Q. 1999, 13, 65. (7) Jirˇicˇny´, V.; Staneˇk, V. An Experimental Set-Up to Measure Flow Transients in a Counter-Current Packed Column. Chem. Biochem. Eng. Q. 1996, 10 (2), 55. (8) Jirˇicˇny´, V.; Staneˇk, V. Transients of the hydrodynamics of counter-current packed bed. Chem. Eng. Sci. 1990, 45, 449. (9) Jirˇicˇny´, V.; Staneˇk, V. A versatile correlation for liquid holdup in a two-phase counter-current trickle bed column. Chem. Eng. Commun. 1985, 35, 253.
Received for review September 1, 1999 Revised manuscript received May 24, 2000 Accepted May 3, 2001
IE990659H
