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Gassed power dynamics of disc turbine impeller
Chemical Engineering Science Vol. Printed in Great Britain.
0009-2509183 Pergamon
38, No. L1.pp. 1909_1!J141983
GASSED POWER DYNAMICS M. GREAVES*,
OF DISC TURBINE
K. A. H. KOBBACYt
$3.0 + -00 Press Lid.
IMPELLER
and G. C. MILLINGTON
School of Chemical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, England (Received 23 September 1982; accepted
14 April 1983)
Abstract-Measurements have been made of the dynamic power response of a disc turbine impeller, fallowing sudden changes in the input gas flowrate. Transient and frequency response results are presented for a number of impeller speeds. A very satisfactory representation of the smoothed-transient response is obtained using a simple linear first order model. The power change that occurs over the transient period is explained by the process of gas cavity formation described by Warmoeskerken, Feijen and Smith. It is found that the speed of formation and stripping of gas cavities depends on the impeller speed and gassing rate.
tNTRODUCTlON
The investigation described in this paper represents a more extended and detailed evaluation of the gassed power dynamics of a disc turbine than that attempted in an earlier study by Greaves and Economides[l]. They identified three main response regions. At low speeds, prior to the start of gas recirculation, the frequency response was characterised by a single “high” frequency component. Under fully established gas recirculation conditions, only a single “low” frequency component was present. In the transition region between these two regions, the response contained both “high” and “low” frequency components. A method was suggested for estimating the rate of gas recirculation based on the measured time constants and the mean residence time of the gas in the tank. Very little attention has otherwise been directed to this aspect of gas-liquid mixing operations. This is somewhat surprising, since the two-phase process depends critically on maintaining a dynamic equilibrium. From a practical viewpoint, gas-liquid reactors are subjected to sudden disturbances, such as changing gas supply or impeller speed. These may arise through start up or shutdown, or even during normal operation. Such effects can impose high loadings on the impeller drive shaft, or may seriously distort the dispersion and mass transfer processes in the vessel. At a fundamental level, there is an important need for more detailed understanding of the cavity formation processes occurring behind the impeller blades. This is because the power characteristics are mainly determined by the gas filled cavities behind the blades. Warmoeskerken et 01.[2,3] have recently provided a more unified description of this complex process, though it still lacks quantitative understanding. The rate at which these processes take place is a& important element in this understanding and requires the additional perspective of dynamic measurement. Various experimental methods are available for the study
of process
dynamics.
which has been widely used for obtaining the frequency response of heat exchangers[4] and other chemical process systems[5]. The utility and convenience of the method lies in the fact that it can yield the full frequency response of a system relatively quickly, using one or two well-chosen pulses. Alternatively, step_testing provides a simple way of obtaining transient response information. The objective is to carry out a system identification, either in the frequency or time domain: The dynamical power responses in this paper refer to a 0.2m diameter vessel. This is small by industrial standards, but scale considerations aside, the gassing effect on power will be the same. Also we have deliberately restricted the investigation to a single coalescing fluid (tap waler) in order to concentrate attention on the basic hydrodynamic-impeller condition. It is important to bear in mind, however, that many industrial fluids are, to varying degrees, coalescence-retarding. Such systems produce smaller bubble size, thereby lowering the speed for onset of gas recirculation. Two interesting effects of gas recirculation are identified in the paper. THEORETICAL CONSIDERATIONS
Pulse testing The general technique (see, for example, Murrill et al.[6]) involves disturbing the input, or independent variable. x(t). in a continuous manner over a short period of time and returning it to its original steady state. This produce? a pulse output response, y(t). In our case, the input is the sparged gas flowrate and the response is the impeller power. By definition, the transfer function of the process is:
From the definition of the Laplace substituting s = iw, we obtain
transform,
and after
Pulse testing is a technique
y(t) sin (ot) dt
y(t) cos (wt) dt - i *Author to whom correspondence should be addressed. fPresent address: Reservoir Department, Kuwait Oil Company. Ahmadi, Kuwait.
x(t) cos Cot) dt - i
I-0
x(t) sin (wt) dt (2)
1909
M. GREAVES et 01.
1910
A computer programme was written[7] to sample the input and output responses and calculate the Fourier Integral Transforms (FIT) in eqn (2), and thence, the frequency response, i.e. magnitude ratio and phase angle. These quantities are, respectively, the absolute value and argument of G(iw). When tested against standard responses reported in the literature the computed result gave excellent agreement. The steady state gain is simply obtained from the ratio of the areas under the output and input curves: =Y K =
I OT. I
0
Y Ctl dt .
(3)
x(t) di
The accuracy of frequency response data obtained from pulse tests is primarily determined by low frequency truncation effects and the error incurred in numerical integration. In general, the indicatiop is that[g], truncation tends to make the low frequency portion of the response abnormally flat, whilst the high frequency end is attenuated too sharply. Quantitatively though, this will depend on the type of process. The computational error involved in numerical evaluation of the FIT’s is primarily dependant on the approximating functions used. In turn, this is dependant on the computer sampling interval, At. “Breakdown” in the numerical integration, beyond the upper limit of frequency, is evidenced by degeneration of the frequency response into a series of meaningless oscillations.
-AL.
Figure 1 shows the equipment used for the dynamical tests. They were carried out in a flat-bottomed cylindrical vessel of 0.2 m diameter. The vessel was equipped with 4 wall baffles and a centrally mounted disc turbine impeller. Three such impellers were used, of standard Rushton dimensions, having D/T ratios of 0.375, l/2 and 213. The impeller clearance was fixed at C/H = 0.25 with the liquid level maintained at H = T. Full mechanical details have been reported elsewhere [ 1,131. All of the experiments were performed at atmospheric pressure using filtered tap water. Before a dynamic test was carried out, the operating conditions of the vessel were allowed to reach steady state. The needle valve (Vl) in Fig. 1 was used to adjust the steady state gas flowrate. A solenoid valve (s) was then used to introduce the step or pulse change in gas flow rate, with the needle valve V2 being used to adjust the height. Typical input step and pulse changes are shown in Fig. 2. The input gas flow disturbance and resulting power response were sampled automatically by a microcomputer data acquisition system. This subsequently performed the Fourier transform calculations for the pulse tests. It was not necessary to correct the torque measurements for frictional error since the steady state component was subtracted-out of the &mamic response.
Step tesfing
The shape of the transient response, or process reaction curve, is used to find the parameters of an approximate lumped-parameter model. In process dynamics/control terminology, this usually means a first order plus dead time (FODT), or second order plus dead time model (SODT):
(FODT)
G(s) =
Keerm (7,s
+
1l
(4) Fin. 1. Instrumentation and control of stirred tank for dynamic tests.
The steady state gain K, dead time 7m and the respective time constants can be obtained by standard methods (see, for example, Luyben[9]). Instrument dynamics If the measuring instruments exhibit significant dynamic lag then this must be taken into account in the evaluation of the process response. However, for systems where two measurements are made it has been shown[lO] that the instrument dynamics often cancel out. In our case, both the venturi gas flowmeter (1) and strain gauge torque transducer (2) have time constants less than IS ms. The computer interface is also extremely fast so that, overall, the instrument function effectively reduces to M&)/M,(s) = 1.
3.0
I
0.6
0 Fig.
II
IV
2. Step and pulse input
I
I
0
Time (5)
05
changes in gas flowrate.
Gassed plower dynamics of disc turbine
Frequewy(radss-b
pulse
0.8~
IO-’ to
1.95~
q
3 8 -y J-12-16-
Pulse duration 0 0.765 A 0.44s
Fii. 4. Effect of input pulse duration on power frequency. Input pulse 0.05X 10” to 1.95x 1O-3m-’ SC’,D = 0.1016m, N = 7 rps.
content of a rectangular pulse increases as its width decreases. The maximum frequency limit, calculated according to Clements and Schnelles’ criterionM, is based on a minimum normalised frequency content of 0.3. These are shown in the last column of Table 1, summarking the main test results. The reliable range of frequency is predicted to extend from 5 to over 10 rads s-l. The first observation from Table 1 is that the system dynamics, characterised by the various time constants and dynamic slopes, change with pulse height. This is of some concern, since it indicates a possible nonlinear condition[l2]. There is always the questionable aspect of fitting the slopes to the high frequency portions of the curves and, subsequently, estimation of the time constants from the corner frequency construction. Normally, we would expect that the errors in these estimates could be as high as 25%, but the results obtained here are reproducible to a much higher certainty than this. It is interesting that, at 5rps, there is close agreement of the time constant values. This indicates a basic linearity in that region and the nonlinearity, therefore, occurs at speeds higher than this. The dependency of the power dynamics on impeller speed generally confirms the effect noticed in the earlier
lo
Fig. 3. Effect of signal sampling rate Input
a0
g-4-
Using a sampling period of At = 2Orps, produces convergence up to 15.7radss-‘. This covers the range of interest in this study. On the other hand, reducing the width of the pulse reveals more detail in the frequency response (Fig. 4). This is expected, since the frequency
I.0
1911
Oa--&-,+0-o
RESULTS AND DBCUSSION
Frequency response In all, 22 pulse tests were made using the middle sized impeller (D/T =OS). The purpose of this wide-ranging series of investigations was to identify salient features of the power response and suitable test conditions, i.e. sampling rate, pulse height and duration. Figure 3 shows the effect of sampling rate on the frequency response (log modulus = 20 log magnitude ratio). At a sampling rate of 20 per second, the response starts to degenerate into a series of meaningless oscillations at about 4 rads s-‘. Increasing the rate to SO per second extends the observable range of frequency response. For convergence of the numerical FIT computations, Lees and Dougherty” proposed an upper bound on frequency:
-2000
impeller
on power frequency response. lo-‘m3s-‘, D=O.l016m, N =
I rps.
Table I. Summary of frequency response results for middle-size impeller (D/T = 0.5) Input Impeller speed
pulse (gas flow flawrate)
s-1
x 1041n3s-1
5
0.00 - 2.78
20
2.78
0.62
0.98
0.22
-52
7.0
5
0.80 -1.95
20
1.15
0.69
1.30
0.24
-55
6.1
7
1.95 d2.78
20
1.98
0.76
2.64
0.91, 0.20
-16. -52
5.8
7
0.80 - 2.78
20
1.15
0.44
1.64
0.58, 0.10
-25. -60
lQ.O
9
0.80 - 2.78
20
1.98
0.64
2.14
0.81. O.L!.
-17. -52
6.9
9
0.80 41.95
20
1.15
0.88
1.98
0.57, 0.09
-24, -50
5.0
11
0.80 -2.78
20
1.98
0.43
1.34
0.43
-18
10.2
11
0.80 - 1.95
20
1.15
0.54
1.20
0.29
-47
8.2
Pulse height (x 104m3s-11
Input pulse duration
Frequency
Sampling period Ins
*
output pulse duration s
Time Constant s
Dynamic slope dbfdecade
deviation limiis6
M. GREAVES et al.
N(i’) 0 IO l 12 v 14 a 17
K(J& -24.9 -46’4 -94.3 -242
.,s) OKl 044,0.25 1.340.27 1.75032
Fig. 5. Effect of impeller speed on power frequency response. Input pulse height 0.8 X 10m4 to 1.95 x lOedm3 s-‘, D = 0.106 tn.
Fig. 6. Effect of impeller speed on power frequency response.
study by Greaves and Economides [l]. From Fig. 5 it can be seen that, at the lowest impeller speed (5rps). when the impeller has just overcome the flooding condition[l3], a single time constant can be assigned to the response curve. At 7rps the system is close to the recirculation condition (N = NR), and well into the recirculation region at 9rps (N > NR). The frequency response at both of these speeds is characterised by two distinct slopes. It was claimed that the higher frequency component (low time constant) was associated with the onset of gas recirculation. Substantial gas recirculation is established when the speed is increased to 11 rps and the response curve changes again, so that there is now only a single dynamic slope. The dual dynamic slope condition, which appears in the frequency response al certain impeller speeds, is due to the existence of a peak or resonance effect. This resonance behaviour was not identified by Greaves and Economides because of the rather low sampling rate they used. It is a particular effect which is characteristic of distributed parameter systems[l4]. Similar dynamic behaviour has also been reported for shell and tube heat exchangers[lS] and fixed bed catalytic reactors1161. In Fig. 5, the first peak occurs at 7rps at a frequency of about 6.5 rads s-l. At 9rps, the resonance peak is very sharp and greatly increased in amplitude. The resonance therefore first appears at low frequencies when the rate of gas recirculation is of the same order as, or less than, the gas sparging rate (N = 7, 9 rps). Under these conditions, the impeller is affected by both sparged and recirculated gas and is consequently forced in a distributed manner. The process behaves as though it is subjected to a dual input disturbance; one coming from the sparged gas flow and one from the recirculated gas. No such effect occurs at 5 or If rps, when either the gas flow, is sparged gas flow, or the recirculated dominant. Pulse tests with the smallest impeller (D/T 0.375) confirm the previous observations, except that the resonance peaks tend to occur at lower frequencies (Fig.
6). The size of the peak at 17rps is not unlike a typical underdamped closed-loop control response, except that it lies below the low frequency gain axis. As shown in Fig. 6, there is a continuous increase in the steady state gain and time constant values as the spee*d of the impeller increases. The large drop in gain, which occurs at 20 rps, can be explained by the pregassing of the impeller caused by surface aeration. Figure 7 shows the frequency response for the largest impeller (D/T = 213). The limited frequency range, however, makes it rather difficult to estimate the dynamic slopes. For the smallest impeller [Q(t) = 0.175 x 10s4 m3 s-l, F lo = 0.02371, we observe a resonance peak occurring at about 6 rad SC’. The system is, in fact, just at the point of recirculation (N = NR). Another change in the frequency response occurs when equal positive and negative pulses are used. Figure 8 shows the hysteresis effect obtained. This points to either a serious nonlinearity, or a real process effect. It should be emphasised, however, that nonlinear effects of
Input pulse height 0 to 0.75
I.0
x
IO-’ m3 s-l, D = O.W62 m.
Frequency
(rads
5-l)
IO
Fig. 7. Effect of pulse. height on power frequency response. Ungassed initial condition, D = 0.135 m, N = 3 rps.
Gassed
1913
dynamics of disc turbine impeller
power
F'G
N(s-) pSCOYW> lnltlal
ov K= -8Z’x103Jm-3
*9 q 7
2.75~16~n?;’
-
final
5.3
0.0153
0051
32 5 15.0
0010900351 0009 O-0294
Fwst order dpproxrmatlon
l NPaatiwpvlsn K=9.22~lO~Jrn-~ (Log modulus is positive on ffdlnate)
I
I
I.0
I
IO Frequency
(rads 54)
Fig. 8. Power frequency from positive and negative pulse tests. D = 0.135 m, N = 3 rps.
Fig. 9. Cavity formation response. Step input 0.8 x 10Jm3 s-‘, D = 0.1016m.
this nature could lead to substantial errors in identification when applying linear systems theory. It also underlines the fact that further development of the pulse testing method for identifying the impeller power dynamics will be both difficult and complex. The transient response discussed next does not require any a priori assumption concerning the linearity of the process as a precursor to analysis.
Pg(0) is the initial steady state gassed power (t < 0), the oscillations appear even more pronounced, as shown in Fig. 11. A further set of cavity response tests was made starting from an initially ungassed condition. These are shown in Fig. 12. Interestingly, they alSo show the same type of oscillatory behaviour observed in the cavity stripping responses. The cause of the oscillatory behaviour in the impeller power response is not immediately apparent. Fist though, attention is directed to the sparged gas flow supply, since this may have introduced some sort of forcing condition on the process. Returning to Fig. 2, the fluctuations present at the low flow rate (Q = 0.8 x 10m4m3 s-‘) show a variation of about 4 or 5%. Clearly then. we would expect there to be some contribution from this source. Calculating this on the basis of the total power change that occurs over the transient (the ultimate change) does not, however, fully account for the oscillatory effect. On the other hand, when a positive step change is made, the resulting gas flow (Q=2.68 x
TRAN8lENT RESPONSE
Figure 9 shows the cavity formation response at N = 5, 7 and 9rps, following upwards step changes in gas flowrate. Only at the lowest impeller speed does the power response follow what can be considered to be a smooth path. At the two higher speeds, there is a degree of oscillation in the responses. The counter process of cavity stripping, following a downwards step change in gas flowrate (Fig. lo), reveals a greater degree of oscillation in the response. If this set of responses is plotted, alternatively, on a ratio basis, i.e. Fg(t)/Pg(O), where
l4-
z
l2-
F ZIO-
a9 -Frst
18.9
OCQ940-CO9
order approxlmotion
b &B3 e
6-
h n” 24 B 2
0 t-
Fig. IO. Cavity stripping response. Step input 2.68 x IO-’to 0.8 x W4 m s-‘, D = 0.1016m.
IO_’
to 2.68 x
1914
M.
Fig.
GREAVES
11.Transient power response (cavity stripping) plotted as
- 0.0 0 O-0 l 00
I
3.80 2.75 I.80
Fimt order approximation
OPl”““““““‘ll 0
05
I-O
I.5
et al.
20
2.5
3-o
3-5
4.0
4.3
Fig. 12. Cavity formation response from initially uugassed condition. D = 0.1016 m, N = 7 rps.
10e4 m3 S-‘) is extremely stable and no contribution seems possible under this condition. Another feature which is noticeable concerning these “large” oscillations iu power is that they tend to appear mainly towards the end of the response, i.e. after the initial fast rising portion, This relatively smooth, fast portion of the response is estimated to be about 60% of the power change. Thus, the “large” oscillations are mainly associated with the minor portion of the response (in terms of power change), but they dominate the slower portion, which affects the total response time. The latter tends to increase with impeller speed, from about 1.5 s to 3.5 s for the initially gassed condition, up to 4.5 s when no gas is initially present.
Pg(l)lPg(O)
Vs time.
If the transient power responses are considered in terms of the smooth curves represented on Figs, 9-12, then the response characteristics would appear to be consistent with the description of gas cavity formation given by Warmoeskerken et aL[2,3] for a 4-blade disc turbine. Following an upwards step change in gas flowrate (Fig. 9), the clinging cavities rapidly transition to form large cavities. The largest change in power is associated with this transition from clinging, to three large and three clinging cavities. Three stable large cavities are reported to occur for 0.025 i FOG < 0.035. With increasing gas flow into the impeller, the cavities continue to grow in size and this region is associated with a more gradual reduction in power. At the lowest impeller speed, as FIG tends ultimately to 0.051, the impeller is probably close to flooding. The steady state gains in Table 2, for N = 7 rps. illustrate the relative power change accompanying different step changes in gas flowrate. Moving in the other direction involves the stripping of large gas cavities, which are very stable. It is therefore expected that this process will be slower than cavity formation. We can best estimate the speed of the two processes by calculating their respective time constants on the basis of a first order mode1 approximation. These values (determined from the 63.2% response) are summarised in Table 2. The ratio of time constants between stripping and cavity formation, for the initially gassed condition at N = 5, 7 and 9 rps are, respectively, 2.09, 2.10 and 2.30. This means that the cavity formation process is approximately twice as fast as cavity stripping. The slightly higher ratios obtained at N = 7 and 9rps may reflect an additional effect due to gas recirculation. As mentioned in the section on frequency response, recirculation starts under these conditions at about 7 rps. The time constant values in Table 2 reveal another interesting response characteristic; that of increasing
Gassed power dynamics Table 2. Parameters Impeller speed
Step input h?s-L
s-l
2.68-0.8
5
5
x 104)
I
0.8 -2.68
I
of disc turbine impeller
of 6rst order response approximation
Time Constant, 7 (s)
Time delay (s)
0.675
0.025
0.35
1915
(D = 0.1016 m) Steady-state gain
Approximate model
UIl-3 x 103) 12.7
I
-12.5 (0.35 8 l 1)
9
0.8 -2.68
0.65
0.025
7
0 -3.8
0.42
0.025
7
0 -2.75
0.55
0.025
7
0 -1.8
0.85
0.025
response time as the impeller speed increases. This is perhaps surprising at first sight, but understandable, since cavity size increases with impeller speed[3]. The speed of response is also affected by gas flowrate, and hence cavity size, as evidenced by the values at N = 7 rps. One further property noticeable in the responses is the small time delay effect. It in fact represents the transport delay from the solenoid valve to the distributor and from there to the impeller. The values reported in Table 2 were determined from the intersection of the smooth curves with the time axis. They are therefore subject to a certain degree of estimating error. Since the values are very small, and even zero in one case, it would be necessary to increase the sampling rate to, say, 20 ms, in order to improve on this. We have not fully established the cause of the “large” oscillations present in the power response, except for the contribution arising from the fluctuating gas flow at low flow conditions. Viewing the gassed power change in terms of a normal steady-state Pg/P vs Flc plot. which is highly nonlinear, suggests that the effect could be due to a nonlinear forcing of the system. On the other hand, the “normal motion” of the gas cavities may be responsible. In the previous section on frequency response it was established that the system is basically linear in the region where there is no gas recirculation (N = 5). However, at N = 7 and 9rps, this situation no longer holds and it is possible that additional gas flow entering the impeller through recirculation may be a factor. The responses inFig. 12, which were obtained using different step sizes, also confirm the nonlinear nature of the process. Nevertheless, the first order model approximation does provide a good overall representation of the individual power responses. The alternative, otherwise, would be to resort to the complexities of nonlinear analysis, and this would be very difficult to
12.7 (0.675 B l 1)
-71.4 -25
-31.4 -39.2
-71.4 (0.65 s + 1) -25 (0.42 s + 1) -31.4 CO.55 s + 1) -39.2 CO.85 8 + 1)
carry out in practice. On balance, it seems that the simple linear modelling approach is capable of providing useful insight into the complex power responses in gasliquid systems. Certainly, using narrower ranges of step input would help to minimise nonlinear effects. More important, though, the fast rising portion of the response is much less affected by oscillations and this is the region where important transitional cavity formation effects occur. Results on this aspect are to be presented in a further paper. CONCLUSIONS
The impeller power frequency response exhibits a forced resonance peak condition which depends on impeller speed. It appears that this is due to a dual pulse disturbance effect arising from additional recirculated gas flow. Prior to gas recirculation, the system is basically linear, but developing nonlinear effects occur as recirculation is established. A type of hysteresis nonlinearity is generated in the power recovery response following the input of a negative gas pulse. The limited range of frequency response obtained under certain conditions also increases the difficulty of analysis. The measured transient power response exhibits a series of “large” oscillations. These appear mainly towards the end of the response, so that the earlier, fast rising portion is comparatively smooth. Overall, the individual response behaviour is well-represented by a fist order model approximation. Accordingly, the power responses can be explained by the process of gas cavity formation described by Warmoeskerken, Feijen and Smith. Cavity formation is found to be approximately twice as fast as the reverse process of cavity stripping. In either case, the power dynamics depends significantly on impeller speed and gassing rate. For batch-operated processes, it is not expected that
M. GREAVPSet al.
1916
rapid power changes will greatly affect gas-liquid mass transfer if the fluid viscosity is fairly low. However, in continuous flow reactors, power transients may occur more frequently, depending on the sensitivity of interaction between the gas space above the liquid, and the liquid level itself. With higher viscosity fluids, the dynamic power response will probably be very much slower, in which case, mass transfer will be affected to a greater degree. The mechanical designer will also need to consider the forces likely to be imposed on the impeller shaft during a rapid power transient, in the event that they could exceed the allowable steady-state design loading. Acknowledgements-The authors are grateful to the SERC for financial support towards this project. One of us (KAHK) would also liketo thanktheuniversityof Bathfortheawardof aResearch Studentship. NOTATION
C D FIG G(s) H
I;
N NR
impeller clearance, m impeller diameter, m flow or aeration number, Q/ND3 process transfer function ungassed liquid height, in complex number steady-state gain, Jm-’ impeller rotational speed, s-’ impeller rotational speed at start of recirculation, -1 S ungassed impeller power, W dynamic gassed power of impeller, W steady-state gassed power of impeller, W dynamic gas flowrate, m3sp1 time. s
At T TX TY o w7
computer sampling interval, s tank diameter, m duration of input disturbance signal, s duration of output response, s frequency, rads s-l upper bound frequency, rad 5-l time constant, s
RlUWBENCEs
Ill Greases M. and Economides C. A., Proc. Third European Conf. on Mixing, p. 357 BHRA 1979. [21 Warmoeskerken M. M. C. G., Feijeo J. and Smith J. M., 1. Chem. E. Symp. Ser. 198164 Jl. [31 Warmoeskerken M. M. C. G. and Smith 1. M., Proc. Fourth European Conf. on Mixing, p. 237. BHRA 1982. [4j Cooper A. R. and Guitierrez H. J.. Chem. Engng I. 1979 17 13. 151Hoagen J. O., Measurements and Control Applications, 2nd Edn. I. S. A. 1979. [61 Murrill P. W., Pike R. H. and Smith C. L., Chem. Engng 1969, Feb. 24 105. [71 Kobbacy K. A. H., Ph.D. Thesis, Universitv of Bath 1981. [81 Clemenis W. C. and Schnellle K. B., Ini. Engng Chem. Proc. Des. Dev. I%3 2 94. [9] Luyben W. L.. P&cess Modelling, Simulation nnd Control for Chemical Engineers. McGraw-Hill, New York 1973. 1101 Johnson J. L. and Fan L. T., A.l.Ch. E.J., 1966 12 161 1026. illi Lees S. and Dougherty R. C., J. Basic Engng (Trans.~ASME D), 1967 89 445. I121 Pollock 0. G. and Johnson A. I., Can. I. Chem. Engng 1969 47 565. I131 GreavesM. and Kobbacy K. A. H., 1. Chem. E. Syrup. Ser. 1981 64 Ll. I141 Cohen W. C. and Johnson E. F., Chem. Engng Prog. Symp. Ser. 1961 57 86. [ISI Cooper A. R. and Gutierrez H. I., Chem. Engng J. 1979 17 19. I161 Hansen K. W. and Jorgensen S. B., Chem. Engng Sci. 1976 31 587.
0009-2509183 Pergamon
38, No. L1.pp. 1909_1!J141983
GASSED POWER DYNAMICS M. GREAVES*,
OF DISC TURBINE
K. A. H. KOBBACYt
$3.0 + -00 Press Lid.
IMPELLER
and G. C. MILLINGTON
School of Chemical Engineering, University of Bath, Claverton Down, Bath BA2 7AY, England (Received 23 September 1982; accepted
14 April 1983)
Abstract-Measurements have been made of the dynamic power response of a disc turbine impeller, fallowing sudden changes in the input gas flowrate. Transient and frequency response results are presented for a number of impeller speeds. A very satisfactory representation of the smoothed-transient response is obtained using a simple linear first order model. The power change that occurs over the transient period is explained by the process of gas cavity formation described by Warmoeskerken, Feijen and Smith. It is found that the speed of formation and stripping of gas cavities depends on the impeller speed and gassing rate.
tNTRODUCTlON
The investigation described in this paper represents a more extended and detailed evaluation of the gassed power dynamics of a disc turbine than that attempted in an earlier study by Greaves and Economides[l]. They identified three main response regions. At low speeds, prior to the start of gas recirculation, the frequency response was characterised by a single “high” frequency component. Under fully established gas recirculation conditions, only a single “low” frequency component was present. In the transition region between these two regions, the response contained both “high” and “low” frequency components. A method was suggested for estimating the rate of gas recirculation based on the measured time constants and the mean residence time of the gas in the tank. Very little attention has otherwise been directed to this aspect of gas-liquid mixing operations. This is somewhat surprising, since the two-phase process depends critically on maintaining a dynamic equilibrium. From a practical viewpoint, gas-liquid reactors are subjected to sudden disturbances, such as changing gas supply or impeller speed. These may arise through start up or shutdown, or even during normal operation. Such effects can impose high loadings on the impeller drive shaft, or may seriously distort the dispersion and mass transfer processes in the vessel. At a fundamental level, there is an important need for more detailed understanding of the cavity formation processes occurring behind the impeller blades. This is because the power characteristics are mainly determined by the gas filled cavities behind the blades. Warmoeskerken et 01.[2,3] have recently provided a more unified description of this complex process, though it still lacks quantitative understanding. The rate at which these processes take place is a& important element in this understanding and requires the additional perspective of dynamic measurement. Various experimental methods are available for the study
of process
dynamics.
which has been widely used for obtaining the frequency response of heat exchangers[4] and other chemical process systems[5]. The utility and convenience of the method lies in the fact that it can yield the full frequency response of a system relatively quickly, using one or two well-chosen pulses. Alternatively, step_testing provides a simple way of obtaining transient response information. The objective is to carry out a system identification, either in the frequency or time domain: The dynamical power responses in this paper refer to a 0.2m diameter vessel. This is small by industrial standards, but scale considerations aside, the gassing effect on power will be the same. Also we have deliberately restricted the investigation to a single coalescing fluid (tap waler) in order to concentrate attention on the basic hydrodynamic-impeller condition. It is important to bear in mind, however, that many industrial fluids are, to varying degrees, coalescence-retarding. Such systems produce smaller bubble size, thereby lowering the speed for onset of gas recirculation. Two interesting effects of gas recirculation are identified in the paper. THEORETICAL CONSIDERATIONS
Pulse testing The general technique (see, for example, Murrill et al.[6]) involves disturbing the input, or independent variable. x(t). in a continuous manner over a short period of time and returning it to its original steady state. This produce? a pulse output response, y(t). In our case, the input is the sparged gas flowrate and the response is the impeller power. By definition, the transfer function of the process is:
From the definition of the Laplace substituting s = iw, we obtain
transform,
and after
Pulse testing is a technique
y(t) sin (ot) dt
y(t) cos (wt) dt - i *Author to whom correspondence should be addressed. fPresent address: Reservoir Department, Kuwait Oil Company. Ahmadi, Kuwait.
x(t) cos Cot) dt - i
I-0
x(t) sin (wt) dt (2)
1909
M. GREAVES et 01.
1910
A computer programme was written[7] to sample the input and output responses and calculate the Fourier Integral Transforms (FIT) in eqn (2), and thence, the frequency response, i.e. magnitude ratio and phase angle. These quantities are, respectively, the absolute value and argument of G(iw). When tested against standard responses reported in the literature the computed result gave excellent agreement. The steady state gain is simply obtained from the ratio of the areas under the output and input curves: =Y K =
I OT. I
0
Y Ctl dt .
(3)
x(t) di
The accuracy of frequency response data obtained from pulse tests is primarily determined by low frequency truncation effects and the error incurred in numerical integration. In general, the indicatiop is that[g], truncation tends to make the low frequency portion of the response abnormally flat, whilst the high frequency end is attenuated too sharply. Quantitatively though, this will depend on the type of process. The computational error involved in numerical evaluation of the FIT’s is primarily dependant on the approximating functions used. In turn, this is dependant on the computer sampling interval, At. “Breakdown” in the numerical integration, beyond the upper limit of frequency, is evidenced by degeneration of the frequency response into a series of meaningless oscillations.
-AL.
Figure 1 shows the equipment used for the dynamical tests. They were carried out in a flat-bottomed cylindrical vessel of 0.2 m diameter. The vessel was equipped with 4 wall baffles and a centrally mounted disc turbine impeller. Three such impellers were used, of standard Rushton dimensions, having D/T ratios of 0.375, l/2 and 213. The impeller clearance was fixed at C/H = 0.25 with the liquid level maintained at H = T. Full mechanical details have been reported elsewhere [ 1,131. All of the experiments were performed at atmospheric pressure using filtered tap water. Before a dynamic test was carried out, the operating conditions of the vessel were allowed to reach steady state. The needle valve (Vl) in Fig. 1 was used to adjust the steady state gas flowrate. A solenoid valve (s) was then used to introduce the step or pulse change in gas flow rate, with the needle valve V2 being used to adjust the height. Typical input step and pulse changes are shown in Fig. 2. The input gas flow disturbance and resulting power response were sampled automatically by a microcomputer data acquisition system. This subsequently performed the Fourier transform calculations for the pulse tests. It was not necessary to correct the torque measurements for frictional error since the steady state component was subtracted-out of the &mamic response.
Step tesfing
The shape of the transient response, or process reaction curve, is used to find the parameters of an approximate lumped-parameter model. In process dynamics/control terminology, this usually means a first order plus dead time (FODT), or second order plus dead time model (SODT):
(FODT)
G(s) =
Keerm (7,s
+
1l
(4) Fin. 1. Instrumentation and control of stirred tank for dynamic tests.
The steady state gain K, dead time 7m and the respective time constants can be obtained by standard methods (see, for example, Luyben[9]). Instrument dynamics If the measuring instruments exhibit significant dynamic lag then this must be taken into account in the evaluation of the process response. However, for systems where two measurements are made it has been shown[lO] that the instrument dynamics often cancel out. In our case, both the venturi gas flowmeter (1) and strain gauge torque transducer (2) have time constants less than IS ms. The computer interface is also extremely fast so that, overall, the instrument function effectively reduces to M&)/M,(s) = 1.
3.0
I
0.6
0 Fig.
II
IV
2. Step and pulse input
I
I
0
Time (5)
05
changes in gas flowrate.
Gassed plower dynamics of disc turbine
Frequewy(radss-b
pulse
0.8~
IO-’ to
1.95~
q
3 8 -y J-12-16-
Pulse duration 0 0.765 A 0.44s
Fii. 4. Effect of input pulse duration on power frequency. Input pulse 0.05X 10” to 1.95x 1O-3m-’ SC’,D = 0.1016m, N = 7 rps.
content of a rectangular pulse increases as its width decreases. The maximum frequency limit, calculated according to Clements and Schnelles’ criterionM, is based on a minimum normalised frequency content of 0.3. These are shown in the last column of Table 1, summarking the main test results. The reliable range of frequency is predicted to extend from 5 to over 10 rads s-l. The first observation from Table 1 is that the system dynamics, characterised by the various time constants and dynamic slopes, change with pulse height. This is of some concern, since it indicates a possible nonlinear condition[l2]. There is always the questionable aspect of fitting the slopes to the high frequency portions of the curves and, subsequently, estimation of the time constants from the corner frequency construction. Normally, we would expect that the errors in these estimates could be as high as 25%, but the results obtained here are reproducible to a much higher certainty than this. It is interesting that, at 5rps, there is close agreement of the time constant values. This indicates a basic linearity in that region and the nonlinearity, therefore, occurs at speeds higher than this. The dependency of the power dynamics on impeller speed generally confirms the effect noticed in the earlier
lo
Fig. 3. Effect of signal sampling rate Input
a0
g-4-
Using a sampling period of At = 2Orps, produces convergence up to 15.7radss-‘. This covers the range of interest in this study. On the other hand, reducing the width of the pulse reveals more detail in the frequency response (Fig. 4). This is expected, since the frequency
I.0
1911
Oa--&-,+0-o
RESULTS AND DBCUSSION
Frequency response In all, 22 pulse tests were made using the middle sized impeller (D/T =OS). The purpose of this wide-ranging series of investigations was to identify salient features of the power response and suitable test conditions, i.e. sampling rate, pulse height and duration. Figure 3 shows the effect of sampling rate on the frequency response (log modulus = 20 log magnitude ratio). At a sampling rate of 20 per second, the response starts to degenerate into a series of meaningless oscillations at about 4 rads s-‘. Increasing the rate to SO per second extends the observable range of frequency response. For convergence of the numerical FIT computations, Lees and Dougherty” proposed an upper bound on frequency:
-2000
impeller
on power frequency response. lo-‘m3s-‘, D=O.l016m, N =
I rps.
Table I. Summary of frequency response results for middle-size impeller (D/T = 0.5) Input Impeller speed
pulse (gas flow flawrate)
s-1
x 1041n3s-1
5
0.00 - 2.78
20
2.78
0.62
0.98
0.22
-52
7.0
5
0.80 -1.95
20
1.15
0.69
1.30
0.24
-55
6.1
7
1.95 d2.78
20
1.98
0.76
2.64
0.91, 0.20
-16. -52
5.8
7
0.80 - 2.78
20
1.15
0.44
1.64
0.58, 0.10
-25. -60
lQ.O
9
0.80 - 2.78
20
1.98
0.64
2.14
0.81. O.L!.
-17. -52
6.9
9
0.80 41.95
20
1.15
0.88
1.98
0.57, 0.09
-24, -50
5.0
11
0.80 -2.78
20
1.98
0.43
1.34
0.43
-18
10.2
11
0.80 - 1.95
20
1.15
0.54
1.20
0.29
-47
8.2
Pulse height (x 104m3s-11
Input pulse duration
Frequency
Sampling period Ins
*
output pulse duration s
Time Constant s
Dynamic slope dbfdecade
deviation limiis6
M. GREAVES et al.
N(i’) 0 IO l 12 v 14 a 17
K(J& -24.9 -46’4 -94.3 -242
.,s) OKl 044,0.25 1.340.27 1.75032
Fig. 5. Effect of impeller speed on power frequency response. Input pulse height 0.8 X 10m4 to 1.95 x lOedm3 s-‘, D = 0.106 tn.
Fig. 6. Effect of impeller speed on power frequency response.
study by Greaves and Economides [l]. From Fig. 5 it can be seen that, at the lowest impeller speed (5rps). when the impeller has just overcome the flooding condition[l3], a single time constant can be assigned to the response curve. At 7rps the system is close to the recirculation condition (N = NR), and well into the recirculation region at 9rps (N > NR). The frequency response at both of these speeds is characterised by two distinct slopes. It was claimed that the higher frequency component (low time constant) was associated with the onset of gas recirculation. Substantial gas recirculation is established when the speed is increased to 11 rps and the response curve changes again, so that there is now only a single dynamic slope. The dual dynamic slope condition, which appears in the frequency response al certain impeller speeds, is due to the existence of a peak or resonance effect. This resonance behaviour was not identified by Greaves and Economides because of the rather low sampling rate they used. It is a particular effect which is characteristic of distributed parameter systems[l4]. Similar dynamic behaviour has also been reported for shell and tube heat exchangers[lS] and fixed bed catalytic reactors1161. In Fig. 5, the first peak occurs at 7rps at a frequency of about 6.5 rads s-l. At 9rps, the resonance peak is very sharp and greatly increased in amplitude. The resonance therefore first appears at low frequencies when the rate of gas recirculation is of the same order as, or less than, the gas sparging rate (N = 7, 9 rps). Under these conditions, the impeller is affected by both sparged and recirculated gas and is consequently forced in a distributed manner. The process behaves as though it is subjected to a dual input disturbance; one coming from the sparged gas flow and one from the recirculated gas. No such effect occurs at 5 or If rps, when either the gas flow, is sparged gas flow, or the recirculated dominant. Pulse tests with the smallest impeller (D/T 0.375) confirm the previous observations, except that the resonance peaks tend to occur at lower frequencies (Fig.
6). The size of the peak at 17rps is not unlike a typical underdamped closed-loop control response, except that it lies below the low frequency gain axis. As shown in Fig. 6, there is a continuous increase in the steady state gain and time constant values as the spee*d of the impeller increases. The large drop in gain, which occurs at 20 rps, can be explained by the pregassing of the impeller caused by surface aeration. Figure 7 shows the frequency response for the largest impeller (D/T = 213). The limited frequency range, however, makes it rather difficult to estimate the dynamic slopes. For the smallest impeller [Q(t) = 0.175 x 10s4 m3 s-l, F lo = 0.02371, we observe a resonance peak occurring at about 6 rad SC’. The system is, in fact, just at the point of recirculation (N = NR). Another change in the frequency response occurs when equal positive and negative pulses are used. Figure 8 shows the hysteresis effect obtained. This points to either a serious nonlinearity, or a real process effect. It should be emphasised, however, that nonlinear effects of
Input pulse height 0 to 0.75
I.0
x
IO-’ m3 s-l, D = O.W62 m.
Frequency
(rads
5-l)
IO
Fig. 7. Effect of pulse. height on power frequency response. Ungassed initial condition, D = 0.135 m, N = 3 rps.
Gassed
1913
dynamics of disc turbine impeller
power
F'G
N(s-) pSCOYW> lnltlal
ov K= -8Z’x103Jm-3
*9 q 7
2.75~16~n?;’
-
final
5.3
0.0153
0051
32 5 15.0
0010900351 0009 O-0294
Fwst order dpproxrmatlon
l NPaatiwpvlsn K=9.22~lO~Jrn-~ (Log modulus is positive on ffdlnate)
I
I
I.0
I
IO Frequency
(rads 54)
Fig. 8. Power frequency from positive and negative pulse tests. D = 0.135 m, N = 3 rps.
Fig. 9. Cavity formation response. Step input 0.8 x 10Jm3 s-‘, D = 0.1016m.
this nature could lead to substantial errors in identification when applying linear systems theory. It also underlines the fact that further development of the pulse testing method for identifying the impeller power dynamics will be both difficult and complex. The transient response discussed next does not require any a priori assumption concerning the linearity of the process as a precursor to analysis.
Pg(0) is the initial steady state gassed power (t < 0), the oscillations appear even more pronounced, as shown in Fig. 11. A further set of cavity response tests was made starting from an initially ungassed condition. These are shown in Fig. 12. Interestingly, they alSo show the same type of oscillatory behaviour observed in the cavity stripping responses. The cause of the oscillatory behaviour in the impeller power response is not immediately apparent. Fist though, attention is directed to the sparged gas flow supply, since this may have introduced some sort of forcing condition on the process. Returning to Fig. 2, the fluctuations present at the low flow rate (Q = 0.8 x 10m4m3 s-‘) show a variation of about 4 or 5%. Clearly then. we would expect there to be some contribution from this source. Calculating this on the basis of the total power change that occurs over the transient (the ultimate change) does not, however, fully account for the oscillatory effect. On the other hand, when a positive step change is made, the resulting gas flow (Q=2.68 x
TRAN8lENT RESPONSE
Figure 9 shows the cavity formation response at N = 5, 7 and 9rps, following upwards step changes in gas flowrate. Only at the lowest impeller speed does the power response follow what can be considered to be a smooth path. At the two higher speeds, there is a degree of oscillation in the responses. The counter process of cavity stripping, following a downwards step change in gas flowrate (Fig. lo), reveals a greater degree of oscillation in the response. If this set of responses is plotted, alternatively, on a ratio basis, i.e. Fg(t)/Pg(O), where
l4-
z
l2-
F ZIO-
a9 -Frst
18.9
OCQ940-CO9
order approxlmotion
b &B3 e
6-
h n” 24 B 2
0 t-
Fig. IO. Cavity stripping response. Step input 2.68 x IO-’to 0.8 x W4 m s-‘, D = 0.1016m.
IO_’
to 2.68 x
1914
M.
Fig.
GREAVES
11.Transient power response (cavity stripping) plotted as
- 0.0 0 O-0 l 00
I
3.80 2.75 I.80
Fimt order approximation
OPl”““““““‘ll 0
05
I-O
I.5
et al.
20
2.5
3-o
3-5
4.0
4.3
Fig. 12. Cavity formation response from initially uugassed condition. D = 0.1016 m, N = 7 rps.
10e4 m3 S-‘) is extremely stable and no contribution seems possible under this condition. Another feature which is noticeable concerning these “large” oscillations iu power is that they tend to appear mainly towards the end of the response, i.e. after the initial fast rising portion, This relatively smooth, fast portion of the response is estimated to be about 60% of the power change. Thus, the “large” oscillations are mainly associated with the minor portion of the response (in terms of power change), but they dominate the slower portion, which affects the total response time. The latter tends to increase with impeller speed, from about 1.5 s to 3.5 s for the initially gassed condition, up to 4.5 s when no gas is initially present.
Pg(l)lPg(O)
Vs time.
If the transient power responses are considered in terms of the smooth curves represented on Figs, 9-12, then the response characteristics would appear to be consistent with the description of gas cavity formation given by Warmoeskerken et aL[2,3] for a 4-blade disc turbine. Following an upwards step change in gas flowrate (Fig. 9), the clinging cavities rapidly transition to form large cavities. The largest change in power is associated with this transition from clinging, to three large and three clinging cavities. Three stable large cavities are reported to occur for 0.025 i FOG < 0.035. With increasing gas flow into the impeller, the cavities continue to grow in size and this region is associated with a more gradual reduction in power. At the lowest impeller speed, as FIG tends ultimately to 0.051, the impeller is probably close to flooding. The steady state gains in Table 2, for N = 7 rps. illustrate the relative power change accompanying different step changes in gas flowrate. Moving in the other direction involves the stripping of large gas cavities, which are very stable. It is therefore expected that this process will be slower than cavity formation. We can best estimate the speed of the two processes by calculating their respective time constants on the basis of a first order mode1 approximation. These values (determined from the 63.2% response) are summarised in Table 2. The ratio of time constants between stripping and cavity formation, for the initially gassed condition at N = 5, 7 and 9 rps are, respectively, 2.09, 2.10 and 2.30. This means that the cavity formation process is approximately twice as fast as cavity stripping. The slightly higher ratios obtained at N = 7 and 9rps may reflect an additional effect due to gas recirculation. As mentioned in the section on frequency response, recirculation starts under these conditions at about 7 rps. The time constant values in Table 2 reveal another interesting response characteristic; that of increasing
Gassed power dynamics Table 2. Parameters Impeller speed
Step input h?s-L
s-l
2.68-0.8
5
5
x 104)
I
0.8 -2.68
I
of disc turbine impeller
of 6rst order response approximation
Time Constant, 7 (s)
Time delay (s)
0.675
0.025
0.35
1915
(D = 0.1016 m) Steady-state gain
Approximate model
UIl-3 x 103) 12.7
I
-12.5 (0.35 8 l 1)
9
0.8 -2.68
0.65
0.025
7
0 -3.8
0.42
0.025
7
0 -2.75
0.55
0.025
7
0 -1.8
0.85
0.025
response time as the impeller speed increases. This is perhaps surprising at first sight, but understandable, since cavity size increases with impeller speed[3]. The speed of response is also affected by gas flowrate, and hence cavity size, as evidenced by the values at N = 7 rps. One further property noticeable in the responses is the small time delay effect. It in fact represents the transport delay from the solenoid valve to the distributor and from there to the impeller. The values reported in Table 2 were determined from the intersection of the smooth curves with the time axis. They are therefore subject to a certain degree of estimating error. Since the values are very small, and even zero in one case, it would be necessary to increase the sampling rate to, say, 20 ms, in order to improve on this. We have not fully established the cause of the “large” oscillations present in the power response, except for the contribution arising from the fluctuating gas flow at low flow conditions. Viewing the gassed power change in terms of a normal steady-state Pg/P vs Flc plot. which is highly nonlinear, suggests that the effect could be due to a nonlinear forcing of the system. On the other hand, the “normal motion” of the gas cavities may be responsible. In the previous section on frequency response it was established that the system is basically linear in the region where there is no gas recirculation (N = 5). However, at N = 7 and 9rps, this situation no longer holds and it is possible that additional gas flow entering the impeller through recirculation may be a factor. The responses inFig. 12, which were obtained using different step sizes, also confirm the nonlinear nature of the process. Nevertheless, the first order model approximation does provide a good overall representation of the individual power responses. The alternative, otherwise, would be to resort to the complexities of nonlinear analysis, and this would be very difficult to
12.7 (0.675 B l 1)
-71.4 -25
-31.4 -39.2
-71.4 (0.65 s + 1) -25 (0.42 s + 1) -31.4 CO.55 s + 1) -39.2 CO.85 8 + 1)
carry out in practice. On balance, it seems that the simple linear modelling approach is capable of providing useful insight into the complex power responses in gasliquid systems. Certainly, using narrower ranges of step input would help to minimise nonlinear effects. More important, though, the fast rising portion of the response is much less affected by oscillations and this is the region where important transitional cavity formation effects occur. Results on this aspect are to be presented in a further paper. CONCLUSIONS
The impeller power frequency response exhibits a forced resonance peak condition which depends on impeller speed. It appears that this is due to a dual pulse disturbance effect arising from additional recirculated gas flow. Prior to gas recirculation, the system is basically linear, but developing nonlinear effects occur as recirculation is established. A type of hysteresis nonlinearity is generated in the power recovery response following the input of a negative gas pulse. The limited range of frequency response obtained under certain conditions also increases the difficulty of analysis. The measured transient power response exhibits a series of “large” oscillations. These appear mainly towards the end of the response, so that the earlier, fast rising portion is comparatively smooth. Overall, the individual response behaviour is well-represented by a fist order model approximation. Accordingly, the power responses can be explained by the process of gas cavity formation described by Warmoeskerken, Feijen and Smith. Cavity formation is found to be approximately twice as fast as the reverse process of cavity stripping. In either case, the power dynamics depends significantly on impeller speed and gassing rate. For batch-operated processes, it is not expected that
M. GREAVPSet al.
1916
rapid power changes will greatly affect gas-liquid mass transfer if the fluid viscosity is fairly low. However, in continuous flow reactors, power transients may occur more frequently, depending on the sensitivity of interaction between the gas space above the liquid, and the liquid level itself. With higher viscosity fluids, the dynamic power response will probably be very much slower, in which case, mass transfer will be affected to a greater degree. The mechanical designer will also need to consider the forces likely to be imposed on the impeller shaft during a rapid power transient, in the event that they could exceed the allowable steady-state design loading. Acknowledgements-The authors are grateful to the SERC for financial support towards this project. One of us (KAHK) would also liketo thanktheuniversityof Bathfortheawardof aResearch Studentship. NOTATION
C D FIG G(s) H
I;
N NR
impeller clearance, m impeller diameter, m flow or aeration number, Q/ND3 process transfer function ungassed liquid height, in complex number steady-state gain, Jm-’ impeller rotational speed, s-’ impeller rotational speed at start of recirculation, -1 S ungassed impeller power, W dynamic gassed power of impeller, W steady-state gassed power of impeller, W dynamic gas flowrate, m3sp1 time. s
At T TX TY o w7
computer sampling interval, s tank diameter, m duration of input disturbance signal, s duration of output response, s frequency, rads s-l upper bound frequency, rad 5-l time constant, s
RlUWBENCEs
Ill Greases M. and Economides C. A., Proc. Third European Conf. on Mixing, p. 357 BHRA 1979. [21 Warmoeskerken M. M. C. G., Feijeo J. and Smith J. M., 1. Chem. E. Symp. Ser. 198164 Jl. [31 Warmoeskerken M. M. C. G. and Smith 1. M., Proc. Fourth European Conf. on Mixing, p. 237. BHRA 1982. [4j Cooper A. R. and Guitierrez H. J.. Chem. Engng I. 1979 17 13. 151Hoagen J. O., Measurements and Control Applications, 2nd Edn. I. S. A. 1979. [61 Murrill P. W., Pike R. H. and Smith C. L., Chem. Engng 1969, Feb. 24 105. [71 Kobbacy K. A. H., Ph.D. Thesis, Universitv of Bath 1981. [81 Clemenis W. C. and Schnellle K. B., Ini. Engng Chem. Proc. Des. Dev. I%3 2 94. [9] Luyben W. L.. P&cess Modelling, Simulation nnd Control for Chemical Engineers. McGraw-Hill, New York 1973. 1101 Johnson J. L. and Fan L. T., A.l.Ch. E.J., 1966 12 161 1026. illi Lees S. and Dougherty R. C., J. Basic Engng (Trans.~ASME D), 1967 89 445. I121 Pollock 0. G. and Johnson A. I., Can. I. Chem. Engng 1969 47 565. I131 GreavesM. and Kobbacy K. A. H., 1. Chem. E. Syrup. Ser. 1981 64 Ll. I141 Cohen W. C. and Johnson E. F., Chem. Engng Prog. Symp. Ser. 1961 57 86. [ISI Cooper A. R. and Gutierrez H. I., Chem. Engng J. 1979 17 19. I161 Hansen K. W. and Jorgensen S. B., Chem. Engng Sci. 1976 31 587.