Chen, J. C., ASME Paper 63-HT-34, 1963. Collier, J. G., Hewitt, G. F., Trans. Inst. Chem. Eng. 39, 127 (1961). Dukler, A. E., C h m . Eng. Progr. Symp. Ser. 56 (30), 1-10 (1960). Elliot, L. C., Dukler, A. E., paper SWDj21, First International Symposium on Water Desalination, Washington, D. C., Oct. 3-9, 1965. Fulford, G. D., Aduan. Chem. Eng. 5 , 151-236 (1964). Isbin, H. S., et al., Trans. A S M E J . Heat Transfer 83, 149 (1961). Jakob, Max, “Heat Transfer,” Vol. 1, pp. 618-20, Wiley, New York, 1953. Ludviksson, V., Lightfoot, E . N., A.I.Ch.E. J . 14, 620 (1968). McAdams, W. H., Drew, T. B., Bays, G. S., Jr., Trans. A S M E 62, 627 (1940). Mitchel, W. T., Quinn, J. A., Chem. Eng. Sci. 23, 503 (1968). Norman, W. S., Binns, D. T., Trans. Inst. Chem. Engrs. 38, 294 (1960). Norman, W. S., McIntyre, V., Trans. Inst. Chem. Engrs. 38, 301 (1960). Penman, T. O., Tait, R. W. F., Ind. Eng. Chem. Fundamentals 4, 407 (1965).
Sack, Melvin, Chem. Eng. Progr. 63 (7), 55 (1967). Sinek, J. R., Young, E . H., Chem. Eng. Progr. 58 (In), 74 (1962). Thomas, D. G., A.Z.Ch.E. J . 14, 644 (1968). Thomas, D. G., Ind. Eng. Chem. Fundumentals 6, 97 (1967). Tong, L. S.,“Boiling Heat Transfer and Two-Phase Flow,” p. 48, Wiley, Kew York, 1965a. Tong, L. W., “Boiling Heat Transfer and Two-Phase Flow,” p. 123, Wiley, New York, 1965b. Unterberg, Walter, Edwards, D. K., A I.Ch.E. J . 11, 1073 (1965). Wilke, W., VDI Forsch. No. 490 (1962). Wallgren. B. O., Ind. Eng. Chem. Fundamentals 6, 479 (1967). Young, R. K., Hummel, R . L., “Higher Coefficients for Heat Transfer with Nucleate Boiling,” 7th National Heat Transfer Conference AIChE-ASME, August 1964. RECEIVED for review July 23, 1969 ACCEPTED January 19, 1970 Research jointly sponsored by the Office of Saline Water, U. S. Department of the Interior, and U. S. Atomic Energy Commission under contract with the Union Carbide Corp.
DYNAMIC RESPONSE TO CONCENTRATION CHANGES OF A PARTLY FILLED HORIZONTAL TANK R O B E R T
M . H U B B A R D ,
G E R A R D
C .
L A H N ’ ,
J O S E P H
A N D
1.
W I L L I A M
DALFERES W .
I l l ’ ,
Z A L E W S K 1 3
Department o f Chemical Engineering, University of Virginia, Charlottesuille, Vu. 22901
Natural mixing in a partly filled horizontal tank due to flow rate and relative position of inlet and outlet w a s studied on three tanks from 5.7 to 36 inches in diameter. Length-diameter ratio w a s 3 to 1. Three liquid levels were maintained and flow rate was varied. Concentration transfer functions usually included a small dead time or delay before any change in output occurred. Dead time increased as tank diameter increased. Bypass or short-circuiting from inlet to outlet without mixing occurred when outlet w a s directly under inlet. Time constants are related to mean residence time. Relations are recommended for concentration transfer functions for many arrangements of inlet and outlet.
THEchemical plant often includes a surge tank between major process units. Such tanks are useful in smoothing out concentration and level changes, preventing excessive deviations in feed to subsequent process equipment. One object of this study was to determine concentration dynamics of partly filled horizontal tanks in which mixing occurred only as the result of inflow rate and the relative Present address, Union Carbide Chemicals Co., Taft, La. Present address, Esse Research and Engineering co., Flor$am Park, N. J. Present address, General Electric Co., Schenectady, N. Y. 1
2
positions of inlet and outlet nozzles. The principal object was to do this on several tanks of progressively larger diameters to permit extrapolation of results to surge tanks of commercial size. The principles of dynamic response and pioneering work in this field have been described in the important monograph by Hougen (1964). Considerable work has been done, particularly by Danckwerts (1953, 1954), on flow systems in which mixing was described by the distribution of residence times. Levenspiel (1962) and Levenspiel and Bischoff (1963) proposed several models to represent flow Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
323
of fluids through vessels. Gutoff (1960) described the effectiveness of mixing tanks in smoothing cyclic fluctuations between process units, but mixing by agitation occurred in his tanks. Stafford (1962) determined local residence times in a small vertical vessel in which mixing occurred only as the result of position and velocity of a submerged inlet stream. T o the authors' knowledge, concentration dynamics of a partly filled horizontal tank has not been studied heretofore.
1-P PTis + 1 M C=PM+M= Tis + 1 Tis + 1
where If
T3 = PTi
V/q = Ti - Ts = TI(1 - P )
M a t hematical Models
For each study made by the authors, the process unit consisted of a horizontal tank containing a constant volume of liquid, because level was kept constant, but through which liquid flowed a t several rates. Under these conditions, when concentration of the solution flowing into the tank was changed in some manner the concentration of the outflow liquid could be expressed as a differential equation, the order of which depended on the nature of the mixing process in the tank. Only four simple mathematical models were considered for the process. Let
V = constant volume of liquid contained in the tank, cu ft q = constant inflow and outflow rate, cu f t per second m = solute concentration in inflow, moles per cu f t c = solute concentration in outflow, moles per cu f t The first mathematical model considered was that of complete and instantaneous mixing of liquid entering with that in the tank. Assuming all variables represent only changes from the initial steady-state condition, the Laplace transform of the fundamental differential equation of this process produces the first-order transfer function
C/M=
1
Ts + 1
~
where C and M are the transformed concentrations, s is the Laplace transform variable, and T = V / q = mean residence time of liquid in the tank, seconds, called the time constant. Under some conditions this first-order element appeared as a component part of a final transfer function. Under other conditions, a model described as secondorder
since (1 - P ) is that fraction not bypassed but completely mixed, it is evident that the bypass stream takes no time at all to flow through the tank. A fourth model was that of plug flow without mixing in which liquid took, V / q = L seconds to travel through the tank. The transfer function of this process is C / M = e-Ls where L is the holdup time, seconds, also called the dead time or transportation lag. Transportation lag also occurred in the flow of liquid through a tube from a point where a change occurred to the point where this change was measured. Although this form of transportation lag existed in the experimental work, the lag times were calculated and deducted from observed times. However, the transfer function of dead time in the tank was observed in almost all experiments as a definite lag between time of the inlet change and time the first change in output composition was observed. The final transfer functions of all tanks studied under all conditions of operation were expressed as combinations of several of these mathematical models. If all models are combined without interaction
C/M =
(T?s+ 1) e-Ls (T*s+ 1)( T 1 S + 1)
This theoretical relation was used, with empirical relations for L , to calculate time constants for comparison with observed values. In general, agreement was good; 58% of the results agreed with the theoretical models within 10%. Experimental Work
was a component part of a final transfer function. Practically, this was possible when the liquid entering the tank mixed completely with only a portion of liquid in the tank, and then this new solution mixed completely with the balance of material in the tank. Now
Ti
+ T?= V / q
A third model assumed that some fraction P of the inflow bypassed all liquid in the tank and flowed directly to the outlet, but fraction (1 - P ) was completely mixed. This process may be pictured as
I
324
'
I
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
Two model tanks and one small commercial-size tank were used. All had a length-diameter ratio of approximately 3 to 1. On the model tanks, inlet nozzles were mounted on the top at the center and one tank diameter to one side of center; also a nozzle was placed in the center of one dished head. Outlet nozzles were placed a t three points on the bottom, at the center, and one tank diameter on each side of center. The model tanks were made first from polyethylene and then from transparent acrylic plastic sheet with cylindrical body and dished heads to simulate the customary tank shape closely. The largest was a steel tank of commercial manufacture. The shape, characteristic dimensions, and location of nozzles for all three tanks are shown in Figure 1. For each tank size, Table I shows the flow rates, values of Reynolds number in inlet nozzle, values of the Froude number for horizontal flow in the tank, and residence time cal-
PLASTIC TANKS Table 1. Experimental Conditions in Tanks
H o u Rate, Gal Min
Inlet Nozzle Inlet Diameter, Reynolds Inch No.
Tank Level
Froude No. in Residence Tank Time, x lo6 See SMALL TANK:
0.125
5,700
v, '/L \
8.8 3.1 1.4
L A R G E TANK :
0.250
3,140 4,720 8,250
$5 !L
'2
3.1 7.0 22.0
0.52
9,600
x I/;
i/Ik
10.5 3.8 1.7
0.52
14,400
?/* 92 5j8
19.7 6.9 3.2
1.61
49,600
V&
92 j/h
8: 36.0 I N .
0 . 5 2 IN.
3 F
T
.
~
INLETS
222 340 452
3 F
T
.
2 IN. PIPE cz ~ OUTLET
Figure 1. Experimental tanks 148 236 302
N OZZ L E CON FIGURATIONS
I
Series 5. Steel Tank (Dalferes),Low Flow 21.4
:
1/4 I N . I L A H N I
8.5 FT.
Series 4. Large Model Tank, High Flow 2.03
5 . 7 IN.
STEEL TANK 245 160 86
Series 3. Large Model Tank (Dalferes), Low Flow 1.36
i
A = 12.0 IN.
C
Series 2. Small Model Tank (Lahn) 0.212 0.317 0.555
A
C = 1/8 I N . ( Z A L E W S K I ) ,
200 300 400
INSIDE D I A M E T E R OF I N L E T N O Z Z L E
B = 17.2 I N .
Series 1. Small Model Tank (Zalewski) 0.189
B2'
108 3.8 1.7
394 586 780
23.2 8.2 3.8
268 396 530
P
Series 6. Steel Tank, High Flow 31.6
1.61
73,400
% % 5/%
culated from quantity of liquid in the tank and flow rate. Three liquid levels were maintained in each tank. The various nozzle configurations and their designations are shown in Figure 2 . Transfer functions were determined on each tank size using the pulse technique, although Lahn (1965) used sinusoidal excitation to check the results obtained by Zalewski (1965) on the same size tank using the pulse method. The flowing fluid used by each was a dilute sodium chloride solution, and the tracer introduced was a more concentrated salt solution. Constant-head tanks were used to store working solutions, and flow rates were measured by rotameters. Levels in the tanks were kept at K, %, and K of full depth by control of the outflow rate. Concentrations were determined by measuring electrical conductivity of solutions using flow-through cells located at the inlet and outlet connections of the tank. An a.c. Wheatstone bridge circuit with amplifier and rectifier, the same as used by Clements and Schnelle (1963b), produced a d.c. voltage proportional to resistance of the conductivity cell. Electrical outputs from the circuits associated with the inlet and outlet cells were recorded simultaneously on a Sanborn Model 150 twochannel galvanometer recorder. Calibration of each conductivity cell produced a linear relation on a log-log plot of molal concentration of solution against cell resistance. Dalferes (19681, working with both the large plastic
(2) (5 66
Figure 2. Designation of nozzle configurations
tank and the steel tank, used water as the flowing liquid and a dye solution as a tracer. A Bausch & Lomb Spectronic 20 colorimeter, modified to accept a flow-through sample cell, measured transmittance of outlet solution from the model tank and of a sample of the outflow from the steel tank. The colorimeter was also modified to provide a potential output, proportional to transmittance, that was recorded on a Texas Instruments Co. Servo/ Riter potentiometer recorder. The wavelength selected was 610 mk, that of maximum absorption of the dye solution used, Calcocid Blue-Black, Ex. Conc., a product of the American Cyanamid Co. This solution followed Beer's law for concentrations from 0 to 11 ppm, for which transmittance varied from 100% to 18.4%. Zalewski, working on the smaller model tank through which 0.002 molal salt solution flowed, used a mechanically operated hypodermic syringe to inject from 0.4 to 1.6 ml of 1 molal sodium chloride solution in about 6 seconds Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
325
into the inlet stream. The input trace was almost a rectangular pulse; the output trace was recorded until solution concentration returned to the initial value in about 10 minutes. Working on the larger model tank, Dalferes trapped 248 ml of 600-ppm dye solution in a loop of tubing which, by turning a four-way valve, was made part of the stream flowing into the tank. Shape and duration of this input pulse were determined using 60-ppm dye concentration which could be measured by the colorimeter. Several trials were made to be sure the technique gave reproducible results; after this, input pulses were measured only to check operation and not during a pulse test of the tank. The pulse lasted about 6 seconds. Its shape was asymmetric with a steep trailing edge; therefore it was more favorable with respect to a wider range of input frequencies excited. The input pulse generated for the steel tank was obtained by mechanically injecting 1500 ml of 2400-ppm dye solution in about 5 seconds into the stream of water flowing into the tank. This addition did not materially affect inlet flow conditions. The shape and duration of the input pulse were recorded by repeating the operation using a solution containing only 150-ppm dye and sampling the mixture a t the inlet to the tank. Lahn, working on the small tank, used sinusoidal variation of the input stream. Identical conductivity cells and bridge-amplifier-rectifier circuits were used for inlet and outlet streams; the cells were placed at the tank nozzles to minimize transportation lags. Sodium chloride solutions of two different concentrations were stored in constanthead tanks. Tubing connections led from each to a Research Controls (Precision Products and Controls, Inc.) Model 75 S %-inch size pneumatic control valve having size F linear trim. One valve used air-to-open, while the other operated air-to-close. A mechanically driven pneumatic sine generator, built by the Taylor Instrument Cos. and producing an output in the range 6- to 12-psi air pressure, was used to operate these valves simultaneously. The result was a sinusoidal variation of the sodium chloride solution concentration in the feed to the tank with little variation in feed flow rate. The inlet flow rate was adjusted by varying the elevation of the two constant-head tanks; total outflow was measured by a rotameter and restricted to maintain a given level in the tank. Inlet and outlet concentrations were recorded simultaneously on two channels of the Sanborn Model 150 recorder. The linearity of the system studied was demonstrated by the fact the output trace was sinusoidal. Since transparent plastic tanks were used, photography was an obvious method of recording behavior of liquid in the tank. Lahn experimented with dye injection to detect bypass and took some still pictures. Dalferes obtained some motion pictures to show the effect of inflow through different nozzle positions. Visual and photographic observation confirmed the possibility of a second-order lag which requires rapid mixing in one small portion of the tank, followed by slower mixing of the solution formed with liquid in the rest of the tank.
configurations of inlet and outlet location and three flow rates. I n the pulse test measurements of Zalewski and Dalferes, the value of dead time for the element of the transfer function was measured directly from the chart, and the output curve was assumed to start at the end of the dead time period. The output chart records were converted into Bode diagrams using the method derived by Draper, McKay, and Less (1953) and described in detail by Clements and Schnelle (1963a). Following a Fortran program provided by Schnelle (1964), Zalewski prepared a program for a Burroughs B 5000 computer using the trapezoidal approximation for integration. The result was a printed record of amplitude ratio and phase angle for an assumed range of angular frequency of a sinusoidal input, generally from to 0.5 radian per second. Significant values were obtained and curves were reproducible over a t least two decades within this frequency range. Once values obtained from the computer output were plotted as a Bode diagram, curves were fitted by trial until the sum of first-order lag and lead elements agreed with the experimental points. Figure 3 shows a Bode diagram drawn through two sets of data taken under the same conditions in series 6 on the largest tank having configuration 111. The curve was drawn as the sum of two first-order lagging elements having the corner frequencies indicated, from which time constants were determined. Using a dead time obtained from the original chart record and not the Bode diagram, the transfer function derived was
CIM =
e-3zs (290s + 1)(120s + 1)
I .o
0
5a w
0.1
P
ao I
= ,0034 1
1
0061
l ! l ! I l ,
TI
,001
0.01
01
k X
3 6 " D I A M E T E R TANK CONFIGURATION IU
a
.OI003 = Iill
I
I
1/2 L E V E L
I
-210
Results
All experimental results were converted into Bode diagrams. The data of Lahn provided values of amplitude ratio and phase angle directly from the recorder charts. Using the small model tank half full, he tried three 326 Ind. Eng. Chem. Process Des. Develop.,
Vol. 9, No. 2, 1970
-240 .0001
1
1
1 l l l I ! l
,001
1
1
I 1
1 1 1 '
1
I
I I I I I
0.01
FREQUENCY, ( R A D S I S E C )
Figure 3. Bode diagram of second-order system
0.1
Since the mean residence time was 396 seconds and the sum of the time constants plus dead time is 442 seconds, agreement with the theoretical model is +12%. Figure 4 shows the Bode diagram drawn for data obtained on the large model tank in series 4 having configuration VII. The curve represents a single firstorder lag with a time constant 300, but a dead time was also observed. The transfer function derived was
C M =
orimeter on the dye stream from the 12-inch tank when outlet was below inlet. Tests on the steel tank were limited to two configurations; existing nozzles had to be used and the outlet was not directly under either inlet. I t was thought that dynamic similarity between tank sizes would be obtained with equal values of Froude number for liquid flowing through the horizontal open channel represented by the tank between inlet and outlet.
Froude number = u2/gd
e-''
where
(300s + 1)
Since the residence time was 302 seconds, agreement with the model is +2%. The results of all experimental work are given in Table 11, a listing of the time constants and dead time obtained. Agreement between known mean residence time V l q and the proper sum of the time constants and dead time given by Equation 2 is shown in the last column of this table as a per cent deviation. Satisfactory checks were obtained in material balances between the dye added and dye leaving the large model tank used by Dalferes. Duplicate runs were made for several experimental conditions by all investigators, and results were generally satisfactory. Dalferes varied the number of points resulting from the real output plus noise with points read from a smoothed curve; the tabulated values of amplitude ratio and phase angle were essentially the same. I t was more difficult to obtain good experimental results whenever the outlet was directly below the inlet. On the small tanks, air bubbles were carried out in the exit stream. These were probably responsible for the larger error observed in all results for this type configuration ( I , V). Air bubbles completely prevented use of the colI O
u = average liquid velocity, ft per sec cl = characteristic depth, assumed depth a t tank center,
ft g = acceleration of gravity, f t per
set'
Turbulence in the tank was generated by the impact of the inlet stream and quiet and uniform flow was not reached even though Froude numbers were very small. Only trends with changing Froude number were observed; no definite correlations were obtained. I t was not possible to maintain dynamic similarity of flow in the nozzles. Reynolds number in the nozzles increased with tank size, while Froude number in the tanks remained the same. Recommended Transfer Functions
Final correlations were obtained from which the following recommendations are made for transfer functions for three types of nozzle configurations. Configurations I and V with outlet directly below inlet. For this type, the transfer function recommended is
(3)
I I V i '
IT -Clp
C O N F l GUR AT I O N S
1 i
cc
z
i
a W
1
I
1 ' 1 1 ' 1 1
v
W
I
60
I
I
I
-I
? I
/
1 , l l , l
W v,
r
0
n
0
~
*
I
I
I
IO 20 TANK DIAMETER
CONFIGURATIONS
-
".t /I I
v,
'
a I
i
5 / 8 LEVEL
a -180 -210 -240
1
1
1 I I / I l l
1
1
I I I I I I I
1
1
I l l l l
40
INCHES
PI + !UU
d
w 40
30
I
I
I
W a >
I
v,
m O
OO
10 20 TANK D I A M E T E R
-
30
40
INCHES
Figure 5 . Correlation for dead time
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 2, 1970
327
Table II. Experimental Results
Config. fmm Fig. 2 I
I1
Series from Table I
Residence Time or V/q, Sec
T I ,Sec
1
200 300 400
218 156 270
0
200 300 400 222 340 452 148 236 302
1 3 4
I11
IV
1
1
3 4
T J ,Sec
L , Sec
Agreement with Theory, 7c
0 0
56 29 48
0 0 0
-26 -58 -44
200 270 333 180 350 370 100 190 250
0 0 0 60 0 0 70 0 0
0 0 31 0 0 0 0 0 0
9
+ 4 -19 -24 +12 + 5 -17 +19 -16 -15
200 300 400 245 160 86 222 340 452 148 236 302 394 586 780 268 396 530
146 244 286 200 110 100 160 220 320 120 110 170 250 360 620 230 290 550
26 50 67 0 40 25 60 70 120 30 100 100 100 130 100 70 120 60
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
19 27 22 28 7 6 19 30 23 15 31 18 53 60 50 42 32 32
200 300 400 222 340 452 148 236 302
156 250 416 200 370 440 150 190 270
62 13 0 26 0 0 20 30 0
0 0 9 0 0 0 0 0 0
9 3 3 8 7 6 6 8 9
or
T 1= 1.17 V q seconds
(4)
For this general type of nozzle configuration, experimental data were obtained on only the smallest tank; deviations from values of time constants agreeing with the model were appreciable. An average ratio T 1 / T 1was %, but agreements between calculated values and observed values of both T I and T3 were best when T S = % T I . Scatter of points was great; therefore relation 4 is not shown graphically. No dead time was observed in any experimental run. Configurations 11, 111, and IV, all having inlet on top of the tank. For this type, the transfer function recommended is:
C M =
e.- L s
(Tis + 1)(7'2s + 1)
328 Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
0
3 2 8 6 5 6 8 6
-1
+ 7 - 6 - 7 - 1 +52
+8 - 6 + 2 +12 + 2 - 5 + 2 - 6 -1 +28 +12 +21 +13 -11 + 2 + 5 +11 -1 +19 - 3 - 7
Dead time increased with tank diameter; a linear correlation such as that of Equation 6 shown in Figure 5 is all that is justified. The vertical spread in dead time values for any size tank results from the effect of parameters ignored in the relation given. Dead time increased as inlet and outlet nozzle separation became greater and decreased as flow rate increased. Figure 6 shows agreement between observed values of TI + T 2 and the result calculated from Equation 7. Since
V , q = Ti + T2 + L Ti + TI = 1.3 Ti = V / q- L
(7)
(5)
where
L = 6 + 1.1 D seconds D = tank diameter, inches T e= 0.3 T I seconds
T 2 ,Sec
(6)
Similar agreement was obtained between observed and calculated values of TI only (Equation 8 and Figure 7 ) . Configurations VI, VII, and VIII, all having inlet in the center of one head. For this arrangement, the transfer function recommended is
~
Table II. Experimental Results (Continued)
('onfi,g. from Fit. 2
vI
Series from Table I
1
3 4 VI1
1
3 4
VI11
1
Residence Time or V q Sec
7'1, Sec
200 300 400
227 416 358
0
200 300 400 340 452 236 302
T.,Sec
Agreement ii ith Theor\,
7' , Sec
L . Sec 0
0
59 71 59
0
-26 +l5 -25
213 286 384 280 450 180 200
0 0 0 30 0 30 40
21 4 5 0 0 0 0
3 8 10 12 14 7 12
- 2 - 4 -5 - 5 + 3 - 8 -16
200 300 400 222 340 452 148 236 302
213 286 400 230 310 380 200 240 300
0 0 0 0 0 0 0 0 0
27 11 10 0 0 25 10 0 0
2 8 5 8 11 8 4 7 7
- 6 - 6 - 1 + 7 - 5 -20 +31 + 5 + 2
200 300 400 245 160 86 222 340 452 148 236 302 394 586 780 268 396 530
192 278 370 250 165 140 140 300 500 140 190 300 330 590 620 260 430 520
0 0 0
0 9 14 7
11 5 6 9 5 3 17 13 10 3 9 10 40 15 32 30 14 23
+ 1 - 9 - 9 + 3 + 2 +59 +25 +18 +10
0
0 0 0
7 0 0 12 0 0 5 0 40 0 0 0 0
0
I
where
The relation of Equation 10 between dead time and tank diameter is shown on Figure 5 . No effects of nozzle separation or liquid level were evident. When inflow rate or Frdude number increased, dead time decreased; no real correlation was obtained. Agreement between T I observed and T I calculated using Equation 11 is shown in Figure 8. I n many instances the observed transfer functions were different from those recommended for the configurations tested. Bypass was evident in about half of the runs when inlet was in the tank head but principally when tank size was smallest. Since bypass seldom occurred with the largest tank, the recommended transfer function does not include a first-order leading element.
As an example of the use of the relations given, a horizontal surge tank 4 feet in diameter by 1 2 feet long
I
1
+li
-15 + 1 + 9 - 3 -16 + 8 +12 + 4
I
1
1
4
i
I
I
:/ I
I I 400 CALCULATED 1.3 T I = I
200
Practical Application
0
n
20 0 0 30 0 0 60 0 0 0 0
( (
I 600
1 9
I
I 800
L
Figure 6. Correlation for inlet on top
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
329
CONFIGURATIONS
IL, IU
Ip. Table 111. Time Constants for Tank 4 Feet in Diameter and 12 Feet long
I Vulm of T I ,Seconds, for Flou
Tank Content
Cu Ft
V,
40gpm
60gpm
100gpm
$4 full 3/8 full % full 5/x full % full
36 54 72 90 108
370 571 773 975 1187
236 370 505 639 713
128 209 290 370 45 1
where
V = ‘/z t a n k volume = % x 144 = 72 c u ft and
q = 60 gpm = 0.134 cu f t per sec Therefore
T I = - -72
0.134
0
2 00
0
CALCULATED
Figure
400
600
TI
33 = 538 - 33 = 505 seconds
The dynamic response of output concentration C to a change in input concentration M is predicted by the transfer function
7. Correlation for inlet on top
--33s
C/M = CONFIGURATIONS
PI, EII TUI
e
505 s + 1
Similar calculations for T I can be made for other tank conditions; results for other assumptions of inflow rate and liquid level may be determined (Table 111). I t should be satisfactory to use an average value of time constant, say for a half-full tank, corresponding to normal flow rate for the process. Literature Cited
Figure 8. Correlation for inlet in end
with ASME torispherical heads is to be used as a tank ahead of a distillation column to attenuate feed concentration changes. The inlet nozzle is in one head, while the outlet is in the center of the bottom. Equation 9 applies to this Type VI1 tank. Normal flow in and out is to be 60 gallons per minute; assume the tank will usually be half full. For this fairly large tank, bypass is assumed not to occur, but dead time is determined from Equation 10 as
L =2
+ 0.65 x 48 = 2 + 31 = 33 seconds
The time constant of the first-order lagging element is determined from Equation 11:
Ti = V,’q - L 330 Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 2, 1970
Clements, W. C., Schnelle, K . B., IND.ENG.CHEM.PROCESS DESIGNDEVELOP. 2, 94 (1963a). Clements, W. C., Schnelle, K. B., I S A Journal 10, No. 7, 63 (July 1963b). Dalferes, J. L., 111, “Effect of Tank Size on Transfer Function of a Partly Filled, Horizontal Cylindrical Tank,” M. S. thesis, University of Virginia, Charlottesville, Va., 1968. Danckwerts, P. V., Chem. Eng. Sci. 2, 1 (1953). Danckwerts, P. V., Ind. Chemist 30, 102 (1954). Draper, C. S., McKay, W., Lees, S., “Instrument Engineering,” Vol. 2, p. 659, McGraw-Hill, New York, 1953. Gutoff, E. B., A.1.Ch.E. J . 6, 347 (1960). Hougen, J. O., Chem. Eng. Progr. Monograph Ser. 60 (4) (1964). Lahn, G. C., “Frequency Response Study of a Partly Filled Horizontal Cylindrical Tank,” M. S. thesis, University of Virginia, Charlottesville, Va., 1965. Levenspiel, O., Can. J . Chem. Eng. 40, 135 (1962). Levenspiel, O., Bischoff, K. B., Aduan. Chem. Eng. 4, 95 (1963). Schnelle, K. B., private communication, 1964. Stafford, R. A., “Local Residence Time Distributions within a Tubular Vessel Equipped with an Eccentric Entrance Port,” M.S. thesis, Lehigh University, Bethlehem, Pa., 1962. Zalewski, W. W., “Transfer Function of a Horizontal Partly Filled Tank,” M. S.thesis,University of Virginia, Charlottesville, Va., 1965. RECEIVED for review August 14, 1969 ACCEPTED December 31, 1969
