development
R FURTHER DEVELOPMENT OF TRANSIENT EQUATIONS, APPLICATIONS, AND EXPERIMENTAL CONFIRMATION AND EDGAR L. PIRET OF MINNESOTA, MINNEAPOLIS 14, MINN.
DONALD R. MASON' UNIVERSITY
T h i s work was undertaken to extend the usefulness of continuous stirred tank reactor systems, because it was felt that these systems are not being fully exploited in the chemical industry. It has been demonstrated experimentally that present mathematical knowledge is sufficient to predict the performance of continuous stirred tank reactor systems during transient (nonsteady-state) periods of operation, when no reaction is occurring or when first-order reactions are being carried out. Several simple methods of minimizing or eliminating off-quality product during starting-up operations have been presented. This is the first general and practical analysis of transient stages in stagewise reaction systems, and the methods outlined in this two-paper series could be applied to a study of nonsteady-state operation in multiple eflect evaporation and other mathematically linear stagewise processes.
transient states (5, 7 ) have been based on purging operations In these the reactant material is mixed with some inert material, thus producing an offquality product This method of starting or stopping a continuous process gives rise to expensive recovery operations, or results in the waste of reactant material. The analysis of other transient-state procedures is therefore desirable.
S GESERAL, there are three basic types of systems in which
A most general equation has been previously derived ( 1 0 ) to describe the reactant concentration in any vessel, n, of an equalvolume continuous stirred tank reactor system, C,,, as a function of time, t, rrhen single or simultaneous first-order irreversible reactions with velocity constant, k , are being carried out. The holding time, e, is defined as the reactor volume, V,divided by the volumetric feed rate, F . Q is a dimensionless group equal to ice 1, which arises from the derivation of (fO),and has the significance of a "figure df merit" for the reactor t o which it refers. The A's represent the initial concentrations in the respective vessels when t is zero. This equation is for the concentration in the nth reactor.
APPLXChTIONS O F TR iNSIENT EQUATIONS The three distinct types of transient states that are discussed in their application t o first-order irreversibly reacting continuous stirred tank reactors are changes in the concentration and volumetric feed rate t o the system, methods for starting up the system, and methods for shutting down the system. The cases considered do not exhaust the possible variations which may prove useful in any particular circumstance, but may serve as a point of departure in devising other useful procedures. CFIAXGES IN CONCENTRATION AND RATE O F FEED STREAM TO SYSTEM
chemical reactions may be carried out: batch systems, tubulartype continuous systems, and continuous stirred tank reactor (CSTR, or C-star) systems. This last type has alternately been called a continuous batch system, a uniflow system (8), a homocontinuous reaction system (e),and an open reaction system ( S ) . In continuous stirred tank reactor systems, the feed material is intioduced t o the first vessel in the series, and the reaction is carried out in the various tanks. The concentration within any one tank is assumed to be uniform, and this is usually acconiplished by using agitation (9). The steady-state design considerations for these systems have been presented by several investigators, one of the most recent being a contribution by Eldridge and Piret (4). In addition t o the steady-state condition of operation, there are certain transient, or nonsteady-state conditions which arise whenever the system is started up, changed from one set of operating parameters t o another, or shut down. At such times the steady-state operating equations do not adequately describe the behavior of the system. The total retained volume of material in an industrial continuous stirred tank reactor system is usually large; and the contents are often valuable, so that starting and stopping operations must usually be controlled in some definite manner. These periods of shutdown may occur frequently-for example, t o clean heat transfer surfaces. The utility of the general transient equations for continuous stirred tank reactor systems which were developed previously (10) is demonstrated here. Furthermore, additional specific transient equations for first-order irreversibly reacting systems are derived in order to describe adequately the conditions which prevail for certain methods of starting up and shutting down continuous stirred tank reactor systems. The only available analyses of the 1
Present address, Bell Telephone Laboratories, Inc., Mountain Avo iY J
l l u r r a y Hill,
+
Its applicability may be illustrated readily as follows: The steady-state conditions prevailing in a continuous stirred tank reactor system for a volumetric feed rate, F,, and a feed stream reactant concentration, C,, , produce a concentration in any vessel, i, which is defined by:
At the instant the feed rate and concentration are changed to new steady values represented by F and C,,, the original steady-
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May 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
1211
state concentrations appear in the transient Equation 1 as the initial conditions. Combining Equations 1 and 2 gives:
c
"
Thus from Equation 3 the concentration in any reactor and a t a n y time can be computed. If only the volumetric feed rate to the system is changed, no distinction is made between the new feed concentration, Ca,, and the original, Ca;, for they are indeed identical. Likewise, if only the feed Concentration is changed, no new values of Q or 0 are defined, and Equation 3 may be simplified accordingly. These types of changes may be computed in a similar manner for unequal volume and combination volume systems, and the same method of approach is applicable for consecutive and reversible reactions, b y using the appropriate equations of (IO). STARTING UP THE SYSTEM
By controlling the starting-up period of operation in such a way as t o minimize purging, the loss of reactant material by shortcircuiting or by-passing may be reduced or avoided entirely. On the other hand, if this type of controI is not practical in a particular system, it may be desirable and possible t o predict the time at which the effluent concentration from a system first meets product specifications. To accomplish these objectives six different procedures, of varying importance, for the starting up of these systems have been analyzed. These are summarized in Table I, with references to the appropriate equations and to the plots that have been prepared t o facilitate their ready evaluation. Method A. Purging Accompanied by Reaction in a System. First, the vessels are to be considered as being initially filled with inert fluid having a density comparable with that of the reactant fluid. Hence the initial concentrations are zero, and the operation is started by introducing the reacting feed stream into the first vessel of the series. A general graphical solution t o this problem cannot be given, for the plots must be determined b g the size and sequence of the vessels in unequal volume systems. However, for systems of equal volume vessels, the equation describing the changes in concentration with respect t o time is Equation 3i in Table IV of (IO)which, after making a slight rearrangement, becomes:
O t -4
Low Range Chart
Figure 1.
Method A. Low range chart relating fractional approach of effluent concentration ratio to steady-state value, and factor ( Q t / O ) . in constant Q stagewise reaction systems having up to eleven vessels in series
tional approach of the effluent concentration t o it,s steady-state value :
Method B. Complete Batchwise Operation of Each Vessel to Its Continuous Stirred Tank Reactor Steady-State Concentration. This type of starting-up operation can be used only with
TABLEI. STARTING. UP OB CONTINUOUS STIRRED T A N KREACTOR SYSTEMS
.....
(n
- 1)l
*
For the special case of purging only, the value of Q becomes unity, and Equation 4 reduces to:
(2)
= 1
- e - t / O [ l + 04. +
y +. . . . .
Now Equation 5 is the same a s Equation 3d in Table I11 of (IO), which has been plotted by Kandiner (6). His graphs have been reproduced as Figures 1 and 2 t o represent Equation 4, but the axes have been changed to (Qt/O) and Q"(Ca,/Ca,). This latter axis represents the frac-
List of appropriate equations and plots for evaluating various methods of starting up first-order irreversible reactions in continuous stirred tank reactor systems Systems of Vessels One n vessels, n vessels, n vessels, vessel equal unequal combineda volumes Methods of Starting Up only volumes systems A Purging accompanied by re- Figures 1 and 2, also Equations Special action of system equations in Tables from derivatables in tions 111, I V , V (10)
B.
Batchwise operation only of each vessel to its CSTR steady-state concentration C. Each vessel filled a t steadystate rate then batchwise to CSTR boncentration D. Each vessel filled a t different rates to CSTR concentration, no batch operation E. Each vessel filled a t different rates, then batchwise to CSTR, concentration F. Progressive filling of empty system a t steady-state rate from inlet a
Figure 3 Equatio; 10 FiF::ation I C
4,
L"
Figure 3, Equation 10 Figure 4, F3uation Figure 5 , Equation
Figre 6, quat on
Figure 6, Equation
1Cl I..
I
'l I
Figure 3, Equation 10
I"
Figre 5, quation
17
,in\ \-",
Figure 3, Equation 10
Require
17
specific
iA.3n
Equations Equations 14 21. 24, 27-33; Figurei7, Figures 7, 9, 10 9, i o
I
Systems having equal and unequal volume vessels, or nonisothermal systems, or both.
INDUSTRIAL AND ENGINEERING CHEMISTRY
1212
Vol. 43, No. 5
which may be solved for (t/6&.
Equation 10 is plotted in Figure 3 for several vesBels of a continuous stirred tank reactor system. For unequal or combination volume systems, Figure 3 may be used if pseudo values of either n or Q are evaluated t o correspond t o an equal volume system. For example. the outlet concentration ratio from the third vessel of a system a t the steady state is described by
To obtain this same concentration ratio, a hypothetical constant Q system may be visualized in which the particular value for QpSeudois derived by
2
0
4
6
8
10
12
14
16
18
20
-9 Qt
Figure 2.
High Range Chart
Method A. High range chart relating fractional approach of effluent concentration ratio to steady-state value, and factor ( Q t / B ) , in constant Q stagewise reaction systems having up t o eleven vessels in series
reactions which are initiated with a reactant or catalyst material, or in some other rapid manner, or in cases where the vessels can be filled essentially instantaneously. By propeFly choosing the times of initiation of a batch reaction in each vessel it is possible to have the vessels attain simultaneously the concentrations which should prevail a t the steady-state operating condition in a continuous stirred tank reactor system. At that instant the feed stream can be introduced a t the correct rate t o place the unit in steady-state operation. This method is advantageous, because no short-circuiting or by-passing loss of material occurs. To calculate the times a t which the reaction must be initiated in the different vessels, none of the previously derived transient equations are needed. Instead, the familiar first-order irreversible batch reaction equation for single or simultaneous reactions is used. E
dt
a
h
-kC,
I n order to regulate the transient period so that the desired steady-state ratio prevails in the third vessel when the system is placed on stream, the reaction in it should be initiated beforehand by a number of t / O units indicated by the n = 3 locus of Figure 3 at the abscissa Q = Qpaeudo. The value of 6 is implicit in the value Of &pseudo.
The reaction velocity constant, k , in Qpseudo must have the same value as in Q 3 , as the h a 1 reacting conditions of the vimalized systems must coincide with the true system. A pseudo value of n, using Q3 as the basis, could be derived and used in a similar fashion. Method C. Filling of Each Vessel a t Steady-State Rate. When the method of filling each vessel at the steady-state rate, followed by batchwise operation to its continuous stirred tank reactor steady-state concentration, is used to start an empty system, the last vessel, n, is filled with reacting material a t the steady-state feed rate, and then allowed to react batchwise to attain the proper steady-state concentration for operation as part of the continuous flow system. Vessel n - 1 is next filled with reacting material, and so on back up the line until the first vessel
(7)
By integrating from zero time and a n initial concentration of C,,, the integrated form of the batch rate equation which is t o be used in subsequent developments is:
The concentration expression in Equation 8 must now be set equal t o the steady-state expression for each vessel in the system, giving:
Figure 3. .Method B.
Graphical Representation of Equation 10
Chart for calculating time required for vessel n t o attain its stagewise steady-state concentration ratio by batchwise operation only
INDUSTRIAL AND ENGINEERING CHEMISTRY
May 1951
1213
The integrated form of the batch rate Equation 8 may be adapted t o describe the second period of operation, so that we now have relationships between the concentration ratios and the times of reaction for each of the two periods. These concentration ratios must be combined, and related to the stead state ratios for each vessel in system, in order to determine the times a t which the feed must be introduced into the various vessels before the unit is placed on stream. This combination is:
tli
7.
Equation 15 is then solved explicitly for (t/O),, the desired unknown, Figure 4. Method B.
Graphical Representation of Equation 16
Chart for calculating totaltime reguired for vessel n to attain its stagewise steady-state ooncentration ratio by reaction during filling, followed by batchwise operation
is filled last. This is similar to Method B, except that the times of starting the filling of the vessels are controlled rather than the times of initiating the reaction in the vessels. The total time required for the concentration in any vessel t o reach its proper steady-state value may be divided into two separate periods: a period of length 0 during which time the vessel is being filled, and a subsequent batchwise operation period. The rate equations for the two periods are different. For the first period, 8, the rate equation for the accumulation of the mass of reactant in the vessel is:
This is a linear differential equation which may be solved by any standard method to give:
To convert W,, to a concentration unit it must be divided by the volume in which i t is contained, This volume is changing during the filling operation and is equal to Ft, so that the concentration of reactant in the vessel then is:
c -- Ca 2 (1 - e - k t ) kt
+
(t/e)n = 1
1
E
+
~n Q 1 - e-ke In ke ~
(16)
Equation 16, which applies t o equal volume systems, is plotted in Figure 4 for various values of the parameter n. The loci of the parameter n in Figure 4 are separated by less than one t / O unit except when the group Q is unity. Therefore if any reaction occurs in the system a t all, there are periods of time when feed must be provided to-two or more vessels simultaneously. In fact, when Q is greater than about 3.5, feed must be entering three vessels a t once for a part of the time, and even four vessels must be fed concurrently if Q is greater than about 6.6. This is definitely a limitation to the method, for a feed system must be provided having a t least twice, and in some cases even three or four times, the steady-state capacity. Method D. Filling Each Vessel at Different Feed Rates, with No Batchwise Operation. When each vessel is filled a t different feed rates, with no batchwise operation to give the continuous stirred tank reactor steady-state concentrations when the vessels are just full, the feed must be introduced to each vessel a t a time and rate such that all vessels attain the filled state simuItaneousIy and have in them the requisite steady-state
STARTING
.9
UP OF C S T R
SYSTEM BY REACTION
.a
DURING FILLING ONLY,USlNG
.?
FERENT FEED RATES
(13)
"I
4 At the instant the vessel is filled, the elapsed time from the start of the operation is the nominal holding time, 6, of the vessel. If the quantity e is substituted for t in Equation 13, the initial concentration, A:, which prevails in the vessel a t the time i t is just completely filled, is:
'
A I
= (1
c.,
- e-M) ke
.3 .2 *I
I25
175 2.0
2.5
30
3.5 40 4.5 5.0
6.0
7.0 8.0 9.0 IO
Q Figure 5.
(14)
1.5
Correlation of Value of Qn Required during Filling Period for Vessel n and Steady-State Value Q. Method D.
No batchwise operation occurs
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 43, No. 5
wherein 8, and 8,, retain the same meanings as in E uation 17. These values of en determine the vchmetric feed rates to the individual vessels, which must be started a t a time (n l)O, before the system is placed on stream.
+
The graphical representation of Equation 19 is given in Figure 6, and is more clearly presented by using h.6, as the independent variable instead of Q o as was done on the preceding figures. Lines have been drawn in to bhow the per cent overload above the steady-state pumping rate required of the feed system during this particular type of starting-up operation. Because of the large overload required for high values of kO,, this method is most practical for the lower values. Method F. Introduction of Feed Stream at SteadyState Rate to First Vessel of Empty System. Now the method may be considered wherein the reacting feed stream fills the first vessel, then overflows into the .I .I5 .2 .3 /1 .6 .8 ID 1.5 2 3 4 6 8 IO second vessel until it is filled, then overflows to fill the I( 00 third vessel, and so on progressively throughout a bystem of equal volume vessels. The steady-state Figure 6. Graphical Representation of Equation 19 operating concentration does not prevail in the effluent Xethod E. Correlation of values of kOn and kOo required for vessel n to attain at time ?le when the last vessel is just filled, but the ita stagewise steady-state concentration ratio, by filling at different feed rates and subsequent batchwise operation deviation is not nearly so great as occurs under Method A for the combined purging and reacting situation. Moreover, there has been no overload on the feed operating concentrations. The entire unit is then placed on section, it is operationally simple, and no fancy schemes are restream. Just prior to the time of placing the system on stream, quired to correlate the feed rates and times. In tact, this method reactant material is being fed to all the vessels a t once. The times may be the only practical method to use on some installationsand rates of filling the vessels are determined by equating the exfor instance, in columnar reactors. However, its mathematical pressions for the steady-state operating concentration of each analysis is the most complicated. vessel with the initial concentration expression, (A;)n, in EquaIn the subsequent derivations, the time variable is measured tion 14. For vessel n this is: from the instant that the feed stream is introduced into the first vessel. The effluent from the first vessel would normally be described by Equation 3h in Table IV of ( I O ) , which iR: The term 6, is defined as the nominal holding time for each vessel in an equal volume system a t the steady-state condition, and the term en represents the holding time required during the filling operation for any vessel, n. Equation 17 is a transcendental difference equation of type 2b as discussed by Tiller ( I I ) , and may be solved graphically. Figure 5 has been constructed for this purpose, wherein the values of AiIC,, and (I/&,)" are plotted with n as a parameter, versus the group Q. To solve Equation 17 graphically, a vertical line is drawn a t the steady-state operating value of Q0,so that it intersects with the n loci curves. Horizontal lines are drawn through these intersections and projected to the right until they intersect the AiIC,, curve, which is the uppermost curve on the figure. The second set of intersections determines the values of &, from which the required feed rates and times of starting to fill the different vessels can be calculated. Method E. Filling Each Vessel a t Different Feed Rates, Followed by Batchwise Operation to Steady-State Concentration. The necessity for filling more than one vessel a t a time may be eliminated by the proper use of volumetric feed rates which are greater than the steady-state value. I n order to accomplish this, it will be assumed for convenience t h a t the batchwise operation period in the f i s t vessel is of the same length as the time required to fill that vessel; for the second vessel, the batchwise period will be taken to be twice as long as the filling period; and for vessel n the batchwise period is to be n times as long as the filling period. As a result of these assumptions, the concentration ratio expression for the batch period is:
(2)istah e-nkon
(18)
By combining this batch period expression with t h a t for the filling period, and equating to the steady-state value, we have:
However, no effluent appears until the first vessel is just filled at time 8, so that the time variable in Equation 20 must be changed accordingly. The initial concentration at time e has been shown by Equation 14 t o be equal to A ; . These modifications of Equation 20 give:
During the time that the second vessel is filling, from 0 to 28, its feed concentration is described by Equation 21, and the rate equation for the filling process must compensate for this changing concentration in the feed. The result is:
where J!7a2 represents the mass of reactant A in the second vessel. By using C, from Equation 21 and solving Equation 22 for N,, there is obtained:
Equation 23 must be divided by the volume F(t -,,e) to convert to concentration units. The initial concentration, Az, a t the time the vessel is just filled is obtained by letting t assume the value of 28, and the resulting expression for is:
May 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
1215
The reactant concentration in the effluent from the second vessel would normally be given by Equation 1 by letting n equal 2. This is:
*
However, no effluent appears until after time 28, a t which time the second vessel is just filled, so t h a t the time variable in Equation 25 must be changed accordingly. T h e initial concentration of A in the second vessel at time 28 has been shown by Equation 24 to be equal to A ; , and the initial concentration of A in the first vessel at time 28, A ; , must be reckoned from Equation 21. This gives:
These modifications of Equation 25 give:
where Tz is used t o represent the quantity ( t - 20). By proceeding in a similar manner, it is possible to evaluate the initial concentrations in subsequent vessels a t the times they are just filled, Without presenting the details, the expressions for A i l ’ , A Y ” , and A;‘“’ are given below. One additional prime has been added t o the A , values for each unit of 8 after the starting time at which it has been evaluated.
No assumptions have been made which detract from the conipleteness of the mathematical analysis, and because of the number and nature of the variables this starting procedure serves as a n excellent experimental verification of the theoretically developed transient equations. Because the calculation of the theoretical curves from these equations is a somewhat tedious job, some of the curves t h a t have been evaluated in the experimental program for the first five vessels are given in Figure 7 , although they are incomplete. Because an infinite number of stages n-ill approach a tubular reactor, the deviation from steady state in subsequent reactors will be less than that calculated for the fifth vessel. If overreaction is not harmful, the feed stream may be shut off after the system is filled, until the steady-state composition in the last reactor is attained by batchwise operation, at which time the feed should be resumed. SHUTTING DOWN THE SYSTE.M
T h e transient problenis during the shutting-down operation usually are not nearly BO complicated as those encountered during the starting-up period. Two different methods of bringing the system t o a nonreacting state are considered. By changing the feed from a reacting solution to an inert stream, the contents of the system can be diluted and purged. Equations 5j, 5k, 5m, and 5p of Table I V (IO)permit the calculation of the reactant concentration in the effluent from any particular system. By shutting off the feed stream, each individual vessel may be allowed to react batchwise t o a point where the reaction is essentially complete. T h e integrated form of the batch rate, Equation 8, must be applied to each individual vessel in much the same manner as was done in the case of Method B for the startingup situation. T h e time required for tthis finishing-off process depends on the desired degree of completion of the reaction. If C,, is defined as the tolerable concentration of reactant in the product stream, this will define the fraction f of the feed concentration which remains in the product material.
f
= (Ca,/Cao)
(34)
Using this definition we can now write:
*
B y using the correct initial values of the concentrations and substituting them in Equation 1, the reactant concentration from a n y vessel may be calculated as a function of time. The expressions for the third, fourth, and fifth vessels are presented in Equations 31,32, and 33.
Reactant concn. in product Steady-state concn.
[
-
1-
The explicit solution for the quantity ( t / 8 ) , is:
By knov-ing Q, the tank number n, and the allowable concentration of reactant in the product, the time required t o complete the reaction hatchwise can be calculated directly from Equation 36.
USE O F STARTING-UP PROCEDURES A numerical example may be given to illustrate the use of the
equations developed in the previous sections for the various methods of starting up a continuous stirred tank reactor system.
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VOl. 43, No. 5
If a first-order irreversible reaction, such as the hydrolysis of sucrose, is to he carried out in a five-vessel equal volume system, and if tho reaction velocity constant, k , for the process is known to be 0.50 per hour in the presence of a n acid catalyst material, what are the limitations of each method of starting up? Assume that the volume of each tank is 50 gallons and that the volumetric feed rate to the system is 50 gallons per hour, so that the nominal holding time 6 for each tank is 1 hour. The value of k$ is then 0.50 and the group Q is 1.50. Method A. The transient condition will be assumed to be terminated when the react a n t concentration in the effluent from the system has attained a value which is 99.0% of the steady-state operating value; in other words, when the value of Q"(C,,/C,,) is equal to 99.0%. The time required can be found from Equation 4 or by the use of Figure 2. From Figure 2 the value of Q t / $ a t which the curve for vessel 5 (n = 5 ) crosses the 99.0% line is equal to 11.6, hence t / $ is 11.6/1.5, or 7.73 units of 8 , or 7.73 hours. During this time a bypassing loss of reactant material from the system has occurred. Method B. The hydrolysis reaction may be initiated by the addition of the acid catalyst material, and this method is applicable. A vertical line is drawn on Figure 3 a t the value of Q of 1.50 to determine the times a t which the acid catalyst must be added to the individual vessels before the system is placed on stream. When the vessels are initially empty and are to be filled with neutral sucrose solution a t the steady-state feed rate, the results are summarized in Figure 8 and Table 11. When the feed stream is started up after the batch operating periods the steady-state condition should prevail and the concentration in the effluent from the last vessel should not vary with time. By starting with a clean empty system, i t has taken 5.8 hours to place the unit on stream without the loss of any material. Method C. Under these circumstances the acid-containing feed stream is introduced into the vessel instead of first filling it with neutral sucrose solution and initiating the reaction with the acid. Again a vertical line is drawn at a value of Q of 1.50, now using Figure 4. The results are summarized in Figure 8 and Table 111. The adoption of this starting-up method would require only about 4.6 hours to attain a steady-state condition, hut feed section must be capable of operating under conditions of 100% overload for part of the time. Method D. All vessels are to he filled with acid-containing sucrose solution a t different feed rates in such a manner that they are all just completely filled a t the same time, and have in them the requisite steady-state concentrations. This problem may be solved graphically with the aid of Figure 5, and the results are presented in Figure 8 and Table IV. The starting-up operation by this method would require about 16 hours and is much slower than the previous methods. I t s use would be dictated more by other advantages which may accrue, and has been considered only to show its relationship to the other methods.
Figure 7.
Experimental Curves for Five Vessels
Method F. Plots of concentration ratios obtained by progressive filling ua. k0 for five-vessel constant Q stagewise reaction system, showing maximum deviations from steady state
May 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
TIMECHARTFOR METHODB TABLE 11. OPERATOR'S
Time 8 : o o A.M.
9 :00
A.M.
11 : 22 A.M. c
12 :00
NOON
(Complete batchwise operation) Hours before Nature of Operations Steady t/e to Be Performed Units State Start filling fifth vessel 5.80 5.80 with neutral sucrose s o h tion Fifth vessel just filled. 4.80 4.80 Start filling fourth vessel ..~ Add acid catalyst to fifth 4.05 4.05@ tank Fourth vessel just filled. 3.80 3.80 Start filling third vessel Add acid catalyst to fourth 3.24 3,24a tank Third vessel just filled. 2.80 2.80 Start filling second vessel Add acid catalyst to third 2.43 2.4aa tank 1.80 Second vessel just filled. 1.80 Start fillinn first vessel Add acid catalyst to second 1.62 1. 62a tank 0.80a First vessel just filled. 0.80 Shut down feed stream. Add acid catalyst to first tank Start feeding acid-contain0 . 0 0.00 ing sucrose solution to first vessel a t steadystate rate I
12:11 P.M.
1:oo
P.M.
1 :48 P.M.
a
From Figure 3.
Method E. The solution for this method is obtained graphically from Figure 6, and the results are given in Figure 8 and Table V. This is the fastest method of those t h a t have been considered, and t h e maximum overload on the feed section for this particular problem is only 85%. Method F. B y referring t o Figure 9 it can be seen that it requires about 9 units of t / O , or 9 hours, after the feed is fist introduced into the empty system before the effluent from the fifth vessel is within the desired 1% of the steady-state value. This is only 4 hours after the effluent appears at the outlet of the system, and there is no overload on the feed section durin the starting-up period. T h e concentration of the first portion of &e effluent is not as far removed from the steady-state value as is the case under Method A for purging accompanied by reaction.
TABLE 111. OPERATOR'S TIMECHARTFOR METHODC (Reaction during filling and batchwise operation) Hours before Steady t/0 Nature of 0 erations Time State Units to Be Pergrmed 8:oo A . M . 4.57 4.57= Start filling fifth vessel with acid sucrose solution 8:49 A.M. 3.75 3.75a Start filling fourth vessel, in addition to fifth vessel 9:oo A.M. 3.57 3.57 Fifth vessel just filled, stop feed to this vessel 9:36 A.M. 2.97 2.97a Start filling third vessel, In addition to fourth vessel 9 : 49 A.M. 2.75 2.75 Fourth vessel just filled, stop feed to this vessel 10325 A.M. 2.15 2.15a Start filling second vessel, in addition to third vessel 10:36 A.M. 1.97 1.97 Third vessel just filled, stop feed to this vessel 11 :14 A.M. 1.33 1.33a Start filling first vessel in addition to second ves: 8e.l ..~ 11:25 A.M. 1.15 1.16 Second vessel just filled, stop feed to this vessel 12:14 P.M. 0.33 0.33 First vessel just filled, shut down feed stream 12 :34 P.M. 0.00 0.00 Start feeding acid-containing sucrose solution to first vessel a t steady-state rate a From Figure 4.
METHOD TANK NO.
ri
1217
STARTING TIMES, WOO A.M.
i
i FEED RATE:
I
50 G P H
l
l
FILL ALL VESSELS
rTl
I
FILL ALL VESSELS AT 50 GPH. NOTE OVERLAPS
I I I I
I FILL' THIS VESSl
3 2 I
;ri! 2
FEED RATES: 61 6
GPH
69.4
GPU
71.8
YPH
I
!
76.9
I
GPH
92.3 GPH I
8:OOA.M.
NOON
4:OoPM
8NPM
MIDNIGHT
Figure 8. Comparison of Methods for Starting Up a FiveVessel Continuous Stirred Tank Reactor System for ke = 0.50 Each vessel, 50 gallons
Black areas (solid areas) represent filling periods; dotted areas represent
periods during which effluent is more than 1% off the stagewise steadystate concentration; thickness of lines are proportional to the feed rates
Figure 8 gives a graphical summary and comparison of the various methods of starting up as applied to the problem under discussion. Before any particular method for starting up a system is adopted, a complete analysis such as this should be carried out for each different system and operating condition. T h e results may vary widely from one example t o another. EXPERIMENTAL
The hydrolysis of acetic anhydride in water solution was used as a typical &&-order irreversible reaction in the laboratory study. Method F for the progressive filling of initially empty systems has been carried out experimentally, b y introducing the feed stream at its steady-state rate into the first vessel and allowing the overflow to fill succeeding vessels in a progressing manner. This particular situation was not overly complicated t o carry out experimentally, it afforded the most complete check on the theoretical derivations, and is a case which should have a great practical utility, as the transient concentrations are never far removed from the steady-state values. This is clearly shown in Figure 7. Apparatus. The apparatus was comprised of five 1800-ml. Ace Glass high pressure borosilicate glass autoclaves which were operated in series. The paddle agitator guided by the groundglass bearing directed the li uid downward, and was rotated at 210 or 325 r.p.m. (Table Baffles were placed inside the vessel. Reaction solution entered the bottom of the reactor through a feed tube (inside diameter 3.5 mm.), and left a t the top. A mercurial thermoregulator operated, through a relay system, a 250-watt infrared lamp directed on the side of the vessel to maintain a constant temperature (*0.01" C,), as read on a thermometer. Cold water was sprayed through a perforated ring
VI).
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1218
TABLE Iv. Tank
&,from Fig. 5
5
8.85 6.15 4.30 2.90 1.87
KO. 4 3
2
1 1
OPERATOR'S
TIMEC H S R T
FOR h l E T H O D
(Reaction during filling only) Start - Fwd .Rate, kBn Bn Gal./Hour 7.85
15.70
5.15
10.30
3.30 1.90 0.87
6.60 3.80 1.74
3.18 4.85 7.68 13.16 28.74
D
TABLE V.
Vol. 43, No. 5
REACTIOS DURINQ FILLIXG OF ONLYO m VESSICL AT A TIUE CALCULATIOXS Feed Rate, On, Hours Ga!./Houi
PRELIiIINARY
Time
Tank Xo.
kBn
5
0,370'' 0 . 360a
8 : O O A.M. 1:24 P.M. 5:24 P.31. 7:54 P . H . 9 : 58 P.M.
Feed at steady-state rate, 50.00 11:42 p . x placing unit on stream Maximum feed rate period is from 9 :58 P.M. until 11:42 P.u., during which time feed section must provide 57.51 gallons per hour.
4 3
O.34Sa
2
0.325: 0.271
1 a
67.57 69.44 71.84 76.92 92.25
+ 1)01~,
Hours
4.440 3.600 2.784 1.950 1,084
From Figure 6.
Time
just below the flange clamp and drained off into a funnel, to provide cooling when required. In this manner, the temperature in each vessel was individually controlled. This is the apparatus described by Eldridge and Piret (4). Operatin Procedure. In starting a run, t.he x-ater and acetic anhydride #ow streams were set to the correct rates and temperature, and allowed to discharge to the drain for a t least an hour in order to ensure a steady-state condition in a packed mixing column preceding the reactors. The agitators were started and all the liquid-filled vessels adjusted to the desired operating temperatures, which was then maintained by the thermoregulators using the cold water sprays when necessary. Immediately before starting the run the liquid was siphoned from the vessels, and analytical samples were collected from the feed stream. The feed stream was then introduced into the first vessel and allowed to fill a specified number of vessels progressively. Analytical samples were collected from the effluent from the last vessel a t different intervals of time, and the feed stream was not interrupted until the effluent sampling was complete, after which time more feed stream analytical samples were collected. Only the effluent from the end of the system was analyzed; otherwise the flow through the system would be disturbed b y collecting samples, and produce undesired transients in the system. To cover the range, experiments were run using from one to five vessels in the system. The analytical procedure of Caudri ( 1 ) and Vles ( l a ) was used to determine the concentrations of unreacted acetic anhydride in the analytical samples.
0,740 0,720 0.696 0.650 0,542
(n
8:OO
AX.
8:44
A.M.
8:50
A.M.
9:33
A.M.
9:39
A.31.
10:21
A.M.
1O:"O
A.M.
11:08
A.M.
11:21
A.M.
11:54
L.M.
12:25 P.>r.
OPERATOR'B T I N E CHART FOR h'lETHOD E Hours before Nature of Operation Steady State t o Be Performed 4.440 Start filling fifth vessel a t 67.57 gal./hour 3.700 Fifth vessel just filled, stop feed 3.600 Start filling fourth vessel a t 69.44 gal./hour 2.880 Fourth vessel just filled, stop feed 2.784 Start filling third vessel a t 71.84 ga!./hour 2.086 Third vessel just filled, stop feed 1,950 Start filling second vessel a t 76.92 gal./hour 1.300 Second vessel just filled, stop feed 1.084 Start filling first vessel a t 92.25 gal./hour 0.542 First vessel just filled, stop feed 0.000 Start feed to system a t 50.0 gal./hour t o place unit on stream"
Simpler fornu of these equations have been experirncntally verified independently (6).
SUhI31ARY In the two papers of this series, the general transient eqdations have been derived which are applicable for purging or for firstorder reactions in continuous stirred tank reactor systems. The Laplace transformation method was used to solve the rate equations, which are differential difference equations. I n addition,
Experimental Results. The operating conditions in the system for the various runs are given in Table VI. The experimental points are plotted in Figires 9 and 10 for values of k0 of 0.506 and 1.965, respectively, and the solid lines were obtained from the theoretical equations which describe the progressive filling process. Discussion of Results. By using systems of from one to five continuouy stirred tank reactors in series, the basic v a l i d i t y of t h e t r a n s i e n t equations which have been derived for first-order irreversible reactions has been established (Figures 9 and 10). The average deviation of the transient data from t h e t h e o r e t i c a l curves is 1.4%, w h i c h is s o m e w h a t greater than the 0.4% as rep o r t e d by E l d r i d g e a n d Piret ( 4 ) for the steady-state running conditions on the same apparatus. This is understandable, for their data t/e were not affected by the e+ perimental operating variaFigure 9. Experimental Verification of Method F tions peculiar to the starting Ice = 0.506 up of the system. Solid curves represent derived concentration ratios in the vessels as functions of time
May 1951
INDUSTRIAL AND ENGINEERING CHEMISTRY
1219 NOMENCLATURE
A 19
30
1.965
ci
1.25
ol,OO
1.50
1.75 2.00
Figure 10.
2.50
3,O
60
4,O
6,O
ZO
8L, 9.0
118 Experimental Verification of Method F
v,
system A, = initial concentration of A in vessel n of a CSTR system A:, A2) , A : ” , etc. = initial concentrations of -4 in vessels 1, 2, 3, etc., when t h e y h a v e j u s t been filled progressively a t time ne C, = concentration of A in feed stream C,, = concentration of A in and from vessel n e = Naperian logarithm base, 2.71828. . . . = fraction of C,, remaining in product stream k = f i r s t - o r d e r irreversible reaction velocity constant n = vessel number n in a CSTR system Q = dimensionless g r o u p ,
kV k e + l = F + I
128 = 1.965 Solid curves represent derived concentration ratios in vessels as functions of time
several reductions t o practice \Yere outlined in various methods for starting up and shutting down these systems. A most complex case wherein the empty vessels are filled in a progressing manner with the reacting feed stream has been analyzed theoretically and carried out experimentally. T h e validity of the most general form of the derived equations is thus established and its applicability t o simpler situations follows.
= reactant component in
= time T , = t - n8, time translated from t = 0 = effective volume of reaction vessel, tank, Or oonlPartment number n in a continuous stirred tank reactor system = IUSS of reactant A in vessel n = (v-/F),nolninal holding time for vessel n
t
~v,,
e,
Subscripts
.4; f = a final state or condition vessel i, n in a CSTR system o = feed stream or a n initial condition in the system u = component
n
2,
=
ACKNOWLEDGMENT
LITERATURE CITED
The authors wish t o express their appreciation for assistance rendered by E. I. du Pont de Nemours & Go., Inc., the Graduate School and Engineering Experiment St,ation of the University of the Bell Laboratories, Inca, and the Veterans Administration.
(1) Caudri, J, F. M.,R ~trav. ~ chim., . 49, 1 (1930). (2) Denbigh, K. G . , Trans. Faraday Soc., 43, 648 (1947).
(3) Denbkh, K. G., Hicks, M., and Page, F. M-t Ib@i 44, 479 (1948).
(4) E]drjdge, J, W., and Fir&, E. L.,Chem. Eng. Progress, 46, 290-9 (1950).
(5) Johnson, J. D., and Edwards, L. J., Trans. Faraday SOC.,45,286 (1949).
(6) Kandiner, H. J., Chem. Eng. Progress,
TABLE VI. No. of Tanks
*
1 1
2 2 2 2
3 3 3
3 4
Run No.
44, 383 (1948).
EXPERIMENTAL OPERATING CONDITIONS Agitator
k per
e,
Min. 0 115 0.0807
Min. 4.40 6.25
k0 0.506 0.504
Av. Total R.P.hi. Acid, N 325 0.154 325 0.169
Av. Av. % Absolute DeviaDeviation tion, % Legend 0.895 0.622 0 0.386 0.268 0
7A
Tz??’* 20.0 15.0
6B 7B 8
5B
20.0 20.0 20.0 20.0
0.115 0.115 0.115 0.115
4.40 4.40 4.40 4.41
0.506 0.506 0.506 0.507
210 325 325 210
0.151 0.156 0.157 0.154
0.863 1,030 0.818 0.986
10, 7C 5C 4Ca
25.0 20.0 20.0 20.0
0.158 0.115 0.115 0.115
3.125 4.40 4.41 4.583
0.494 0.506 0.507 0.527
325 325 210 210
0.167 0.155 0.154 0.155
0.881 1.575 1.580 0.893
0.300 0.487 0.558 0.300
9 5D 4Da
20.0 20.0 20.0
0.115 0.115 0.115
4.40 4.41 4.583
0.506 0.507 0.527
325 210 210
0,155 0,155
0.250 0.400 0,333 0,250 0.386
15
0.400 ’
0.525 0,414 0.500
6 5
16 13
20.0 20.0
0.115 0.115
4.375 4.40
0.503 0.506
325 325
0.158 0.158
1.011 1.756 1.491 1.599 2.434
1 1
19 12
3 30 0 .. 0 0
0.2208 0.2208
8.90 8.95
1.965 1.976
325 325
0.161 0.157
1.002 1.269
0.379 0,477
2 17 30.0 0.2208 8.85 18 30.0 0.2208 8.90 2 e Not plotted in Figurea 9 and 10.
1.954 1.965
325 325
0.159 0.158
1.372 1.014
0.189 0.132
4
4
0.154
A
8 None
0.
(7) Kirillov, N. I., J . Applied Chem. (U.S.S.R.), 13, 978 (1940). (8) Ibid., 18,393 (1945). (9) MacDonald, R. W., and Piret, E. L.,
Chem. Eng. Progress, to be pub-
liahed.
(10) Mason, D. R., and Piret, E. L., IXD. ENG.CHEM.,42, 817-25 (1950). (11) Tiller, F. M., Chem. Eng. Progress, 44, 299 (1948). (12) Vles, S. E., Rec. trau. chim., 52, SO9 (1933).
None
0 0
None
0 0 -
8 8
RECEIVEDSeptember 15, 1950. Presented before the Division of Industrial and Engineering Chemistry at the 118th Meeting of the .&VERICAN CHEMICAL SOCIETY,Chicago, Ill. For detailed material supplementary to Figure 7 of this article, order Dooument 2912 from the American Documentation Institute, 1719 N St., N.W.,Washington 6, D. C.. remitting 50 cents for microfilm (images 1 inch high on standard 35-mm. motion picture film) or 80 cents for photocopies (6 X 8 inches) readable without optical aid.
