K L
steady-state gain liquid rate: Ib. mo1es;hr. rn slope of equilibrium curve .V number of trays in column P = exponent in Equation 16 R = reflux rate. Ib. moles hr. s = Laplace operator t = time, hours V = vapor rate, Ib. moles, hr. x = mole fraction of component in liquid = mole fraction of component in vapor y z = mole fraction of component in feed GREEKLETTERS cy = relative volatility = time constant assuming first-order system T w = frequency. radians, min. W X = breakpoint frequency, radiansimin.
literature Cited
= = = =
SVBSCRIPTS
D
= distillate
E‘
= = = =
feed rectifying section of column s stripping section of column it’ bottoms product 1. 2. etc. = tra)- number or constant designation A bar over a variable indicates that it is a steady-state quantity: a variable without the bar is a perturbation from ateady state. 7
(1) Baber, M. F., Edwards, L. L., Harper, W. T.. Witte, M. D., Gerster. J. A , , Chem. Eng. Progr. Symp. Se7. 57, No. 36, 148 (1961). (2) Baber. M. F., Gerster. J. A , , A.I.Ch.E. J . 8, 407 (1962). FUND4MENTALS (3) Bollinger, R. E., I,amb, D. E., IND.ENG.CHEM. 1, 245 (1962). (4) Calvert. S., Coulman, G., Chem. Eng. Progr. 57, 45 (1961). (5) Dobson, J . G.. Interkama 1960. Dussrldorf, 1960. (6) ,Grrster, J. A , ? Hill, A . B.. Hochgraf. N. N.. Robinson, D. G., ‘’ rray Efficiencies in Distillation Columns,” Final Report from UniLersity of Delaware to A.1.Ch.E. Research Committee, American Institute of Chemical Engineers, New York, 1958. (7) Lamb, D. E.. Pigford, R. I,., Rippin, D. W., Chem. Eng. Progr. Symp. Ser. 57, No. 36, 132 (1961). (8) Lupfer, D. E., Parsons, J. R., Chem. En which was first given bk- Danckiverts ( 3 ) , and the actual residence-time distribution curve, the value of the reaction rate constant, k : can be calculated from any conversion level. Such calculations can be done by a graphical integration technique, but they are more conveniently done on a n analog or digital computer. The value for T.the total amount of tracer added. is computed from the residence.time curve. This is simply the area under the curve. To facilitate computer solution Equation 1 is modified as follows:
wher'e f ( e ) d O is the fraction of material having a residence time de. Expressed mathematically, between e and e
+
t f(0) =
~
''
E
9
''
sr" 2lP lm ~
td0
In this particular study Equation 2 was programmed for solution on an analog computer. T h e values for f ( 0 ) are obtained from the residence-time distribution curve, and by trial and error the value of the reaction rate constant. k : required to give a specified conversion is obtained. \,Vhen the reaction rate constant is calculated in this manner, it is the true value of the rate constant. The value obtained depends only upon the chemical and kinetic factors affecting the process, and is independent of the reactor mixing or performance level.
TIME, MINUTES
Figure 1 .
=
Typical residence-time distribution curve
Contacting Efficiency
B(B)
Figure 2. Generation curve by pulse input
0 2 Mixed
of
residence-time
Tanka i n Series
0 6 Mixed Tanks i n S e r i e s
+ M i x e d - P h a s e Reactor, F i g u r e 4 * M i x e d - P h a s e Reactor. Fxgure 5 A M i x e d - P h a s e Reactor
Reactor, F i g u r e 7
UMlxed-Phase
0
I
0
10
I
20
\
Figure 6
I
30
I
40
1
50
I 60
I
70
I
80
\
-''\h 90
j
100
CONVERSION, 'b
Figure 3. 382
Variation of contacting efficiency with conversion
l&EC PROCESS DESIGN A N D DEVELOPMENT
Often in industry two similar reactor systems d o not perform in the same way-for example, a pilot plant reactor and a commercial reactor, or even two identical commercial reactors, may not give the same process results. Such differences are usually the result of differences in kinetics, catalyst effciency, reactor flow patterns, etc. T h e separation of the purely kinetic and physical effects is difficult; thus it is frequently useful to have a method for determining the contacting efficiency in a given reactor or system of reactors !series, parallel? etc.). I t is best to make sure of the contacting in reaction equipment used in pilot plant or commercial operation. whenever any doubt exists. 'Ihus. a supposed plug flow reactor may in fact deviate considerably from true plug flow, because of faulty catalyst packing, etc., or a multistage reaction system may actually behave as if it had fewer stages than supposed. The analysis of residence-time distribution curves and the application of first-order kinetics--just described-furnish a convenient means for obtaining a quantitative index of contacting efficiency. This can be done as follows : Assume a hypothetical first-order reaction. Next, assume a n arbitrary high conversion level of 90% (C,!Ci = 0.1). Normally this level of conversion is sufficient to detect variations in residence-time distribution. From the residence-time distribution curve of the reactor under study, determine from Equation 2 the "pseudo"-reaction rate constant. k: required to give 90y0conversion in the hypothetical reaction.
This value of k is associated, in the given reactor, with an average residence time determined from the following equation :
Lrn Lrn etde
=
(4)
~
tde
Now, a true plug reactor can be considered as the standard (1007, contacting efficiency). So the next step is to calculate the residence time required to get 907, conversion, in a hypothetical plug-flow reactor for the same reaction rate constant solved from Equation 3. This is obtained by solving, for residence time (Op), the well-known equation for a first-order reaction in a plug-flow reactor.
TIME, MINUTES
n
Figure 4. Residence-time distribution curve for pilot plant reactor
Then “contacting efficiency” is simply defined as 1007, (Equation 4)-that is, times the ratio of Op (Equation 5) to Contacting efficiency
OP = -
Ow
X 10070 I
1
The contacting efficiency as defined is calculated entirely from the residence-time distribution curve. The method using an assumed hypothetical first-order reaction just described gives a sensitive method for detecting small variations in residence- time distribution. Thus, differences in liquid flow patterns between different reactors can be determined from their respective residence-time distribution curves. T h e actual conversion obtained in the test reactor, which is often a function of many variables such as temperature and catalyst activity, need not be known. This is a real advantage for the proposed method, since reactor contacting is separated from the effect of process variables which are frequently so hard to measure accurately. This method can also be used to characterize equally well all types of reactors-plug flow, well-mixed, etc. Figure 3 shows how contacting efficiency varies with conversion for various reactors. Shown for comparison are the contacting efficiencies for several equal-sized mixed-tank reactors connected in series. Residence-time curves for the mixed-phase reactors are those indicated by the figure numbers given in the paper. The contacting efficiency approaches 1007, for all types of reactors a t low conversion and efficiency becomes lower and differences in contacting efficiency become greater as higher conversions are approached. Reactors which have the same contacting efficiency will have identical residence-time d.istribution curves. The residence-time distribution curves used to calculate reactor contacting efficiency can also be used to calculate reaction rate constants for the particular reaction under study when reaction kinetics are first-order or pseudo-first-order. I n this case, no assumptions about a reactor contacting model are necessary. Fortunately, in many catalytic processes the kinetics are first-order or pseudo-first-order.
I
1
I
-I -
-
-
1 6 . 8 MINOTES CONTACTING EFFICIENCY
Qav
100
63.5%
-
-
-
A -
A
TIME, MINlPTES
Figure 5. Residence-time distribution curve for commercial desulfurization reactor during period of poor operation
-n
I
I
1
I
-
Application to a Commercial Desulfurization Reactor
Hydrodesulfurization of hydrocarbons is a typical mixedphase, heterogeneous catalytic process. Generally speaking, H z and oil are fed downflow over a fixed bed of catalyst, and sulfur is removed from r.he oil as HQS. Laboratory studies have shown that this desulfurization reaction is essentially firstorder over the range of desulfurization normally encountered. The example of the application of residence-time distribution
10
TIME. MINLPTES
Figure 6. Residence-time distribution curve for commercial desulfurization reactor during period of poor operation VOL.
3
NO. 4
OCTOBER 1 9 6 4
383
0.1
FEED:
GAS OIL
0.05
0.04
A
95%.
0
40.50%
8.3 0
10
20
30
40
50
60
analysis presented involves a commercial hydrodesulfurization reactor which did not perform according to pilot plant predictions. As was to be expected, the question arose: Why was desulfurization lower than expected? Was the poor desulfurization the result of a; physical effect--e.g., channeling of liquid in the catalyst bed-or was it a chemical effect-e.g., a less active catalyst, coking of the catalyst bed, poisons, etc.? A series of carbon-14 tracer experiments was made on the commercial reactor and on the pilot unit. Typical curves obtained are shown in Figures 4 to 6. Analysis of these curves by the techniques previously described showed that the poorer reaction performance was due to poor contacting. Contacting efficiencies of around 40 to 60y, were calculated for the commercial reactor, and around 90% for the pilot plant reactor. The very low contacting efficiencies suggested severe channeling and/or bypassing of liquid in the desulfurization reactor. An investigation indicated that the catalyst bed was being disturbed during startup. Shifting and channeling of the catalyst bed \vere probably occurring. Steps were taken to pack the catalyst bed more uniformly, and the startup procedure was revised so that chances of faulting the catalyst bed were minimized. Additional tracer tests were run, and a typical residence time distribution curve from these tests is shown in Figure 7. Analysis of these new data showed that the revised startup procedure had indeed improved contacting. Contacting efficiencies immediately rose to 70 to 80%, and actual desulfurization results were more in line with pilot plant predictions. The reaction rate constants calculated from Equation 2 are independent of reactor mixing and contacting efficiency. T h e only factors affecting the value of the reaction rate constants are those chemical in nature-e.g., type of feed stock, treat gas composition, catalyst activity, pressure, and temperature An Arrhenius plot of the reaction rate constants, calculated for the same catalyst on two different feed stocks, gas oil and Diesel fuel, for different reactor space velocities and contacting efficiencies is shown in Figure 8. .4s can be seen, the rate constants line up very well as a function of temperature. In addition. they reflrct the ease of desulfurization for the two feed stocks. The reaction rate constant correlation was obtained from reactors of widely different contacting efficiencies. The commercial unit and pilot plant reaction rate constants 384
l & E C PROCESS D E S I G N A N D D E V E L O P M E N T
COSTACTIKG EFFICIENCY 8 7
8.5
8.4
8.6
8.9 8 8
9 1
9.0
9.5
9.3 9.2
9.4
9.8
RECIPROCAL ABSOLUTE TEMPERATURE
T I Y E , MINUTES
Figure 7. Residence-time distribution curve for commercial desulfurization reactor during period of good operation
COhTACTING EFFICIENCY
i / * ~io4
Figure 8. constgnts
Effect of temperature on reaction rate
on exactly the same catalyst and feed stock are compared in Figure 9 . Within the experimental error, there is no significant difference in the commercial unit and pilot plant reaction rate constants. Thus, any differences in conversion at the same operating conditions between the pilot plant and commercial unit must be attributed to differences in contacting between the two units.
Models of Reactors and Higher Order Reactions
When reaction kinetics are other than first-order, reaction rate constants cannot be calculated directly from a residencetime distribution curve. In these instances, it is necessary to have' a n actual contacting model of the reactor under study. However, differences in residence-time distributions can still be determined by the method previously discussed. Physical models of reactors are helpful in pinpointing contacting problems and are absolutely necessary when considering higher order reactions. Zwietering ( 6 )a n d Cholette and Blanchet ( 7 , 2) have covered this in recent publications. Once a reactor contacting model has been determined, kinetics of any order may be handled. A method of determining physical models from residence-time distribution curves has been given by Turner (5). In our studies, we have determined that a simple parallel path model is a good one. In this model, a down-flow, fixedbed mixed-phase reactor is visualized: as shown in Figure 10. Liquid is pictured as flowing through the reactor in a large number of parallel paths. Flow through each path is plug flow, but because of different resistances to flow in the various paths, the same amount of liquid does not necessarily flow through each differential cross section of the reactor. As a consequence, what is pictured is a very large number of parallel paths of liquid flow, with the possibility that the liquid velocity in any path is different from that in any other path; thus, there is a residence-time distribution of the liquid flowing through the reactor. From such a physical picture and firstorder kinetics it is possible to derive Equation 1 . This derivation is given in the Appendix. The model bvhich we have chosen is consistent with our data and is in line with other experimental observations we have made-for example, tem-
I t
PILOT PLAKT L I N E
TIME
5c
2
APPROXIMATE 2 d LIMITS
0.07
0.06
PAYSICAL MODEL
c
\
0.05
-
0
0.04
2 0.03
P
$
2
0 0 2
P I L O T PLAYT COVMERC I REACTOR
\
\
'F ,
TEMPERATURE,
\
640
8.5 L l1 8.7 8.9
II 9.1
600
I
I
9.3
I I ! 9.5
I
I
9.7
I
I
Figure 10. Model of reactor from residence-time curve
V O Rx LO* Figure 9. Reaction rote constants for pilot plant and commercial unit
perature variations and flow studies in small packed columns. The model proposed here is similar to the segregated floiv model discussed by DanckLverts ( 3 ) and the zero intermixing model of Greenhalgh. Johnson, and S o r t (4). Once we obtain a satiTfactory model for a reactor, we are not limited to firstorder kinetics, but can study kinetics of any other. T h e only limitation is how well the assumed model represents the actual reactor.
given point in the channel is the same for both channels. This assumes no mixing in the svstem Let us assume that a reactor consists of a large number of plug flow channels through which liquid is flqwinq at diTerent velocities and that the amount of tracer flowinq from the reactor in time A0 is due to flow of liquid through a limited area of reactions. A A , where the true liquid velocity is between z and u Au. It is assumed that AL is small compared to u . By material balance
+
tVAO
Generalizations
The method presented is a satisfactory one for describing the Chemical and physical performance level of a n i reactor effects are easil) reparated. The residence-time distribution curve5 give a large amount of information about the reactor svftem and can be uspd to predict the performance level of a reactor, irrespective of the process conditions or kinetics. \.\'hen the process being studied is first-order or pseudo-firstorder, the reqidence-time distribution curve can be used to calculate reaction rate constants which in turn can be correlated against proceqs variables.
=
t,vAOfAA
where t , is the concentration of the tracer in the slug. Vt,AO is the total amount of tracer added,
Since
V21A0 = TLfAA or
V2tA0 f A A = --
Tu
For a first-order reaction
- dr
Appendix Calculation of Conversion from Tracer-Time Curves
One characteristic of an impulse created by injection of some tracer material into a stream is that the time, AO, it takes to pass any point in a channel is always the same, assuming it is put into a liquid moving at constant velocity a t the injection point. This can readily be seen for channels of greater or less cross section through vvhich all of the liquid flows, since the mass flowing through the channel per unit time is constant. If a reactor is assumed to consist of a large number of parallel channels through which the liquid is flowing at various velocities, it is still true that the time it takes the slug of tracer material to pass any point in any channel is constant and is AO, If the velocity is. say, ten times higher in one channel than in another, then ten times as much tracer material will go through a unit area of the higher velocity channel, but since it goes through at ten times the velocity the time for the slug to pass a
__ = kr dO
or (7) where O is the time it takes the particular stream to flow through the reactor. Let C, be the total reactant leaving the reactor per unit time, C I the reactant in the feed per unit time, and AC, the amount of reactant coming from the liquid flowing through reactor area, A A , per unit time. Then
AC, = R,vfAA Substituting from Equation 6 for j A A R,vV2tA0 R,V2tA0 A c - -__ - ___ 0 -
Tu
VOL. 3
NO. 4
T
OCTOBER 1964
385
Substituting for
e
R, from Equation 7 R iW e - "A0 AC, =
T
For the whole reactor m
C, = X A C , = 0
RiVZ
T
time from tracer concentration-time curve concentration of tracer from tracer concentration-time curve f = fraction of reactor cross section. AA, for liquid flow V = total volume liquid through reactor per unit time or. if gas only, total volume gas per unit time r = concentration of reactants 7' = total amount of tracer injected or to L-10 where to is concentration of tracer i n pulse and 1 0 is duration of pulse C = reactant from reactor per unit time Ri = concentration of reactant in feed R, = concentration of reactant from differential section of reactor oiitlet t
te-"A0 0
T h e reactant flowing into the reactor per unit time is V R I Cf. T h e fractional reduction in reactant is then
=
= =
since A0 is small compared to 0 Literature Cited
T h e integral can be obtained from the data given in the tracer concentration-time curve. Various values of k must be assumed until the right value for reactant reduction is obtained.
Nomenclature AA = very small portion of reactor cross section = true liquid velocity in differential cross section u
(1) Cholette, X.. Blanchet, J..Can.J . Chrni. E n g . 39, 192 (1961). (2) Cholette, A , , Cloutier, L.. [bid., 3 7 , 105 (1959). (3) DanckLverts. P. V.; ChPm. E y . Sci. 2, 1 (1953). (4) Greenhalgh. K. E., Johnson, R. L.. Kott: H. D., Chem. E n g . Progr. 5 5 , 44 (Februarv 1959). ( 5 ) Turner. G . '4.. Chem. En?. Scz. 7 , 156 (1958). (6) Zwietering. Th. K.> I5id., 11, 1 (1959).
RECEIVED for reliew October 15. 1962 ACCEPTED.July 2 , 1964
SCALE-UP OF A VENTURI AERATOR M . L . JACKSON AND W . D . COLLINS Department o j Chemical Engineering, Cniwrsity o j Idaho. lMoscozu. Idaho
Energy losses and oxygen transfer characteristics of two sizes of Venturi devices, designed for air aspiration b y the liquid, were compared. The ratio of linear dimensions was 5, and the ratio of water flow capacity was 25. Nominal sizes are indicated b y inlets of 4-inch piping and 3/4-inch tubing. Best performance of the 4-inch device occurred at the lowest water rate observed, 86 gallons per minute. At this rate 1.4 pounds of oxygen were transferred per unit horsepower-hour of energy input to the liquid. This transfer characteristic compares favorably with other types of aeration devices. However, the 3/4-inch Venturi aerator transferred four times this quantity of oxygen per unit of energy input. In the large scale system transfer factors for water and for very dilute sulfite solutions, corresponding to falling and constant driving force conditions, were essentially equal. Consideration of factors involved in scale-up suggests that in many cases multiple 4-inch Venturis might b e preferable to a single device of much larger capacity.
transfer of oxygen in Bernoulli-type devices for the airThese devices were of the nozzle, orifice, and Venturi types which transfer oxygen by aspirating air into the throat and producing interfacial area between the gas and liquid phases. Energy supplied to effect liquid flow results in the low pressures for air aspiration. The nozzle, orifice, and Venturi devices were compared on a small scale, with the Venturi type giving the highest oxygen transfer rate per unit of power expended in pumping the liquid. Judging by a comparison of the nozzle with conventional devices (2) operating under similar conditions, it was concluded that the Venturi device should offer an economical means of transferring oxygen from air to water. Accordingly, the performance of the Venturi device was investigated on a larger scale. I t was also desired to determine, if possible, the scale-up factors peculiar to this device, so that the feasibility of using very large Venturi devices for oxygen HE
T water system has been studied (5, 6 ) .
386
I b E C PROCESS D E S I G N A N D DEVELOPMENT
transfer might be evaluated. The large Venturi is a full scale model for many purposes, and the work was not directed to provide a basis for scale-up to still larger sizes. Theoretical Considerations
The transfer of oxygen from air to a liquid is important in many instances of waste treatment and also in certain cases of industrial processing, notably those involving biological activity. T h e solubility of oxygen is unfortunately very low, of the order of 7 to 10 p.p.m. at atmospheric temperatures and pressures, which means that not much oxygen can be transferred by a single pass through a transfer device. Most such operations, therefore, involve relatively high capital costs for equipmente.g., trickling filters, spargers. or diffusers in activated sludge plants-and high continuous operating costs. Energy requirements to provide the oxygen from air can be substantial, particularly if compressed air is employed.
