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Ind. Eng. Chem. Res. 2008, 47, 3380-3387
Two Approximate Methods To Estimate Conversion and Yield in Mixed Chemical Reactors Gary K. Patterson* Department of Chemical and Biological Engineering, UniVersity of MissourisRolla, Rolla, Missouri 65401
In many cases, quick methods are needed to evaluate the conversion and yield that may occur in a given reactor with a desired chemical reaction and necessary product. Full experimentation and/or full numerical simulation to evaluate the situation is neither warranted nor needed. In many cases, methods even quicker than computational fluid dynamics (CFD) with closure are needed to approximate yield results for various versions of a reactor. This paper reviews two methods that might be used beyond assuming perfect mixing or perfect plug flow to determine the likely conversion and yield for reactions in tubular reactors with mixing effects and in stirred vessel reactors. These methods can typically be implemented using a simple numerical integration program (such as POLYMATH) and/or a math program (such as MATLAB). In most cases, the results will not perfectly duplicate actual laboratory or pilot results, but simulated comparisons of various reactor setups can be made which are likely to be valid, because they can give the trends that occur for given changes. Such simplified calculations should always be made before more costly laboratory and simulation efforts are made. The approximate methods reviewed and compared are plug-flow mixing in a pipe with or without static mixing elements and several segment mixing in a stirred vessel. Closures for the extent of mixing in the pipes or segments are paired-interaction (P-I) and random coalescence and dispersion (C-D) mixing, both of which have shown good correspondence with the experimental results when used in CFD simulations. Introduction When a chemical reactor is under initial design or when reactor improvements are sought, a decision must be made in regard to to the sophisticationsthat is, the computational difficultysof the method used to estimate conversion and yield. When the degree of mixing has a significant effect on conversion and yield, the choice of method becomes even more important, because perfectly mixed or ideal plug-flow approximations frequently are entirely inadequate, except to determine maximum conversion and yield values. The mostsophisticated methods, which generally involve computational fluid dynamics (CFD) and complex closures of simultaneous partial differential equations, are very time-consuming and expensive to perform. For rapid approximations of likely reactor performance, methods are needed that can yield several results each computational day. The objective of this paper is to present two methods of that type: one that is best-suited for tubular turbulent flow reactors (with or without static mixer elements) and one that is best-suited for stirred vessels with semi-batch or continuous flow operation. The methods presented are intermediate in complexity between the CFD-based methods and ideal reactor methods. Ideal Reactors for Extremal Approximations The ideal plug-flow reactor (or, alternatively, the ideal batch reactor) generally represents the reactor with the greatest conversion of reactions and the greatest yield of the product of the fastest reaction in the mix. In the following, equations for the ideal cases are given by way of introducing the nomenclature and form to be used for the cases where mixing is taken into account. If XA represents the conversion of component A, * To whom correspondence should be addressed. Phone: 573-3416941. Fax: 573-341-4377. E-mail address:
[email protected].
XA )
moles Ain - moles Aout moles Ain
then, in the ideal plug-flow reactor, the conversion of A is given by the solution of
dXA
)-
dL
( )( 1
CA,in
( ) ∑j RA,j
AC
Q
(1)
where L is the length of the reactor, CA,in the feed concentration of reactant A, AC the reactor cross-sectional area, RA,j the reaction rate of component A in reaction j, and Q the total volumetric flow rate. This equation assumes no volumetric expansion or contraction during the reactions. In the ideal perfectly mixed continuous flow reactor, conversion is given by
XA ) -
( )∑ V
QCA,in
RA,j
(2)
j
where V is the reactor volume. The computation of conversion with this equation generally gives the greatest value possible in a stirred flow reactor, although slightly higher conversions are conceptually possible for certain feed points and circulation patterns. For both the ideal plug-flow and ideal perfectly mixed cases, the yield of a given reaction product may be determined by summing the conversions to that product for each reaction that is producing it and subtracting each conversion of that product to another product. Yield (YC) may be defined as
YC )
net moles C produced moles A reacted
10.1021/ie800042t CCC: $40.75 © 2008 American Chemical Society Published on Web 04/26/2008
(3)
Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3381
where C is a desired product and A is a reactant chosen to serve as the basis. For instance, if the reactions are
A+BfC
(k1)
(4)
C+BfD
(k2)
(5)
where k1 and k2 are reaction rate constants, then for an ideal plug-flow reactor,
dCi
)
dL
( )∑ AC Q
(
Ri,j)
(6)
j
for making such estimates for single- and multiple-reaction systems. For the segregated flow model, rate equations such as those previously given for the ideal tubular flow reactor are used to determine the conversion of each reaction, as a function of time in the reactor. Fluid elements are assumed to flow through the reactor as tiny batch reactors that do not mix until they exit from the reactor, each with a resident time determined by a residence time distribution (RTD). This form of segregation could be called “time segregation”, because fluid elements of different resident times do not mix. The average outlet concentration of all those fluid elements for each component i is given by
or
dCi,avg ) Ci(t)E(t) dt
( )
dCA AC ) (-k1CACB) dL Q
(7)
dCB AC ) (-k1CACB - k2CBCC) dL Q
(8)
AC dCC ) (k C C - k2CBCC) dL Q 1 A B
(9)
( ) ( )
( )
AC dCD (k C C ) ) dL Q 2 B C
(10)
LoutAC tout ) Q
CC,out CA,in - CA,out
V CA,out ) CA,in (k C C + k2CBCC) Q 1 A B
()
CD,out
(QV)(k C C ) 1 A B
(QV)(k C C - k C C ) V ) C + ( )(k C C ) Q
CC,out ) CC,in +
D,in
∑j
Rji(λ) +
(Ci(λ) - Ci,in)E(λ) 1 - F(λ)
(16)
where
F(λ) )
dCi
for cases where volume change is negligible. If the reaction is an ideal continuous flow stirred tank, then
CB,out ) CB,in -
dλ
)-
∫ -E(λ) dλ
and λ is the life expectancy for a fluid element in the reactor (or λ ) tmax - t). The equation must be integrated from the maximum time in the reactor to zero time; therefore, the equation becomes
The yield of C may be given by
YC )
where E(t) is the distribution of residence times for the fluid elements and each Ci is determined at each time by the kinetic equations. For the maximum mixedness model, the final conversions of the reactions are given by
dCi
which may be solved easily using a math program such as POLYMATH,1 to a final length given by Lout. The reaction time is related to reactor length by
(15)
1 A B
2 B C
2 B C
(11) (12) (13) (14)
Yield is calculated in the same way as that previously described. Note that for both ideal reaction models, only the reaction kinetics are required for calculation of conversion and yield. Churchill2-5 has reviewed and further developed various asymptotic solutions for conversion and yield in tubular flow reactors where laminar flow profiles, radial transport, heat transport, etc., are included as important variables. Segregated Flow and Maximum Mixedness Models A somewhat more accurate way to determine the extremes of conversion and yield in a reaction system is through the use of segregated flow and maximum mixedness models. Fogler6 reviewed these models and presented examples of using them
d(tmax - λ) -
)
∑j Rji(tmax - λ) +
[Ci(tmax - λ) - Ci,in]E(tmax - λ) 1 - F(tmax - λ)
(17)
Fogler has presented examples for solving these model equations using the program POLYMATH. Note that, for these models, both the reaction kinetic equations and the RTD of the reactor must be known or estimated. If the RTD of the proposed reactor is almost the same as the RTD of the ideal perfectly mixed vessel, then the results of the maximum mixedness model are the same as the ideal perfectly mixed model previously given. The segregated model will typically give higher levels of conversion and higher yields of the fastest reaction than the maximum mixedness model and, therefore, can be used as a better estimate of the maximum conversion and yield for a given RTD. The use of the RTD to make estimates of the conversion and yield of a system of reactions in a given reactor does not account for the effects of mixing of the chemical reactants introduced into the reactor. The models using RTD generally assume that the reactants are magically premixed at the feed point, which is impossible, except for very slow chemical reactions. Slow chemical reactions are not of concern when one is considering mixing effects, because the mixing will have occurred before significant reaction conversion has happened. A general rule of thumb is that the rate constant of the fastest reaction must
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be smaller than the rate constant of the mixing at the point of feed injection for mixing to be an issue in the design of a reactor.
dc2A,avg c2A,avg ) dt (2.05(LS2/)1/3 + 0.5(ν/)1/2 ln(Sc))
(20)
Intermediate Methods for Mixed Reactors To account for the mixing between reactants in an approximate way, the following two methods are introduced here: tubular reactor and stirred tank reactor. For the tubular reactor, the recommended method involves the integration of ordinary differential equations, which is very similar to those presented in the section on ideal reactors, but with terms added to account for the effect of unmixedness between the reactants (here, this is called reactant segregation, which is different from time segregation) on the reaction rates. Closure approximations are required to relate the reactant segregation to the concentrations. The stirred tank reactor performance may be approximated using a segmented model of the vessel where each segment is treated as a continuous-flow mixed reactor with terms added to account for reactant segregation. The solution of the equations for the vessel segments becomes trial and error, because the equations for the entire tank form a set with nonlinear unknowns, so a solution method that assumes linear equations will not work. A more direct and convenient method involves the use of random coalescence and dispersion (C-D) of representative fluid elements to model the mixing and flow in the reactor. Unlike the fluid elements in the segregated flow model previously discussed, these fluid elements mix with each other at prescribed rates. Flow is modeled by allowing the fluid elements to move from segment to segment at prescribed flow rates, based on estimated flow patterns in the reactor. The mixing and flow are done at the ends of time increments during which chemical reactions occur in each fluid element as if it is a tiny batch chemical reactor. The simulation is inherently timedependent, so fed-batch as well as continuous-flow reactors may be simulated. Tubular Reactor Case. The method recommended here to account for reactant segregation mixing effects in a tubular reactor is a generalization of the ideal case previously presented. The method makes use of a simple closure that was developed earlier7-9 and tested extensively using CFD simulations of stirred vessels. It is called the paired-interaction (P-I) closure, because only two-component mixing effects are included, even if the mixing of more than two components is involved. The P-I closure links the concentration correlation ((cAcB)ave) to the average concentrations (CA,avgand CB,avg) and the degrees of segregation (c2A,avg and c2B,avg) for a reaction A + B f C. These variables result in the Reynolds-averaged reaction rate equation for incompletely mixed reactants. For instance, the average reaction rate for A + B f C is
RA,avg ) (k1CACB)avg ) k1(CA,avgCB,avg + (cAcB)ave)
(18)
for a constant reaction rate constant k1, and cA and cB are fluctuations of the concentration about the means CA,avg and CB,avg. The P-I closure is expressed simply as follows:
(cAcB)ave ) -
c2A,avgc2B,avg CA,avgCB,avg
(19)
To use the closure, the degrees of segregation (c2A,avg and c2B,avg) must be known. The most convenient approximation for the segregations is the Corrsin model10 for mixing rate, which is given by
where LS is the scale of segregation, the rate of turbulence energy dissipation rate per unit mass, ν the kinematic viscosity, and Sc the Schmidt number (Sc ) ν/DA) for the fluid (here, DA is the diffusivity). Following the work of Pohorecki and Baldyga,11 the coefficient on the term with the Sc parameter was increased from 0.5 to 4, to more effectively account for viscosity effects in the computations that are presented below. Combining all of the aforementioned equations (eqs 6-10 and eqs 18-20) for a competitive-consecutive chemical reaction (A + B f C; C + B f D) that has been performed in a tubular reactor with static mixers, which gives an almost-constant reactant mixing rate from entrance to exit, gives the following results:
( )(
c2A,avgc2B,avg AC dCA,avg ) - k1CA,avgCB,avg + dL Q CA,avgCB,avg
( )(
)
c2A,avgc2B,avg AC dCB,avg - k1CA,avgCB,avg + ) dL Q CA,avgCB,avg
(
k2CB,avgCC,avg +
( )(
)
)
(
( )(
(21)
c2B,avgc2C,avg (22) CB,avgCC,avg
c2A,avgc2B,avg AC dCC,avg ) kC C dL Q 1 A,avg B,avg CA,avgCB,avg k2CB,avgCC,avg +
)
)
c2B,avgc2C,avg (23) CB,avgCC,avg
c2B,avgc2C,avg AC dCD,avg k2CB,avgCC,avg ) dL Q CB,avgCC,avg
)
(24)
where
( )( )
2 2 AC dci,avg dci,avg ) dL Q dt
The value of may be estimated from the expected pressure drop in the tubular reactor, based on the relation
avg )
( )( ) Q ∆P AC FL
(25)
where ∆P is the pressure drop in length L and F is the fluid density. The term ∆P/(FL) may be estimated using the relation
( )( )
2 Q ∆P ) Kf FL D AC
2
(26)
where K is the ratio of the static mixer pressure drop divided by open pipe pressure drop, f the Fanning friction factor, and D the inside diameter of the open pipe. LS (the scale of segregation) is dependent on the geometry of injection and of the static mixers (if present). For single- or multiple-feed injectors, this scale can be estimated using
L S ) rS )
( ) 2QF πuo
1/2
(27)
Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3383
Figure 1. Schematic of the Vassilatos and Toor tubular reactor with a mixing head.12
Figure 3. Schematic of the Baldyga, Bourne, and Hearn (BBH) reactor with static mixers.16
complex reaction system. The reactor is depicted schematically in Figure 3. Following Baldyga, Bourne, and Hearn,16 the pipe radius divided by two was assumed as an approximation of the mixing scale, LS. The method may be easily modified to compute conversions and yields as a function of the distance downstream for any turbulent plug-flow reactor and set of chemical reactions, if realistic feed conditions can be given. The chemical reactions, with their respective rate constants in the Baldyga, Bourne, and Hearn (BBH) case may be depicted as follows:
A + B f p-R A + B f o-R Figure 2. Comparison of the simulated and experimental conversion results for the Vassilatos and Toor12 acid-base tubular reactor for two reaction rates.
where QF is the volumetric feed rate for the injector in question and uo is the surrounding (ambient) flow velocity into which the injection must blend. The term rS represents the final radius of the injected fluid, and the ambient velocity is denoted as uo. If the scale of passage in the static mixers is smaller than rS, then the smaller scale should be used. For a twisted element static mixer, LS ≈ D/4. The initial value of the segregation values for each reactant that is fed may be calculated based on the relationship for complete segregation. For instance, c2A,avg ) c2B,avg ) CA,avgCB,avg if CA,avg ) CB,avg. Good approximations result from using the following equalities: c2A,avg ) C2A,avg, c2B,avg ) C2B,avg, etc. (a) Example of a Simple Second-Order Reaction in a Tubular Reactor with a Mixing Head Injector. Toor’s group12-14 made measurements of the degree of conversion for several acid-base chemical reactions in a tubular reactor that was equipped with a multijet mixing head. Patterson7 applied the Corrsin model for turbulent mixing to simulate the behavior of this reactor, using the experimental results of McKelvy et al.15 for the turbulence generated in the pipe by the mixing head injector. Except for the mixing head, which was composed of many tubes for injecting the acid and base reactants, the tubular reactor was an empty pipe 3.18 cm in diameter and 56 cm in length from the end of the mixing head. A schematic diagram of the reactor is shown in Figure 1. Figure 2 shows simulation results, compared to experimental measurements, of the conversion for two reactions (k ) 1011 and k ) 12 400 L/gmol s) and the measured values of and LS for a flow rate of 0.266 L/s. Note that the high level of turbulence has subsided to a very low level and the mixing head generated scale of mixing has increased significantly after only 7 cm and the reactions are essentially complete after only 3 cm. The simulation values are very close to the measurements. Similar results were obtained for simulations of other second-order acid-base reactions that were experimentally studied by Toor et al. (b) Example for a Tubular Reactor with Static Mixers. The previosuly described approach may be used to compute the yield values for a twisted-ribbon static-mixer reactor with a
(kR1 ) 12238 m3 kg-mol-1 s-1)
(28)
(kR2 ) 921 m3 kg-mol-1 s-1) (29)
p-R + B f S
(kR3 ) 1.835 m3 kg-mol-1 s-1) (30)
o-R + B f S
(kR4 ) 22.25 m3 kg-mol-1 s-1) (31)
AA + B f Q
(kR5 ) 125 m3 kg-mol-1 s-1) (32)
Details of the actual chemical components are found in the reference material. Because the static mixer has a diameter of 0.04 m, the value of LS is given as 0.01 m, which is one-half of the radius, and is assumed to be constant, although that is a strong approximation. Concentrations as a function of distance from the pipe entrance for all reactants and products were computed for comparison with the experimental results. The values of at various flow rates, which produced increasing mixing rates at increased flow rates, were calculated from the pressure drop data given by the authors and were as follows: Q (m3/s)
(m2/s3)
0.0005 0.0010 0.0015 0.0020 0.0025
1.38 11.0 37.1 88.0 171.9
The Sc values were taken to be 2000(ν/νwater) and the ν values were 0.89 × 10-6 or 3.6 × 10-6 m2/s. The resulting yields for flow rates in the range of 0.0005 m3/s to 0.0025 m3/s are shown in Figure 3 where they are compared with the experimental data. The equations for rates of mixing and chemical reaction were solved using POLYMATH (distributed by the CACHE Corp., an affiliate of AIChE) to obtain the simulated yields. Differential equations, as entered by the user for the case of q ) 0.0005 m3/s and ν ) 0.89 × 10-3 m2/s, were
(1) d(A)/d(t) ) -k1*(A*B + ab) - k2*(A*B + ab) (2) d(B)/d(t) ) -k1*(A*B + ab) - k2*(A*B + ab) k3*(PR*B + prb) - k4*(OR*B + orb) k5*(AA*B + aab) (3) d(PR)/d(t) ) k1*(A*B + ab) - k3*(PR*B + prb) (4) d(OR)/d(t) ) k1*(A*B + ab) - k4*(OR*B + orb)
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(5) d(AA)/d(t) ) -k5*(AA*B + aab) (6) d(S)/d(t) ) k3*(PR*B + prb) + k4*(OR*B + orb) (7) d(Q)/d(t) ) k5*(AA*B + aab) (8) d(sa)/d(t) ) -sa/tau (9) d(spr)/d(t) ) -spr/tau (10) d(sb)/d(t) ) -sb/tau (11) d(saa)/d(t) ) -saa/tau (12) d(sor)/d(t) ) -sor/tau Explicit equations, as entered by the user, are as follows:
Figure 4. Concentrations of the reactants and products, as a function of reactor length, for q ) 0.0005 m3/s and ν ) 0.89 × 10-6 m2/s.
(1) ab ) -sa*sb/(A*B) (2) prb ) -spr*sb/(PR*B) (3) orb ) -sor*sb/(OR*B) (4) aab ) -saa*sb/(AA*B) (5) k1 ) 12238 (6) k2 ) 921 (7) k3 ) 1.835 (8) k4 ) 22.25 (9) k5 ) 125 (10) sc ) 2000 (11) ls ) 0.01 (12) eps ) 1.38 (13) nu ) 0.89e-6 (14) tau ) 2.05*(((lsˆ2)/eps)ˆ0.333) + 4*((nu/eps)ˆ0.5)*ln(sc) (15) yieldQ ) Q/A0 (16) q ) 0.0005 (17) L ) t/(q/(3.14*0.0016)) A0 ) 0.02 B0 ) 0.0166 PR0 ) 10E-6 OR0 ) 10E-6 AA0 ) 0.08 S0 ) 0 Q0 ) 0 sa0 ) 0.00332 sb0 ) 0.00332 spr0 ) 0 sor0 ) 0 saa0 ) 0.00531 In the previously given set of equations, A ) CA, B ) CB, PR ) Cp-R, OR ) Co-R, AA ) CAA, S ) CS, Q ) CQ, sc ) Sc, tau ) τ, ls ) Ls, L is the length from pipe entrance, nu ) ν, eps ) , t is the time, and q is the volumetric flow rate (the units for all parameters are MKS units).
Figure 5. Effect of volumetric flow rate on the yield of the secondary product Q (CQ/CA0) for the BBH reaction system.
Computations for this example were performed using the program POLYMATH; however, any program that integrates sets of stiff differential equations (for instance, those using Gear methods) may be used. For this problem, steady state was attained in the reactor after 0.5 s of integration for all of the flow rates used. This corresponds to reactor lengths that are dependent on the velocity of the feed stream(s). Example profiles of concentrations of reactants A and B and products p-R, o-R, S, and Q are shown in Figure 4. The necessary reactor length, in this case, is 0.2 m, because the reaction time is 0.5 s and the feed rate is 0.0005 m3/s, giving a velocity of 0.4 m/s. The yield of Q in this case is 0.365, based on the initial concentration of component A. Figure 5 shows the effect of volumetric flow rate and kinematic viscosity on the yield of the secondary product Q. The simulated values do not show a viscosity dependence that is as great as that observed for the experimental data at the lower flow rates, although the coefficient on the second term of the Corrsin equation was increased. Also, the simulated yield values are slightly lower than the experimental values at the higher flow rates. Generally, however, the simulation performed quite well for such a complex reaction system. Stirred Reactor Case. As mentioned previously, the stirred reactor is best simulated on an approximate basis using the random C-D method. The reactor may be divided into any number of segments for the simulations. The examples given here will be for a round stirred tank with a liquid height almost equal to the reactor diameter. The tanks were divided into
Ind. Eng. Chem. Res., Vol. 47, No. 10, 2008 3385
Figure 7. Linear regression of the values of I required to match reaction conversion versus reactor length to experimental measurements against measured values of /LS2.
Figure 6. Schematic of a mixed tank with either 6 or 15 tangentially symmetric segments, all of which contain the numbers of fluid elements proportional to the segment volumes.
concentric-ring volume segments, because constant properties were assumed in the tangential direction. This means that the simulation is essentially two-dimensional, although a threedimensional geometry is certainly possible. Figure 6 shows the geometries and flow directions assumed; 6 and 15 segment examples are given. Flow rates are made proportional to impeller discharge rates, based on experimental data.17 A program written in MATLAB18 was used to determine the solutions of the simulations. MATLAB was chosen because of (i) the ease of programming for large numbers of operations, (ii) its ability to do random choices, and (iii) its similarity to FORTRAN in written form. The program as presently written can account for some variation in the ratio of liquid height to diameter, the impeller height (centered side-to-side), the impeller type (radial flow and axial flow), various reaction types (A + B f C; C + B f D; D + C f E; F f G; G f H; H + B f I; A f I) in any combination, and rate constants. Volumetric flow rates for the input and effluent and from segment to segment are estimated based on known experimental data for impeller pumping rates and flow patterns. Because the fluid in the vessel is represented by fluid elements, flow is simulated by moving fluid elements from segment to segment, with the number moved in each case being proportional to the volumetric flow rate. Rates of mixing are determined by use of the same Corrsin equation used above in the tubular reactor case. Estimates of the rates of turbulence energy dissipation per unit volume () are determined from well-known power dissipation in vessels with known impeller configurations. For a Rushton turbine radial flow impeller, for instance, the average value of in the impeller discharge region is ∼20 times greater than the average value in the more-quiescent portions of the vessel. Therefore, values of are set such that the total power dissipation corresponds to the known value with the ratios known from experimental measurements (see the work of Wu and Patterson17). The scalar mixing lengths (LS) are also set based on experimentally known values. The rates of coalescence and dispersion (C-D) are then determined based on a correspondence between the ratio /LS2 (see eq 20) and the C-D rate I, which was determined experimentally by Canon et al.:19
I ) Kcd(/LS2)1/3
(33)
This is illustrated by Figure 7, which shows the linear regression of -2 (ds/sdt) ≈ (/LS2)1/3 with the C-D rate (I) required to
Figure 8. Concentrations and yield of component C for the 36-L Paul and Treybal reactor20 for feed into the top, with a Rushton turbine (D/T ) 1/2) at 300 rpm.
successfully match the chemical reaction degree of completion measured by Vassilatos and Toor.12 The value of Kcd was determined to be ∼0.1. (a) Examples of Stirred-Tank Reactors: Laboratory Size Reactor. Paul and Treybal20 determined the mixing effects on the reaction between tyrosine and iodine to produce monoiodated and di-iodated products in 5-L and 36-L stirred reactors. The reactors were operated in batch-fed mode, and the reactions were given as follows:
tyrosine (A) + I2 (B) f HI + tyrosine-I (C) (k1 ) 35 L g-mol-1 s-1) (34) tyrosine-I (C) + I2 (B) f HI + tyrosine-I2 (D) (k2 ) 3.8 L g-mol-1 s-1) (35) The 36-L reactor was simulated for two impeller speeds (N ) 150 and 300 rpm), impeller diameters D of one-third and onehalf of the tank diameter T (D/T ) 1/3 and D/T ) 1/2, respectively), and feed locations at the vessel top and into the impeller stream. The simulations used the six-element version shown in Figure 6, and a Rushton radial flow impeller was used. The iodine (I2) was fed at a slow rate (1000 s), and the feed concentration was 10 times greater than the resident concentration of tyrosine. The total feed was such that the number of moles of I2 fed equaled the number of moles of tyrosine residing in the reactor. Figure 8 shows an example of the concentrations of reactants and products as a function of time for the case of D/T ) 1/2,
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Figure 9. Comparison of simulated and experimental results for the 36-L Paul and Treybal reactor20 (TF is top feed, IF is impeller feed, RIF is feed at the wall at impeller level, and PBDF is a pitched-blade impeller with down flow). The six-segment C-D model was used for computations.
top feed, and N ) 300 rpm. The plot for the yield of C is made zero until it becomes
