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Non-Ideal Reactors Deviations from ideal reactor behavior Tank Reactors: inadequate mixing, stagnant regions, bypassing or short-circuiting Tubular Reactors: mixing in longitudinal direction, incomplete mixing in radial direction, bypassing (especially in fixed bed reactors) Definitions * Segregated flow - fluid elements do not mix, have different residence times - Need ’Residence Time Distribution’ * Micromixing - adjacent elements mix partially - Need extent of micromixing * Effects of non-ideality are higher for viscous reaction mixtures * Study limited to non-idealities for single reactions in homogeneous reactors here. Residence Time Distribution * residence time is the time it takes for a fluid element to pass through the reactor * age is the time since the element entered the reactor * residual lifetime - remaining time the element will spend in the reactor * age+residual lifetime=residence time F(t): Fraction of the effluent stream that has residence time less than t. F(0)=0 F(inf)=1 E(t)dt = dF is the fraction of the effluent stream that has residence time between t and t+dt E(t) is typically called the residence time distribution function Based on the concentration of tracer species measured at effluent, E(t) can be defined as:


 


    



  if the volumetric flow rate can be assumed to be constant, for a pulse input.    

   for a step input.

Properties of RTD


   !
#"$ (fraction of effluent that has been in reactor for time less   #
#"()+*-,. (for closed system) Mean Residence Time: %'&  1
32 %4&  0 !
#"$ Variance of the RTD: /0
 (fraction of fluid in the reactor that has age between t Internal age distribution function 5

Cumulative RTD: than t)

and t+dt)

RTD for Ideal Reactors Since the mixing conditions are well known for these ideal reactors, the RTD can be predicted without real experimentation: PFR - All elements come out at same time


+67
82 %  69
 is the Dirac Delta function, has a value of : 
>?@69
>A2 % #"$>B <
%  Furthermore, ;=<

at t=0 and is zero otherwise.

CSTR- Material balance on inert tracer can be performed: in-out=accumulation

2 .EFF* G C D EH+E ;JI >$KL
@2M, %  
=NPOQ URS T#U' Thus, %4&  %

Laminar flow reactor: Similarly (see Fogler), the RTD for a laminar flow reactor can be derived as:


 C V XW % ,ZY 
 % 0 Z, Y[#\ V X]^ % ,ZY Reactor modelling with RTD: * chance of a molecule reacting depends on the other molecules it encounters during its stay in

the reactor, => need residence time and mixing pattern * if velocity and local rate of mixing of every molecule in the reactor were known, differential mass balance can be written and integrated to get the final conversion * in the absence of such information approximations are necessary for reactor modelling * so, for reactors with known RTD, some assumptions regarding mixing have to be made. Mixing conditions: extremes

The two extreme situations are: -> incoming fluid remains completely segregated - i.e., it forms small globules that are uniformly dispersed, and such that all molecules in the globule have the same residence time (Segregation Model) -> incoming fluid is completely mixed on a molecular scale (Maximum Mixedness Model) * it turns out that for first order reactions either assumption amounts to the same thing in terms of final conversion, since conversion is independant of local concentration * in case of the CSTR, maximum mixedness also implies perfect mixing. if the RTD of the real reactor is the same as that for a CSTR, then the maximum mixedness assumption on top of it, leads to equations matching that of a CSTR. Estimating Deviations from Ideal Behavior: Zero and One-Parameter Models (i) Wholly Segregated Flow - measure RTD from experiment- calculate conversion by assuming wholly segregated flow (good for PFR; not good for nearly ideal CSTR) (ii) Maximum Mixedness Model - measure RTD - assume maximum mixedness (good for nearly ideal CSTR) (ii) Axial Dispersion Model - assume ideal PFR + axial dispersion occurs - actual RTD is required - good for PFR in turbulent flow (iii) Series of Ideal CSTRs of equal volume - actual reactor response to be used to determine number of tanks (iv) PFR with recycle (not covered in class) - mixing introduced through recycle - extent of recycle determines whether close to PFR behavior (zero recycle) or CSTR behavior (infinite recycle)

Degree of segregation Danckwerts’ two limiting mixing conditions can be further clarified as: (i) Incoming fluid is broken up into discrete fragmants which are small compared to the size of the reactor, and are uniformly dispersed in the reactor. the molecules of the fragments which have entered together remain together indefinitely. (fully segregated fluid) (ii) Incoming material is dispersed on a molecular scale in a time less than the average residence time; the mixture is chemically uniform ’Segregation’ is a measure of mixing in the reactor. It is understood to be the degree of departure of reactor from the perfectly mixed behaviour. - Suppose we superimpose the F vs. time diagram of a perfectly mixed reactor over that for the real reactor. The difference between the two curves can be said to be a measure of segregation in the system. * A useful definition of perfect mixing is: If the age distribution of material in the reactor is the same as that in the outgoing stream, the reactor is perfectly mixed (CSTR). The other extreme is when the exit age is the same for all internal ages (PFR). * If the age of a molecule is a= time elapsed since the molecule entered the reactor

.9_(`a_b
c_d2 _  0 - average over all the molecules, _

being the mean age of molcules currently in

the reactor

* Now let the mean age of molecules occupying some point in the reactor be

.9_(`a_ Q  c
_ Q 2 _  0

- average over all points in the reactor.

_ Q . Then,

Now, for mixed flow, there is no variation in ages of molecules at different points in the reactor, therefore . (Note that this cannot be strictly true except for the CSTR)

.7_7`a_ Q  C

For ’Fully Segregated Flow’ the variance in ages between points in the reactor equals the variance in ages, as molecules that enter at the same time stay together and pass through the reactor all together. This,

.9_(`a_be.7_7`a_ Q

for this case.

Based on these observations, ’degree of segregation’ can be defined as:

fge.9_(`a_ Q ,.7_7`Z_ , such that fhji for fully segregated flow, and fh C for mixed flow.

The degree of segregation which can be estimated from experiments, serves as a useful means of estimating the mixing characteristics of the real reactor. Qualitative analysis of the effect of segregation/mixing:

Consider (a problem from Fogler) a second order reaction in two reactor sequences: a. A PFR followed by CSTR b. A CSTR followed by PFR Let the concentration of reactant at inlet be 1, and the reaction rate constant and residence time in each reactor also 1, all in self-consistent units. Then it can be shown that in case (a) the final conversion is 63.4% while in case (b) the final conversion is 61.8%. Note that it can be further shown the RTD for both (a) and (b) is the same. This difference in conversion in the two set-ups is attributable to the fact that in (a) the mixing between molecules of different ages happens later than in case (b), and this in turn implies that the degree of segregation is higher in (a) than in (b). For second and higher order reactions a higher degree of segregation is suitable, while the opposite is true for negative reaction orders. Estimating conversion * in the general, the complex mixing pattern has to be known in addition to the residence time function for the evaluation of conversion. * in the extreme cases of mixing this estimation is quite simplified: * typically, we can say that estimating conversion using the RTD of the real reactor, along with the completely segregated and maximum micxedness models will give the minimum and maximum bounds for the ’real’ conversion. Completely segregated model * In this case, each ’globule’, i.e., bunch of molecules that enter the reactor at same time, can be thought of as a separate batch reactor. * Each globule is a lump of all the molecules that have the same residence time in the reactor * Let the conversion in the batch reactor be X(t), where t refers to the time spent in reactor, as relevant to the globules, this t is the corresponding residence time Mean Conversion of globules with res. time between t and t+dt = Conversion in batch reactor at time t x fraction of globules that have residence time between t and t+dt. In other words, reactor conversion for this model can be written as:

kl  ;  km
#!
#"$

Maximum Mixedness Model * This limiting case is not as easy to model * If the residence time distribution function for the real reactor matches that for a CSTR, then

in the limit of maximum mixedness, we can assume that the real reactor behaves like a CSTR, and just solve the CSTR equations * It is not possible to apply this model if the RTD is that of a PFR * If the residence time distribution is some other function, then the minimum value of the degree of segregation is not zero, we hypothesize a tubular reactor, with fluid entering along the length of the tube, through side entrances, in such a way that the RTD remains same as that of the real reactor. In addition, plug flow is assumed so that there is complete radial mixing as soon as the fluid enters the reactor. * Then, the differential equation of mass balance can be derived to be:

G o n j2M`qpsrt
uEvpw2xEvp ; m{ R}y |3 zoq ~o E p ; refers to the inlet concentration of the reactant, ` p is the reaction rate, is the RTD, where  the cumulative RTD, and  is a time variable such that its value is 0 at the exit of the reactor and

inf. at the entrance of the reactor.

* Two initial conditions are common for this hypothetical reactor,

  : V Evp€+Evp ;

or

* either can be used

  : V "Evp8,["   C

* to solve this equation, we have to integrate backwards (typically numerically), starting at a large value of



(usually 4-5 times the average residence time), and finally reaching

conversion can thus be calculated.

  C

The

* The process is little more involved than the completely segregated model, but turns out to be really simple if the RTD matches that of CSTR * Note that this model cannot be used of the RTD matches that of a PFR, a quick substitution into the differential equation will show you the inconsistency * This is because the micro- and macro-mixing aspects of a reactor are not exactly independent of each other, if the RTD is that of a PFR then maximum mixedness is impossible as the RTD itself implies virtually no mixing between adjacent elements. * Detailed mathematical analysis can be found in the journal articles by Danckwerts and Zwitterieg, and is closely related to the analysis of degree of segregation for real reactors Tanks-in-series model * this model has one adjustable parameter, and also requires the RTD function * typical application is for tubular reactors * the assumption is that the real reactor can be approximated as a series of equal-volume CSTRs * the job is the find the number of such CSTRs required * if the average res. time and variance are evaluated from from the RTD function, the number

of reactors can be obtained from a simple relation:

‚  „U'ƒƒ

* Further, the conversion can be calculated easily as all the CSTRs are of same size (by repeated application of the CSTR mass balance equations)

k…†i2 {P‡?ˆ { UŠ‰‹Œ

* for first order reactions it is simply



where n is number of tanks from above, and is the residence time in eaach

tank (=t/n)

* Note that for non first-order reactions, numerical solution of the system of algebraic equations from mass balance for each tank, may be required. Axial dispersion model * more complicated to analyse than the tanks-in-series; also requires one parameter to be estimated (dispersion coefficient) * application to tubular reactors * axial dispersion, governed by Fick’s law of diffusion is superimposed on ideal plug flow * dispersion causes the input pulse to broaden as it moves down the tubular reactor, thus it can be interpreted as deviation from plug flow (where the pulse remains infinitesimally narrow) * First an unsteady mole balance on the inert tracer is to be done (for determination of the

Ž Ž'Ž  j  2 ~ Ž   r’‘b“

Ž Ž'ƒ ” •‘ “ ƒ ( is the dispersion coefficient)

dispsersion coefficient from pulse tracer experiments):

* Different types of boundary conditions for the reactor are to be imposed, depending on the situation just outside: Closed-closed implies that outside both entrance and exit of the real reactor, plug flow can be assumed; Open-open implies that the dispersion non-ideality exits just outside the reactor at entrance and exit * the following equations are relevant for estimating dispersion coefficient for each of these BCs (Pe is Peclet Number = D/uL):

„ ƒ  – 0 N 2 – 0 N
@i—2 I R – N  „ ƒ U ƒ – 0 r – ˜ (b) Open-open: U ƒ N Nƒ

(a) Closed-closed:

* Note that the Peclet number for some reactor types can be also be obtained as functions of Reynolds number etc. through correlations (See Levenspiel, correlations in figures). These obviate the need for tracer experiments, but exist only for a handful of reactor types * After estimatoin of the dispersion coefficient, the combined process of axial dispersion and reaction has be solved to obtain conversion, appropriate boundary conditions have to be chosen in this case as well. Analytical solution is only possible in case of first order reactions (Fogler has detailed calculations).

Summary * Non-idealities in reactors are understood from viewpoint of micro and macro-mixing. * Macromixing effects are captures thorugh the residence time distribution function that can be obtained from inert tracer experiments * Micromixing effects are analyzed with two extreme cases - COmplete segregation (no micromixing) and maximum mixedness (complete micromixing) models. * One-parameter models used for tubular reactors are tanks in series model and axial dispersion model - the parameter is to be evaluated using the RTD; conversion to be calculated by re-writing the balance equations for assumed reactor types. * Several other models have been developed for capturing non-ideality. Levenpsiel and Fogler give descriptions of some of these, others can be found in research articles.