Residence Time

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Polymerization Technology – Laboratory Course

Residence Time Distribution of Chemical Reactors

If molecules or elements of a fluid are taking different routes through the volume of a continuousoperated reactor, they will spend different times within such a reactor. The distribution of theseholding times is called the residence time distribution (RTD) of the fluid.The RTD can affect the performance of a reactor and may also have a strong input on theselectivity of a chemical reaction. In case of polymerization reactions the RTD can have an effecton the molecular weight distribution of the polymer formed. This will be mainly the case when themean life time of the active species of the polymerization reaction is in the same order ofmagnitude like the mean residence time of the reactor. In this case polymers with a narrowmolecular weight distribution can only be formed in a reactor with narrow RTD. The RTD in caseof polymerization reactions can also play a major role if the reaction mixture is a segregatedsystem. Segregation in the reaction mixture can easily occur if the reaction mixture is of highviscosity or heterogeneous nature with elements which act as individual micro reactors withoutexchange of mass. The RTD of a polymerization reactor is therefore an important parameterwhich may affect the performance of the reactor but also the properties of the polymer formed.

1. Experimental methods for determination of RTD

Most important for determination of the RTD of a reactor is the application of a suitable tracer. Asuitable tracer should be easy to detect and the total amount of injected tracer should bedetectable at the exit of reactor.The most important methods for the determination of the RTD are the so called pulse and stepexperiments. They are easy to perform and interpret.

The pulse experiments

In this case a certain amount of a tracer is added pulse wise to the fluid entering the reactor andthe concentration-time relation of the tracer at the exit of reactor is recorded. This is shownschematically in Fig. 1. From the balance of material for the reactor the mean time of theconcentration-time distribution can be found.

Mean time (holding time):∑∑

∫

∫== ∞

∞

iii

iiii

0

0

tC

tCt

Cdt

tCdtt

∆

∆[s]



Fig. 1: Tracer concentration-time correlation of a pulse experiment

To find the RTD which is also called the exit age distribution E concentration-time distributionhas to be normalized in such a way that the area under the distribution curve is unity. For doingthis the concentraion readings have to be divided by the area under the concentration curve. Thisis shown in Fig. 2. The relationship between C and E curves only hold s exactly for reactors withso called closed boundary conditions. This means that the fluid only enters and only leaves thereactor one time. No dispersion of tracer at the boundaries of reactor should happen. Very often itis convenient to use a dimensionless θE curve for reasons of comparison of reactors. In this casetime is measured in terms of mean residence time t/t=θ . Then EtE ⋅=θ .

Fig. 2: Transforming the experimental concentration curve into the exit age curve

ReactorV [m3]

Tracer pulse input Tracer pulse output

Flow rate, v [m3/s]

introduced amount of tracer, M [kg]

time [s ]

Area = ∫ ∑∞

=0 i

ii tCCdt ∆ Tracer concentration [kg/m3]

C[kg/m3]

time [s]

Mv

CE=

[1/s]

time [s]

Area = 1 Area = M/v



The step experiment

In this case the tracer is not introduced pulse wise into the fluid entering the reactor but isintroduced in a continuous way by injecting a constant side stream of tracer to the fluid enteringthe reactor and measuring the outlet tracer concentration C versus time as shown in Fig. 3. Themean residence time is given by following equation:

∫∫

∫==

max

max

max

C

0maxC

0

C

0 tdCC

1

dC

tdCt

Fig. 3: Tracer concentration-time correlation of a step experiment

The dimensionless form of the concentration curve is called F curve or transition function. Herethe tracer concentration is rising from zero to unity with time (see Fig. 4).

Fig. 4: Transforming the experimental tracer concentration curve into the F curve (transition function)



2. RTD of plug flow and mixed flow reactors

Fig. 5 is showing the residence time distribution of a plug flow tubular reactor, a single well mixedstirred tank reactor and of a cascade of N equal size well mixed stirred tank reactors which areconnected to each other in series. The most narrow distribution is shown by the plug flow tubularreactor and the cascade of stirred tank reactors with an infinite number of vessels. The broadestRTD results in case of a single stirred tank reactor. The RTD of equal sized stirred tank reactorswith mixed flow is given by the following equations:

( )t

NtN1N

e!1N

Ntt

t1

E−−

−

=

with itNt ⋅= (N = number of reactors).

( )

−++

++−=

−− 1N2t

Nt

tNt

)!1N1

...tNt

!21

tNt

1e1F

Fig. 5: RTD of different ideal reactors: PFTR, HCSTR and Cascade of HCSTRs (fromLevenspiel)

1

plug flow mixed flow 1 2 N

mixed flow in series

1

1

t1

N=15

∞

∞

∞

1

15

15

t

t

t

t

θ



3. RTD of reactors with non-ideal flow pattern

In reality the flow pattern of reactors deviate from plug or mixed flow patterns. This is especiallythe case for polymerization reactors in which a polymer solution or dispersion with high viscosityis flowing through the volume of reactor causing a non-ideal flow pattern. Non-ideal flow patternscan result for example if the reactor volume contains so called dead or stagnant regions or ifbypass or recycle flow is present next to the active flow through reactor regions of plug andmixed flow. If these non-ideal flow patterns are present in a given reactor they can be seen easilyby looking at the corresponding experimental RTD. The following models can be used to describethe measured RTD of real reactors with deviation from ideal flow:

• Compartment Model• Dispersion Model• Tanks-In-Series Model• Convection Model (for laminar flow)

In Fig. 6 compartment flow models are given for a tubular and a stirred tank reactor which arecharacterized by the presence of dead zones and bypass. The dispersion and tanks-in-series

Fig. 6: Compartment models for tubular and stirred tank reactor with dead zone and bypass (from Levenspiel)

E

t

E

E

t

E



model is used in general when small deviations from plug flow are expected. They are oneparameter models. A dispersion number is used in case of the dispersion model whereas thenumber of stirred tanks is used in case of the tanks-in-series model. In Fig. 7 the correlationbetween dispersion and hydrodynamics of different liquids flowing in pipes are given.

Fig. 7: Correlation between dispersion number and Reynolds number of three different liquids flowing in pipes. Dispersion number: uLDax , Reynolds number: ηρud , Schmidt number: Dρη

The convection model is used if a viscous liquid is pumped through a tubular reactor. In generalthe flow is of laminar characteristic with a parabolic velocity profile. Thus the spread in residencetimes is caused only by velocity variations. The velocity profile of a laminar flow is showntogether with the corresponding RTD in Fig. 8.

Reynolds number

Dispersion

number

water



Fig. 8: Parabolic flow velocity profile and residence time distribution of laminar flow in pipe

4. Experimental

RTD of tubular reactor and continuous stirred tank reactor

4.1 Pulse experiments in tubular reactor• Put selector valve of laboratory set to tubular reactor position• Switch on conductimeter. Put selector switch for test data to “Strömungsrohr”.• Settings of conductimeter:

Temp.: 20°C Coeff%K-1: 1

• Settings of recorder: Speed: 200 mm/min

Range: 20 mV Zero line to 90 % scales on the paper

• 3 measurements at various flow rates: 50 l/h with 1,5 ml KMnO4/KCl solution for marking 80 l/h with 1,0 ml KMnO4/KCl solution for marking 105 l/h with 0,6 ml KMnO4/KCl solution for marking (see diagram for flow rates).

Marking solution with 12,5% KCl

L

d u2umax =

0 0,5 1 1,5tt=θ

1

4

3

2

θE



• Before starting each measurement check the flow rate at the rotameter and wait forconstance of the zero line.

• Inject the marking solution quickly and mark the starting point on the recorder paper.• Measurement is finished at constant conductivity.

4.2 Pulse experiments in stirred tank reactor:• Close water cock!!!• Put selector valve to stirred tank reactor position• Flow rate at the rotameter: 60 l/h• When the stirring tank is filled with water, switch on the stirrer.• Stirrer: 50 and 600 RPM• Reactor volume: 1400 ml• Settings of recorder:

Speed: 50 mm/min Range: 20 mV Zero line to 90 % scales on the paper

• 2 measurements at each stirring rate with and without dead water zone (100ml) for each experiment use 2,0 ml of the KMnO4/KCl solution for marking.

• Marking solution with 25% KCl

4.3 Step experiments in stirred tank reactor:• Flow rate at the rotameter: 60 l/h• Stirrer: 200 RPM• Reactor volume: 1400 ml• Settings of recorder: like 2• 2 measurements with and without dead water zone (100ml pipette) for each experiment use 25% KCl solution and the flexible-tube pump for marking.

4.4 Report:• Estimate E and F curves for all pulse and step experiments• Compare hydrodynamic residence time )V/V( R

& with experimental residence time (meantime).

• Determine influence of stirrer speed, dead water zone and flow rate on residence timedistribution and discuss extensive all results

Residence Time

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