1

Ideal reactorsIdeal reactors

• Ideal reactors are useful models which are easy to treat(it is easy to find their performance equation).

• One of them usually represents the best way of contacting the reactants and for this reason we often try to design real reactors so that their flows approach these ideals.

• When the behaviour of real reactors deviates from that of ideal reactors, it can be described by appropriate combination of ideal reactors (modelling of real reactors with ideal reactors).

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OutlineOutline• Ideal reactor types: batch reactor (BR),

continuously stirred tank reactor (CSTR),plug flow reactor (PFR)

Application of continuous/discontinuous reactors

• Performance equations:mass (and energy) balancesreaction time t for BR residence time τ for CSTR, PFRcomparison between PFR and CSTR

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Ideal reactor typesIdeal reactor types

1.Batch Reactor (BR)[German: Absatzweiser Rührkessel (AIK)

diskontinuierlich betriebener Rührkesselreaktor]

2.Continuously operated Stirred Tank Reactor (CSTR)[German: KontinuierlicherDurchflußrührkessel (KIK)

kontinuierlich betriebener Rührkesselreaktor]

3.Plug Flow Reactor (PFR)[German: Strömungsrohrreaktor (IR)]

IN OUT

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Batch reactor (BR)Batch reactor (BR)

• The BR operates in a discontinuous way:the reactants and any additional substances(catalyst, solvent etc.) are loaded into the reactor,well mixed and left to react for a certain period.The resultant mixture is then discharged.

• In the course of this process, the composition in the reactor changes continuously: the BR operates in a unsteady mode with respect to time.

• At any instant the composition throughout the reactor is uniform: the BR operates in a steady mode with respect to space.

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Steady-state operation

with respect to position

Operation mode Operation mode –– ideal BRideal BR

Unsteady-state operation

with respect to time

[A0]

productsA → (irreversible unimolecular reaction)

[ ] [ ] kteAA −= 0

If the reaction is 1st order:[ ] [ ]1AkdtAdrA =−=

[AE]

t=tE

t=0[A0]

t=tE[AE]t

0

[A]

x0

[A]

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Operation mode Operation mode –– ideal BRideal BR

steady mode with respect to position

t0

unsteady mode with respect to time

t1 t2

c1

c1

c1

c0

c0

c0

c2

c2

c2

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Continuously operated stirred tank reactor (CSTR)Continuously operated stirred tank reactor (CSTR)

• The CSTR operates in a continuous way:the reactants are continuously fed into the reactorand products are continuously drawn from the reactor.The content of the reactor is well stirred so that concentration and temperature are uniform throughout (they correspond to those of the exit stream; 100% of backmixing).

• In the course of this process, the composition in the reactor does not change: the CSTR operates in a steady mode with respect to time.

• At any instant the composition throughout the reactor is uniform: the CSTR operates in a steady mode with respect to space.

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Steady-state operation

with respect to position

Operation mode Operation mode –– ideal CSTRideal CSTR

Steady-state operation

with respect to time

[AE]

t0

[A0]

t[AE]

x0

[A0]

In a CSTR the composition in the reactor is always that of the exit stream.

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Operation mode Operation mode –– ideal CSTRideal CSTR

steady mode with respect to position

steady mode with respect to time

t2

ce

ce

ce

t0

ce

ce

ce

t1

ce

ce

ce

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IN OUT

Plug flow reactor (PFR)Plug flow reactor (PFR)• The PFR operates in a continuous way:

the reactants are continuously fed intothe reactor and products are continuously drawn from the reactor.The flow of fluid through the reactor is orderly with no element of fluid overtaking or mixing with any other element ahead or behind. There may be lateral mixing but no mixing or diffusion along the flow path (0% of backmixing).

• In the course of this process, in any point of the reactor the composition does not change: the PFR operates in a steady mode with respect to time.

• At any instant the composition throughout the reactor varies from point to point: the PFR operates in a unsteady mode with respect to space.

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Operation mode Operation mode –– ideal PFRideal PFR

x=0[A0]

x=L[AE]

L

[A0]

[AE]

Unsteady-state operation

with respect to position

Steady-state operation

with respect to time

0t

[A]

x0

[A]

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Operation mode Operation mode –– ideal PFRideal PFRsteady mode with respect to time

)(........)()( 21 eiii tctctc ===

c0 c2c1 c3 c5c4 c6 c7

unsteady mode with respect to position

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Discontinuous and continuous reactorDiscontinuous and continuous reactor• The batch reactor (discontinuously operated reactor) is simple, needs

little supporting equipments, present high flexibility. Large conversions can be reached as the reaction time may be arbitrarily long.Disadvantages are idle periods (for loading, unloading, heating).Therefore it is ideal for small-scale experiments (kinetics studies). Industrially it is used when relatively small amounts of material are manufactured (drugs, dyes, cosmetic articles).

• Continuously operated reactors (PFR and CSTR) need reduced volume at the same production level as in the BR because the plant never runs idle.They are ideal for industrial purposes when large quantities of material are to be processed and when the rate of reaction is high. Supporting equipment needs are great; however extremely good product quality control can be obtained.

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Performance equationsPerformance equations

DefinitionThe performance equations interrelate four parameters:

1. rate of reaction r [mol/m3*s]2. conversion Χ [%]3. reactor volume V [m3] 4. Volumetric feed rate v [m3/s ].

If any one of these quantities is unknown, it can be found from the other three.

The starting point for performance equations are material and energy balances.

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Material balanceMaterial balanceIf the composition within the reactor is uniform (independent of position), the balance may be made over the whole reactor.When the composition is not uniform, it must be made over a differential element of volume and integrated across the whole reactor volume for the appropriate flow and concentration conditions.

The resultant integrated expressions are performance equations.

An equation for conservation of the masses has to be written for each component:

IN = OUT + REAC + ACC

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Energy balanceEnergy balance

In nonisothermal operations energy balances must be used in conjunction with material balances.Again, depending on circumstances, this accounting may be made either about a differential element of reactor or about the reactor as a whole.

IN = OUT + REAC + ACC

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Material Material -- energy balancesenergy balances

IN = OUT + REAC + ACC

We will limit ourselves to the easy case of isothermal operations and therefore we will use only the material balance.

Material balance

Energy balance

The material balance and the energy balance are tied together by their third terms because the heat effect is produced by the reaction as well.

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Performance equation Performance equation -- ideal BRideal BRMake the balance for the limiting component A.In a BR, since the composition is uniform throughtout at any instant of time, the balance can be made in the whole reactor.

[ ]dt

dn

dtnd

dtdn

timemolesACC AA

AAA Χ−=

Χ−== 0

0 )1()/(

( )VrtimemolesREAC A=)/(

( )Vrdt

dn A

AA +

Χ−= 00

∫∫Χ Χ

==A

A

AA

t

rVdntdt

00

0 )(∫

Χ Χ=

A

A

AA r

dct0

0 )(

t = reaction time: performance parameter for BR

Irreversible single reaction A Bnth-order, rA=kCA

n, with n>0

Area = t /cA0

General material balance: IN = OUT + REAC + ACCSimplification for BR: IN = OUT = 0

0 = REAC + ACC

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ReactionReaction time t and residence time time t and residence time ττ

• The reaction time t is the natural performance measure for BR (discontinuous reactor).

τ = V/vo = (reactor volume)/(volumetric feed rate)

If these specified conditions are those of the stream entering the reactor:

• For continuous reactors (CSTR and PFR) we should use the residence time τ = time required to process one reactor volume of feed measured at specified conditions.

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Performance equation Performance equation -- ideal PFRideal PFR

( )dVrdF AA +=0

AA dFFtimemolesOUT +=)/(( )dVrtimemolesREAC A=)/(

)1(0 AAA XFF −= ( )dVrdXF AAA +−= 00

∫=AX

A

AA r

dXcT0

0ττ

∫∫ =AX

A

AV

A rdX

FdV

00 0

v0

v∫===

AX

A

A

AAA rdX

ccFV

000

0

0

ττ ττ

Irreversible single reaction A Bnth-order, rA=kCA

n, with n>0

Area = τ /cA0

General material balance: IN = OUT + REAC + ACCSimplification for PFR: ACC = 0

IN = OUT + REACAFtimemolesIN =)/( molar feed rate (mol/s)

Make the balance for the limiting component A.In a PFR, since the composition of the fluid varies from point to point along the flow path at any instant of time, the balance must be made for a differential element of volume dV.

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Performance equation Performance equation -- ideal CSTRideal CSTRMake the balance for the limiting component A.In a CSTR, since the composition is uniform throughtout atany instant of time, the balance can be made in the whole reactor.General material balance: IN = OUT + REAC + ACCSimplification for CSTR: ACC = 0

IN = OUT + REAC0)/( AFtimemolesIN =

( )AA XFtimemolesOUT −= 1)/( 0

( )VrtimemolesREAC A=)/(

( )VrXF AAA +−= 00

A

A

AAA rX

ccFV

===00

0

0

ττ ττvv0

A

AA r

XcT 0=ττ

Irreversible single reaction nth-order,A B (-rA)=kCA

n, with n>0

Area = τ /cA0

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Graphic comparison of ideal CSTR and PFRGraphic comparison of ideal CSTR and PFR

• For any particular duty and for all positive reaction orders theCSTR is always larger than the PFR.

τ = V/v0

Irreversible single reaction nth-order,A B (-rA)=kCAn, with n>0

• An arrangement of an infinite number of CSTR in series, with total volume V, has the same performance of a PFR of volume V.

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BooksBooks

• Octave LevenspielChemical Reaction Engineering, third edition, Wiley (1999).Chapter 3-4.

• http://www.ltc1.uni-erlangen.de/htdocs/e/index.htmChapter VI.1, VII.1

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VocabularyVocabularyAccumulation AkkumulationBackmixing vollständige RückvermischungBatch reactor absatzweiser Rührkessel (AIK),

diskontinuierlich betriebener RührkesselContinuously operated stirred tank reactor kontinuierlicher Durchflußrührkessel (KIK)

kontinuierlich betriebener RührkesselreaktorConversion UmsatzDiffusion DiffusionEnergy balance EnergiebilanzIdeal reactors ideale ReaktorenMaterial balance StoffbilanzMixing VermischungModelling of real reactors Modellierung realer ReaktorenNonisothermal operations Nicht-IsothermalebetriebsweiseOperation mode BetriebsweisePlug flow reactor Strömungsrohrreaktor (IRReaction time ReaktionszeitReal reactors reale ReaktorenResidence time VerweilzeitSteady-state with position örtlich stationärSteady-state with time zeitlich stationärUnsteady-state with position örtlich instationärUnsteady-state with time zeitlich instationärVolumetric feed rate volumetrische Zuflussrate