1

§ Chapter 6 Design Equations and Reactors

<A> Batch Reactors:

Properties: no input and output streams

Characteristics: (i) Simple

(ii) High conversion (with prolonging the reaction period)

(iii) Temperature control

(iv) Un-avoided side reactions

Applications: (i) small scale production (fine &special chemicals)

(ii) Testing new process (pilot plant)

(iii) Process difficult to be continuously operated

Note that dtdN

Vr A

A1

= (general form), since NA = NA,0(1-XA) ⇒ dNA = -NA,0 dXA

Vrdt

dXNdt

dXV

Nr A

AA

AAA −=⇔−=∴ 0,

0,or ∫ ⋅−

=AX

A

AA Vr

dXNt00,

A. For incompressible fluid, constant volume

2

∫ ∫ −−=

−=∴ A A

A

X C

CA

A

A

AA

rdC

rdX

VN

t0

0,

0, AAA dCdXC =− 0,Q

(i)

(ii) If ⇒=− αAA kCr integration ie. α

AA

A kCdt

dCr =−=− [see P. 130~132 equation 4-3]

B. For V is not constant

(i) At constant T & P => V = V0(1+εXA)

Ar1

−

XA

0,/ ACt

XA

t

Ar−−

1

CA CA,0

3

∫∫ +−=

+⋅−=∴ AA X

AA

AA

X

AA

AA Xr

dXCXVr

dXNt00,0

00, )1()1( εε

If )11(0,

A

AAAAA X

XCCandkCrε

α

+−

==− Q

),,( 0, εAA XCft =∴ (by numerical integration)

(ii) At constant V & T (homework 證明之)

=> ∫ −=⇔=−

P

PA

A rdP

RTt

dtdP

RTr

0

11δδ

Remarks:

A → B + C (irreversible reaction)

If the reaction order > 0 ⇔ ∞→−Ar1

as X → 1

A ↔ B + C (reversible reaction)

If the reaction order > 0 & Xe is the equation conversion ⇒ ∞→−Ar1

As X → Xe

0 XA XA

VA ⋅− γ1

t/NA,0

4

<B> CSTRs or Backmix Reactors

Characteristics: if the concentration of products in the inlet is “0”

Since dtdNVrFF A

AAA =⋅+−0, , under steady-state

AAAAAAAA XFXFFFFVr 0,0,0,0, )1( =−−=−=−

Thus, exitA

AA

A

AA

rXF

rXF

V)(

0,0,

−⋅

=−⋅

=

t CP,0

C

CA = CA,e

CP = CP,e

CA,0

CA,0 、FA,0, 、V0

CA、XA

FA

CA,e = CA XA,e = XA

Advantages: 1. good T control ∴ quality control good

2. Easy to be analyzed

Disadvantages: low conversion, ∴ large reactor

volume is required!

Applications: 1. Large scale production

2. Homogeneous liquid phase reaction

XA= constant at t >0

5

As V = constant ⇒ CA = CA,0(1-XA) ⇒0,

0,

A

AAA C

CCX

−=

∴ )( 0,0

0,

0,0,AA

AA

AA

A

A CCr

VC

CCr

FV −

−=

−⋅

−=

⇒ A

AA

rCC

VV

−−

== 0,

0

τ (Space time) or )(0,

0,

0, AA

AA

A

A

A rCCC

rX

FV

−−

=−

=

(i)

(See ex. 2.2) (ii)

0

Ar−−

1

XA,e

V τ=

A

A

AA

rX

FV

rXF

V

=∴

−= 0,Q

Ar−−

1

CA,e CA,0

0,

0,

A

A

FCV ⋅

=τ (For the constant volume case)

6

(iii) if the reaction rate equations are known

e.g., αααγ )1(0, AAAA XkCkC −==−

A

A

AA rX

CFV

−==

0,0,

τQ (For a constant volume)

0,0,0, )1( AAA

A

A CXkCX

FV τ

αα =−

=∴

補充

kCV

VkCF

VrD

A

A

A

Aa τ==

−=

0,0

0,

0,

0, (For a 1st-order reaction)

0,0,0

20,

AA

Aa kC

CVVkC

D τ== (For a 2nd-order reaction)

Thus, for a second-order reaction in a CSTR

20,

A

AA

kCXF

V = since 0VV = ; )( 0,00, AAA CCVXF −=

∴ 20,

220,

0,2

0,

0 )1()1( AA

A

AA

AA

A

AA

XkCX

XkCCX

kCCC

VV

−=

−=

−==τ

7

)1 chosed bemust sign (minus2

41)21(

241)21(

2)2()21()21(

0,

0,0,

0,

220,0,

≤+−+

=

+−+=

−+−+=⇒

Aa

aa

A

AA

A

AAAA

XD

DD

kCkCkC

kCkCkCkC

X

Q

τττ

ττττ

See p. 142, Fig. 4-6 and Ex 4-2, p. 142 for 1st question

Remark: Da ≤0.1 %10≤⇒ AX

Da ≥10.0 %90≥⇒ AX

constant)1(

)1(0,

0,

0,

0,

0, =⋅=−

=⋅

⇒⋅−

= − kCX

XF

kCVF

VrD A

A

A

A

A

A

Aa

αα

α

τ (Damköhler number)

If V≠ constant ααα

ε)

11(0,

A

AAAA X

XkCkCr+−

==−⇒ 代入

得 αα

ε)

11(0,

0,

A

AA

A

A

XXkC

XFV

+−

=α

αα ε)1(

)1(0,

0,

A

AA

A

A

XXX

FkCV

−+⋅

=⋅⋅

∴ = constant

8

=ατ 0,AkC⋅ (Damköhler number; see p. 137~138 equation 4-8, 4-9, & Da)

<c> PFR or Tubular Flow Reactor

The characteristics of PFR are compatible to that of a batch reactor (空間換取時間)

Advantage: high conversion

Disadvantage: bad T control ⇒ require high rate of heat exchanger

Application: homogeneous gas phase reaction

dVrdxxFxFdt

dNAAA

A ⋅++−= )()( , under steady state

L

x

CA,0 , FA,0 V0 XA,0=0

CA,e , FA,e Ve , XA,e

0

CA,0

x

XA,e

CA,e

XA

9

dVrdFdt

dNAA

A ⋅=∴=⇒ 0

∫ ∫ −=⇒

⋅=−=⇒−=V X

A

A

A

AAAAAAA

eA

rdX

FdV

dVrdXFdFXFF

0 00,

0,0,

,

)1(Q

or ∫ −== eAX

A

A

AA rdX

CFV ,

00,0,

τ or ∫ −

= eA

A

X

XA

AA r

dXFV ,

0,0,

or ∫ −== eAX

A

AA r

dXCvV ,

00,0

τ

If V is constant and CA = CA,0(1-XA), 0,

0,A

AAAAA C

dCdXdXCdC−

=−=

∫∫ −−=

−−

==∴ eA

A

eA

A

C

CA

AC

CA

A

AAA rdCor

rdC

CCFV ,

0,

,

0,0,0,0,

1 ττ

10

(iii) If rate equations are known

a. krA =− (zero order) kXX

rdX

CFV AeAX

XA

A

AA

eA

A

)( 0,,

0,0,

,

0,

−=

−== ∫

τQ

)( 0,,0,

AeAA XXk

C−=∴τ

b. AA kCr =− (1st order) and control V

)11

ln(1

)1(

,

0,

0,0,0,

,

0,

,

0,

,

0,

eA

A

X

X

X

XAA

A

A

AX

XA

A

AA

XX

k

XkCdX

kCdX

rdX

CFV eA

A

eA

A

eA

A

−−

=∴

−==

−== ∫ ∫∫

τ

τQ

)1( 0,,0, ApureAA XCC −= and )1()1( ,0,0,,,,, eAAApureAeApureAeA XXXCXCC −+−=−=

(i)

Ar1

−

XA0 XA,0 XA,e

0,0, AA CFV τ

=

(ii) V = constant

-Ar1

CA,e CA,0

0,

0,

A

A

FCV ⋅

=τ

11

)1

1()(1)(0,

0,,0,,0,

0,

,0,,0,,0,

A

AeAAeAA

A

pureAAeAApureAA X

XXCXX

CC

CXXCC−−

−=

−+=−+=

eA

A

A

AeAA

A

A

eA

A

eA

A

C

CA

A

AAAA

AA

XX

kX

XXC

CkC

Ck

or

CC

kCkCdC

CCFV

CdCdX eA

A

,

0,

0,

0,,0,

0,

0,

,

0,

,

0,0,0,0,0,

11

ln1

11

ln1ln1

ln11 ,

0,

−−

=

−−

−

==

−=−

==⇒−= ∫

τ

τ代入Q

c. AA kCr =− and V is not constant

[ ])1()1(11

111

)11(

)0(11

00,

00,

00,0,

0,0,

++−−=++−=+

−+

=

+−

=−

==

=+−

=−∴

∫∫∫

εεεεεε

ε

ε

τε

AAA

A

X

A

A

A

X

A

AA

AX

A

A

AA

AA

AAA

XXX

dXXX

kCXXkC

dXr

dXCF

V

XXXkCr

AAAQ

即 [ ])1ln()1(1

0,0,AA

AA

XXkCF

V−+−−= εε k

XXor AA )1ln()1( −+−−=

εετ

12

Special case, V = constant ⇒ ε = 0

( )[ ]

kXX

kX AeAA 0,,1ln)1ln( −−−

=−−

=τ

See ex. 2.2, 2-3, & 2-4 (p.42~p.47)

Recycle Reactors (a special case of PFR) Suitable situations: (i) Reactions are auto-catalytic.

(ii) To maintain nearly isothermal operation

(iii) To promote a certain selectivity

(iv) Extremely used in biochemical operations.

Def: moles of species recycled / moles of species removed 3,

,

3,

,

3,

,

i

Ri

A

RA

t

Rt

FF

FF

FF

R ===

Conversion per pass: XA,S = moles of A reacted in a single pass / mole of A fed to the reactor

FA,0 FA,0’

FA,2

FB,2FB,1

FA,1

FB,0

FA,R FB,R FC,R

FB,3

FA,3

FC,3

FD,3

FD,R

R

XA,2XA,3 = XA,0

v2

vR

v0

v1

Overall conversion: XA,O = moles of A reacted overall / mole of fresh feed

For stream 1 and 2 ⇒ PFR; Note: XA labeled in the figure is relative to FA,0 or FA,0’

∫ −= 2,

1,'0,

A

A

X

XA

A

A rdX

FV

where FA,1 = FA,0’(1-XA,1)

Mole balance: FA,1 = FA,0 + FA,R = FA,0 + FA,2[R/(R+1)]----(A)

FA,2 = FA,0‘ [(1-XA,1) – (1-XA,2)] = FA,0‘ (XA,2 – XA,1) = (R+1) FA,3

Since FA,3 = FA,0 (1-XA,3), FA,2 = (R + 1) FA,0 (1-XA,3) (put in (A))

Thus, FA,1 = FA,0 + (R+1)[R/(R+1)] FA,0 (1-XA,3) = FA,0 (1 + R – R XA,3)

Since )1()1()1()1(

3,0,

3,0,3,0,

1,

2,1,,

AA

AAAA

A

AASA RXRF

XFRRXRFF

FFX

−+

−+−−+=

−=

3,

3,

3,

3,3,

11)1()1()1(

A

A

A

AA

RXRX

RXRRXRRXR

−+=

−++++−−+

=

Since XA,0 = XA,3 ∴ )1(1 0,

0,,

A

ASA XR

XX

−+=

Since FA,0‘ (1-XA,3) = FA,2 = (R + 1)FA,3 = (R + 1) FA,0(1-XA,3), ∴ FA,0‘ = (R + 1) FA,0

Moreover, FA,1 = FA,0’(1 - XA,1) = (R + 1) FA,0(1-XA,1) = FA,0(1 + R -R XA,3)

∴ 1 - XA,1 = (1 + R -R XA,3) / (R + 1) ⇒ 110,3,

1, +=

+=

RRX

RRX

X AAA

Thus, the design equation becomes:

∫ −= 2,

1,'0,

A

A

X

XA

A

A rdX

FV

⇒ ∫+ −

+= 0,

0,10,

)1( A

A

X

XR

RA

A

A rdXR

FV

(1)

(2) As R → 0, ⇒ PFR

V/FA,0

R 1R + 1

XA,0[R/(R+1)]XA,0

Mean -1/rAV/[(R+1)FA,0]

(3) As R → ∞ ⇒ CSTR

XA,0 [R/(R+1)]XA,0

-1/rA

V/[(R+1)FA,0] ≈ V / FA,0

V/FA,0

XA,0[R/(R+1)]XA,0

Mean (-1/rA)

< D > Packed-Bed Reactors (very similar to PFR)

∴ '0, AA

A rdw

dXF −=

∴ ∫ −= AX

AA r

dXFw00, '

Advantages: high conversion

Disadvantages: Difficult T control, Channeling of the gas flow, Catalyst replacement, etc.

Applications: Heterogeneous gas-phase reactions

Space time: 0v

V=τ ; Space velocity: V

vSV 01

==τ

CA,0, FA,0 CA,e, F2

V0, XA,0 dw Ve, XA,e

x

CA,0

CA,e

XA,0

XA,e

x

§Steps for Reactor Design (i) General mole balance equation (ii) Design Equation for the specific reactor (iii) rate law determination (iv) stoichiometry with system conditions

(v) combine (iii)+ (iv) ⇒ ( )AA Xfr =− (vi) Evaluation of the design equation (numerical or analytical) See p. 128, Fig. 4-2

§Reactors in Series Properties: (1) The exit stream of one reactor is the feed stream for another reactor.

(2) The conversion XA.i is defined as the total number of moles of A that have

reacted up to the specified “i” reactor per mole of A fed to the first reactor.

(3) There are no side streams withdraw and the feed stream enters only the first

reactor.

⇔ )(022,2,1, statesteadyVrFF AAA =+−

V1

V2

FA,0 XA,0=0

FA,1

XA,1

FA,2

XA,2 V3

FA,3

XA,3

PFR: ∫−−

=iA

iA

X

X A

AAi r

dXFV,

1,

0,

CSTR: A

iAiAAi r

XXFV

−−

= −1,,0,

3,0,0,3,

2,0,0,2,

1,0,0,1,

AAAA

AAAA

AAAA

XFFFXFFFXFFF

−=

−=

−=

2,

1,2,0,222,1,2,0,

)()(

A

AAAAAAA r

XXFVandVrXXF

−−

=−=−∴

- FA,0 / rA

XAXA,3XA.2XA,1

<A> two CSTR in series <B> three CSTRs

XA,1 XA,20

V1/ - FA,0

V2/ - FA,0

- / rA

XA,1 XA,20

V1/ - FA,0

V2/ - FA,0

- / rA

XA XA0

V1/-FA,0

V2/ - FA,0

- / rA

V3/ - FA,0

XA XA0

V1/-FA,0

V2/ - FA,0

- / rA

V3/ - FA,0

V1 V3 V2

Ar1

− - - - - - - - - - - - -

- Ar - - - - - - - - - - - -

AX - - - - - - - - - - - -

Vt = V1 + V2 + V3

<C>N CSTRs

For the nth CSTR,

in – out + generation = accumulation

nAAAnA

nAAAnA

nA

nAnAn

nnAnAnA

XFFFXFFFcensi

rFF

V

statesteadyVrFF

,0,0,,

1,0,0,1,

,

,1,

,,1,

)1(

)(0

−=

−=

−−−−−−

=⇒

=+−⇒

−−

−

−

PFRr

dXFr

dXFV

rX

FVVrXF

V

rXXF

VXXFFF

fA AX X

A

AA

iA

iAAtN

N

i iA

iAA

N

iit

nA

nAAn

nA

nAnAAnnAnAAnAnA

⇒−

=−

=⇒

−∆

==⇒−∆

=∴

−−

=⇒−=−⇒

∫ ∫

∑∑

∞→

==

−−−

,

0 00,

,

,0,

1 ,

,0,

1,

,0,

,

1,,0,1,,0,,1,

lim

)((1))( 代入

For a first-order irreversible reaction

XA0

- / rA

XA0

- / rA

XA0

- / rA

τκκτ

νν

τκ

+=

−=∴

==−−

===−

1

)(,

0,0,

00,

0

AA

A

AA

A

AAAA

CCor

CCC

ntconstavwithCSTRr

CCVCr

τκτκ+

=⇒−=1

)1(0, AAAA XXCCncesi

Hence, for two CSTRs in series, 11

0,1, 1 κτ+=⇒ A

A

CC

From a mole balance on reactor 2

)1)(1(1)(

1122

0,

22

1,2,

2,2

2,1,2

2,2

2,1,0

2,

2,1,2 κτκτκτ

τκ

ν++

=+

=⇒−

=⇒−

=−−

=⇒ AAA

A

AA

A

AA

A

AA CCC

CkCC

CCC

rFF

V

⇒ n equal-sized CSTRs in series operated at isothermal

κκκκττττ ======== nn LLLL 2121 ,

)1()1()1( 0,

0,0,, AAn

a

An

AnA XC

DCC

C −=+

=+

=⇒κτ

)44.160.()1(

11

)1(0,

0,

0,

0,

−+

−=⇒

−==−

=

FigpseeX

rxnorderstF

VCF

VrDwhere

nA

A

A

A

Aa

κτ

κτκ

)1()1(

10,,, rxnorderCCr st

nAnAnA −+

==−∴κτ

κκ (the rate of A disappearance in the nth reactor)

CSTRs in Parallel: n equal-sized CSTRs in parallel.

FA,0 FA,i XA,i

For reactor i )(,

,,0,

iA

iAiAi r

XFV

−= ; Because equal size & the same T

AnAiAAA rrrrr −=−=−==−=−⇒ ,,2,1, LL

iA

iAA

A

AA

iA

iAAAiAi r

XFrXF

Vorr

Xn

FnV

nF

FnV

V,

,0,0,

,

,0,0,,0, )(;

−=

−=

−=∴==

Remark: The conversion achieved in any one of the reactors in parallel is identical to what

would be achieved if the reactant were fed in one stream to one large reactor of volume V.

- 1 / rA

XAXA,e

- 1 / rA

XAXA,e

See p. 163, ex.4-2

<D> two PFR in series

- 1 / rA

XAXA,2XA,1

V1/ - FA,0

V2/ - FA,0

- 1 / rA

XAXA,2XA,1

V1/ - FA,0

V2/ - FA,0

- 1 / rA

XAXA,e

- 1 / rA

XAXA,e iiA

Ai

A

A

A

A

F

CV

F

VC

C

VCτ

ν===

,0,

0,

0,

0,

0,0

0,

Vt= V1 + V2 = 1 PFR

0,0,,0, AAiA

i

FV

CFV

==τ

V1/FA,0

V1/FA,0

<E> one PFR + one CSTR <F> one CSTR + one PFR

- 1 / rA

XAXA,2XA,1

V1/ - FA,0

V2/ - FA,0

- 1 / rA

XAXA,2XA,1

V1/ - FA,0

V2/ - FA,0

- 1 / rA

XAXA,2XA,1

V1/ - FA,0

V2/ - FA,0

- 1 / rA

XAXA,2XA,1

V1/ - FA,0

V2/ - FA,0

<G> CSTR of different sizes in Series: F0, ν

F1,C1ν

F2,C2 ν

V1τ1

V2τ2

V3τ3

F0, ν

F1,C1ν

F2,C2 ν

V1τ1

V2τ2

V3τ3

V1/FA,0

V2/FA,0

V1/FA,0

V2/FA,0

Constant flow volume, liquid phase ⇒ CSTR

nA

nAnA

nA

nAnA

nA

nAnAA

A

Annn

rCC

rCC

rXXC

FCVV

,

,1,

,

1,,

,

1,,0,

0,

0,

)(

)(

−

−=

−

−−=

−

−===

−−

−

ντ

)1(1

,1,

, LLLnAnA

nA

n CCr−

−=⇒

−τ

n

AA

obtaincanweiprelationshCrfrom

τ−−⇒

CA = CA,0(1-XA)

(i) if NAAA CCC ,1,0, ,, L are known ?=⇒ nV

(1) (2)

(3) from equation (1) nA

nAnAn

A

AA

A

AA

rCC

rCC

rCC

,

,1,

2,

2,1,2

1,

1,0,1 ,,

−

−=

−

−=

−

−= −τττ

ντντντ nnVVV ===⇒ ,, 2211

(ii) If CA,0 & nVV ,1 LL are known ⇒ ?? , == nAn XC

(1) (2)

(3) From equation (1)

1,0,

1,

1

11

1

AA

A

CCrV−

−=⇒=

τντ

Ar− ………………..

AC ………………..

Ar− ………………..

nAnA CC ,1, −− ………………..

Ar− ………………..

AC ………………..

CA

CA,1 CA,20

-rA

NM

L

S=1/τ2

S=1/τ1

-rA,1

CA

CA,1 CA,20

-rA

NM

L

S=1/τ2

S=1/τ1

-rA,1

CA,3 CA,2 CA,1 CA,0

⇒ From CA,0 plot a straight line with its slope = 1/τ1 ⇒ L is obtained.

(4) From -rA vs. CA plot ⇒ CA,1 is obtained !

Similarly, then nAAA

A CCMCC

rV⇒⇒⇒⇒

−−

=⇒= L2,2,1,

2,

2

22

1τν

τ is obtained!

(5) ⇒0,

,0,,

A

nAAnA C

CCX

−=

(H) Determine the minimum volume of two CSTRs in series

0,

1

AFV

0,

2

AFV

A

XA XA,2 XA

Ar1

−

0

AFV

FVX

rii

FV

FVVVVi

AAA

A

AAt

=

+−

⋅−

+⇒+=

0,

2

0,

12,

0,

2

0,

121

1)(

min)(

Max it! (yellow area)

( )( )αα

ααα

AAA

AAAA XkCrXkCkCr

−=−∴−==−

1111

0,0,

( )

−−

−=⇒

−=−

−=

AAA

AAA rrXA

XkCr

reactionorderstxe

11111

1,1.,.

2,0,

α

( ) ( )

( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( )( ) ( )

xXxX

cbbxcbxx

XkCXXkCX

XkCXXkCXXX

XXkCXXX

XkCkC

XXkCXkCdX

dA

XkCXkCXA

AA

AAAAAA

AAAAAAAA

AAAAAA

AA

AA

AAAAA

AAAAA

−=∴=−⇒

<−±

=<⇒=+−

=−+−−+−−⇒

=−+−−+−−−−⇒

=−+−−−−⇒

−+

−−

−==⇒

−−

−=⇒

11

12

400

011111

0111111

01111

111

110

11

11

22

2,0,2,0,2

2,0,2,0,2,2

2,0,2,2

20,

0,

0,2,0,

0,2,0,

(I) Comparison of PFR in Series & in parallel

∫∫∫

∫∫

−=

−

+−

=+=∴

−=

−=

2,2,

1,

1,

2,

1,

1,

00,00,211,

0,200,1 ,:)(

AA

A

A

A

A

A

X

A

AA

X

XA

AX

A

AAt

X

XA

AA

X

A

AA

rdXF

rdX

rdXFVVV

rdXFV

rdXFVseriesIna

PFRsforVV

VVVVVVVVV

Vr

dXFVV

V

VVVVV

Vr

dXFVV

Vr

dXFVVVb

parallelseries

tt

X

A

AA

X

A

AA

X

A

AAt

A

AA

=∴

=+=⇒=++

=−+

⇒

=++

=−+

⇒−

=+=

∫

∫∫

1,212,22121

200,

21

2

12121

100,

21

100,211,

)(

)()(

2,

2,2,

XA,2 V1 V2

XA,1

XA=0

FA,0

V1

V2

FA,0

XA=0

0.21

1AF

VVV+

0.21

2AF

VVV+

XA,2

(a)

(b)