§ chapter 6 design equations and reactors batch reactorsjcjeng/topic 6_n.pdf · § chapter 6...
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§ 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)