Chemical Reaction Engineering Lecture-4 Module 1: Mole Balances, Conversion & Reactor Sizing (Chapters 1 and 2, Fogler)
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Chemical Reaction Engineering
Lecture-4
Module 1: Mole Balances, Conversion & Reactor Sizing (Chapters 1 and 2, Fogler)
Topics to be covered in today’s lecture
• Conversion (X)
• GMBE in terms of conversion (X) for the following reactors – Batch Reactor
– Continuous Stirred Tank Reactor
– Plug Flow Reactor
• Compare Volume of CSTR and PFR
• Introduction to Levenspiel Plots
Chapter 2. Conversion and Reactor Sizing
Conversion (X)
• Conversion: quantification of how a reaction has progressed
fedAspeciesofMolesreactedAspeciesofMolesX A ""
""= (A: limiting reactant)
)1(0 XNN AA −⋅=
Batch Reactors
AO
AAO
NNNX −
=
Continuous (or Flow) Reactors
0
0
A
AA
FFFX −
=
)1(0 XFF AA −⋅=
can be omitted
• Maximum conversion for irreversible reactions: X = 1.0 • Maximum conversion for reversible reactions: X = Xe
? 0
A
AA
CCCXWhen −
=
aA + bB → cC + dD ; A + b/a B → cC + dD Limiting Reactant
Batch Reactor Design Equation
Vrdt
dNA
A ⋅=0
0
A
AA
NNNX −
= )1(000 XNXNNN AAAA −=−=
VrdtdXN AA ⋅−=0
For a constant-volume batch reactor VrdtdXN AA ⋅−=0 A
A rdt
dC−=
Therefore, a batch reactor has been widely used to investigate the rate law equation.
VrdtdXN AA ⋅−=0 ∫∫ =
⋅−=
X
AA
X
AA XCf
dXCVr
dXNt0 0
00
0 ),(
X
VrN
A
A
⋅−0
0 t
constant-volume reactor
NA = NA0 X
Design Equation in Differential Form
Design Equation in Integral Form
Design Equation for Flow Reactors
X = f (t) for Batch Reactor
X = f (V) for Flow Reactor FA = FA0 (1 – X) [moles/time]
FA = CA0 0υ
CA0 : morality for Liquid System CA0 = PA0/RT0 = yA0P0/RT0 for Gas System
),()()( 0
000
XCfXF
rXF
rXFV
A
A
exitA
A
A
A =−
=−
=
CSTR Design Equation
)(0
A
AA
rFFV
−−
=0
0
A
AA
FFFX −
= )1(000 XFXFFF AAAA −=−=
000 AA CF υ= υυ =0For incompressible fluid
)(0
A
A
rF−
X
CA0
CA
FA0
υ0 CA
FA
υ
X0
X
Area = Reactor volume
Design Equation for CSTR
∫∫ =−
=X
AA
X
AA XCf
dXFr
dXFV0 0
00
0 ),(AA r
dVdXF −=0
PFR Design Equation
AA r
dVdF
=0
0
A
AA
FFFX −
= )1(000 XFXFFF AAAA −=−=
000 AA CF υ= υυ =0For incompressible fluid
)(0
A
A
rF−
X
Area = Reactor volume
PFR
CA0
FA0
υ0
X0
CA
FA
υ
X
No radial gradients
Design Equation for PFR
∫ −=X
AA r
dXFW0
0 'AA r
dWdXF '0 −=
PBR Design Equation
000 AA CF υ= υυ =0For incompressible fluid
)'(0
A
A
rF−
X
Area = Catalysts weight
No radial gradients
Design Equation for PBR
Design Equation in terms of Conversion
REACTOR DIFFERENETIAL ALGEBRAIC INTEGRAL FORM FORM FORM
VrdtdXN AAO )(−= ∫ −=
X
AAO Vr
dXNt0
)( AAO rdVdXF −= ∫ −=
X
AAO r
dXFV0
CSTR
PFR
ExitA
AO
rXFV
)()(
−=
BATCH
Batch and Levenspiel Plots
][])(
[ 0 Xr
FVA
ACSTR ×
−=
Continuous Stirred Tank Reactor (CSTR)
)(0
A
A
rF−
X
∫=
= −=
xx
x A
APFR dX
rFV
0
0
X
)(0
A
A
rF−
Plug Flow Reactor (PFR)
Isothermal system
∫=
= ⋅−=
xx
x AABatch Vr
dXNt0
0 )(X
VrN
A
A
⋅− )(0
Batch Reactor
Class Problem # 1
The following reaction is to be carried out isothermally in a continuous flow reactor: A → B Compare the volumes of CSTR and PFR that are necessary to consume 90% of A (i.e. CA=0.1 CAO). The entering molar and volumetric flow rates are 5 mol/h and 10 L/h, respectively. The reaction rate for the reaction follows a first-order rate law:
(-rA) = kCA where, k=0.0001 s-1
[Assume the volumetric flow rate is constant.]
Solution to Class Problem #1
0
2
4
6
8
10
12
14
16
18
20
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Conversion (X)
F A0/(
-r A)
• Calculating Reaction rate in a CSTR Pure gases reactant A(CAo=100 millimol/liter) is fed at steady rate into a
mixed reactor (V=0.1 liter) where it dimerizes (2A→R). For different gas feed rates the following data are obtained
Find the expansion factor, conversion of each run
rate of equation for this reaction - rA = k CA
n
Run number 1 2 3 4
liter/hr 30.0 9.0 3.6 1.5
CA, out, millimil/liter 85.7 66.7 50 33.3 0υ
Class Problem # 2
Steady state Mixed Flow Reactor
Input =output + disappearance by reaction + accumulation
0
FA0= FA0(1-XA) + (-rA)V
Replacing in equation (1)
(1)
FA0XA= (-rA)V (2)
(3)
SOLUTION ( Ref. Chemical Rxn Eng, O. Levenspiel pp104-106 ) For this stoichiometry, 2A → R,
The expansion factor is
and the corresponding relation between concentration and conversion is
or
The conversion for each run is then calculated and tabulated in Column 4 of Table E1.
V=V0(1+εAXA) for linear expansion (4)
(5)
(6)
Given Calculated
Run CA,out XA
1 30.0 85.7 0.25
2 9.0 66.7 0.50 4500
3 3.6 50 0.667 2400
4 1.5 33.3 0.80 1200
TABLE E1
From the performance equation, Eq. (3), the rate of reaction for each run is given by
(7)
These values are tabulated in Column 5 of Table E1. Having paired values of rA and CA (see Table E1) we are ready to test various kinetic expressions.
Instead of separately testing for first order (plot rA versus CA), second order (plot rA versus CA
2), etc., let us test directly for nth-order kinetics. For this take logarithms of –rA = kCAn,
giving
This shows that nth-order kinetics will give a straight line on a log(-rA) versus logCA plot. As shown in Fig. E1, the four actual data are reasonably represented by a straight line of slope 2, so the rate equation for this dimerization is
(8)
(9)
FIGURE E1.
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