most of reactions are not carried out isothermally - vscht.czbernauem/ark/lectures/chapter 6.pdf ·...
TRANSCRIPT
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6. Energy balance on chemical reactors
Most of reactions are not carried out isothermally
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BATCH or CSTR heated (cooled) reactors
Inlet
tubing
Mixer
Coolant
inlet
Coolant
outlet
Reaction
mixture
Outlet
tubing
Double
shell
Internal
heat
exchanger
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The balance of total energy involves: Internal energy mechanical energy (kinetic energy) potential energy .... Transformation of various kinds of energy Balance of total energy Main reason to study energy balances : assesment of temperature of reacting system (reactor)
Rate of change
of total energy
Input x Output
of total energy
by convective
flux
Input x Output of total energy
by molecular flux
Work done
by external
forces
Work done by molecular interactions
R.B.Bird, W.E.Stewart, E.N.Lightfoot : Transport Phenomena, 2nd Edition, J.Wiley&Sons, N.Y. 2007
kin pE U E E
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R
V
V
E e dV
Application of the 1st law of thermodynamics on the open homogeneous reacting
system
Heat flux [W] Rate of work done on surroundings [W]
eo,e1 – specific total energy of inlet (outlet) streams [J/mol]
VR – volume of reaction mixture [m3]
Single phase
reacting
system
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Rate of work done by the reacting system on the
surroundings consists of:
• Flow work of inlet stream(s)
• Flow work of outlet stream(s)
• Work provided by stirrer
• Work done by volume change
• Work done by electric, magnetic fields
1 1o o
dEF e Fe Q W
dt
W
1 1 1 1 1
o o o mo o
m
s
R
f
V P F V P
V P FV P
W
dVP
dt
W
1 1 1 1R
o o mo o m s f
dVdEF e V P F e V P Q P W W
dt dt
1, molar energies of inlet and outlet streams [J/mol]oe e
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Neglecting potential and kinetic energies ( ), we have
1 1R
o mo m s f
dVdUF h Fh Q P W W
dt dt
If 0, 0s fW W
1 1R
o mo m
dVdUF h Fh Q P
dt dt
From enthalpy definition
RR
dVdH dU dPV P
dt dt dt dt
1 1R o mo m
dH dPV F h Fh Q
dt dt
E U
1, molar enthalpies of inlet and outlet streams [J/mol]mo mh h
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Introducing partial molar enthalpies of species
We have finally
1
1 1
1
No o
o mo i i
i
N
m i i
i
F h F H
F h F H
1 1
N No o
R i i i i
i i
dH dPV F H F H Q
dt dt
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1, , , ,
1 1, ,
j j j i
j j
N
i
iP n T n i T P n
N N
P i i m R P i i
i iT n T n
H H HdH dT dP dn
T P n
H HC dT dP H dn V c dT dP H dn
P P
BATCH reactor
R
dH dPV Q
dt dt
Enthalpy is a function of temperature, pressure and composition
Total heat capacity [J/K] Partial molar enthalpy [J/mol]
3Molar density [mol/m ] Molar heat capacity [J/mol/K]
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We know that (from thermodynamics)
, ,
1
j j
RR R p
T n P n
VHV T V T
P T
,
1
j
Rp
P nR
V
V T
where the coefficient of isobaric expansion is defined as
1
1N
im P R R p i
i
dndH dT dPc V V T H
dt dt dt dt
and we obtain
Finally by substitution of in the energy balance of the batch reactor
1
Ni
m P R p R i
i
dndT dPc V V T H Q
dt dt dt
,
1
NRi
R ki V k
k
dnV r
dt
idn
dt
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Using definition of the enthalpy of k-th reaction
we have
1
N
r k ki i
i
H H
,
1
NR
m P R p R R r k V k
k
dT dPc V V T V H r Q
dt dt
Isobaric reactor ( ) 0dP
dt
,
1
( )
NR
m P R R r k V k
R
k
dTc V V H r Q
dt
V f t
we need state equation !
Homework 8: Energy balance of ideal gas isobaric batch reactor
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Isochoric reactor ( ) 0RdV
dt
Homework 9: Energy balance of ideal gas isochoric batch reactor
,
1
( )
NRp
m R V R r k k V k
k T
dTV c V H T V
d
P f
Qt
t
r
,
the coefficient of isobaric expansion the coefficient of isothermal compressibility
1 1
the volume variation due to k-th
j
R Rp T
P n TR R
V V
V T V P
1 , ,
, ,
chemical reaction
, where is the partial molar volume of species i
the specific heat capacity at
j i
j j
N
k ki i i
i i T p n
RV P P R
V n P n
VV V V
n
VP PC C T C TV
T T T
constant volume
2
, j
p
p P R
V n T
C TV
Variation of the pressure can be derived from total differential of volume
1,
1
,
1j
j
NiR
iNiP n p i
i
iR T R T
T n
dnV dTV
T dt dt dndP dTV
Vdt dt V dt
P
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Summary of energy balance of BATCH reactor
0dP
dt ,
1
( )
NR
m P R R r k V k
R
k
dTc V V H r Q
dt
V f t
0RdV
dt
,
1
( )
NRp
m R V R r k k V k
k T
dTV c V H T V
d
P f
Qt
t
r
If liquid (condensed) systems
Rate of change Rate of heat generation Rate of heat
of reaction mixture by chemical reactions loss (input)
enthalpy
Heat flux :
0,p p Vc c ,
1
NR
m P R R r k V k
k
dTc V V H r Q
dt
2 1
2
the overall (global) heat transfer coefficient [W.m .K ]
the heat exchange area [m ]
temperature of external co
(
oling (heating) f
)
luid
H
H
e
eQ S T T
S
T
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Isothermal
reactor
Adiabatic
reactor
Limiting cases
,
1
0NR
m P R R r k V k
k
dTc V V H r Q
dt
,
1
0NR
m p R R r k V k
k
dTQ c V V H r
dt
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Example Adiabatic reactor with 1 reaction, constant heat capacities
Energy balance on adiabatic BATCH reactor
Molar balance of key component
m p R R r V
j jo
j j V
dTc V V H r
dt
dc dXc r
dt dt
1
1 1 1
1 1 1
,
Assuming that
( )
N
m R p i pi
i
N N No oi
m R p i pi i pi i j j pi
i i i j
o oN N Nj j j jo o
i pi i pi i pi p
i i ij j
pi p
o
j r o j
o
o o
j i pi j p
V n c y c
V c n y c n c n n X c
n X n Xn c c n c c
c const c const
y H T XT T
v y c y c X
1
1
( ) the adiabatice rise of temperature
o j jN
j
i
o
j r o
j No o
j i pi j p j
i
T X
y H T
v y c y c X
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Trajectories of T(t) and Xj(t)
To
T(t)
Xj(t)
T(t)
Xj(t)
t
Xj = 1
1
( )
o
j r o
j No o
j i pi j p j
i
y H T
v y c y c X
Exothermal reaction
0r H
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Homework 10
The reversible reaction
A1 + A2 A3
is carried out adiabatically in a constant-volume BATCH reactor. The
kinetic equation is
Initial conditions and thermodynamic data
Calculate X1(t),T(t).
1/2 1/2
1 2 3
3 1
1
5 1
2
(373 K)=2x10 s E 100 kJ/mol
(373 K)=3x10 s E 150 kJ/mol
f b
f
b
r k c c k c
k
k
3
1 1
3
2 2
298 3
0.1 mol/dm 25 J/mol/K
0.125 mol/dm 25 J/mol/K
40 kJ/mol 40 J/mol/K = 373K
o
p
o
p
o o
r p
c c
c c
H c T
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Example
Acetic anhydride reacts with water
(CH3CO)2O + H2O 2CH3COOH
in a BATCH reactor of constant volume of 100 l. Kinetics of
reaction is given by
Data
In neglecting variation of heat capacities with temperature,
calculate T(t) and X1(t) for an non-adiabatic and adiabatic case.
-1 -1 -1 1 3
1
1
0.3 mol.l , 3.8 kJ.kg .K , 209kJ.mol , =1070 kg.m
.S =200W.K 300 K, 0, ,
initial mass fractions, - mass heat capacities (kJ.
o
pM r
pio o
H e p pM i pMi pMi
i i
o
i pMi
c c H
cT T c c w c c
M
w c
-1 -1 1
1 1
kg .K ), - molar weight (kg.mol )
8.31446 J.mol .
iM
R K
46500
7 3 1
12.14 10 mol.m .minRTVr e c
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0.00
0.20
0.40
0.60
0.80
1.00
1.20
304
306
308
310
312
314
316
0 10 20 30 40 50 60
t [min]
T [K] X1
By numerical integration we get
1
11
1 465000.1 209E3 2.14E7xexp - 300 (1 ) 200 (300-T)
(1070 3.8E3 0.1) (8.31446 )
465002.14E7 exp - (1 )
(8.31446 )
dTX
dt T
dXX
dt T
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Continuous (perfectly) stirred reactor (CSTR)
1 1
N No o
R i i i i
i i
dH dPV F H F H Q
dt dt
1
1 1
1N
im P R R p i
i
N No o
R i i i i
i i
dndH dT dPc V V T H
dt dt dt dt
dPV F H F H Q
dt
,
1
, 1,NR
oii i R ki V k
k
dnF F V r i N
dt
Energy balance on CSTR
Total differential of enthalpy
Molar balance on CSTR
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,
1 1
1 1
N NR
o
m P R p R i i i R ki V k
i k
N No o
i i i i
i i
dT dPc V TV H F F V r
dt dt
F H F H Q
We get
,
1
1
NR
m P R R p R r k V k
k
No o
i i i
i
dT dPc V V T V H r
dt dt
F H H Q
1
N
r k ki i
i
H H
Enthalpy
change for
k-th reaction
,
1 1
+NR N
o o
m P R R r k V k i i i
k i
dTc V V H r F H H Q
dt
The CSTR usually works at constant pressure (no pressure drop)
0dP
dt
( )T mQ S T T
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Steady state
0idndT
dt dt
,
1 1
+ 0NR N
o o
R r k V k i i i
k i
V H r F H H Q
1
o
o
T
o o
i i i i pi
T
T
o
r k r k p kT
N
p ki piki
H H H H c dT
H H c dT
c c
Assuming ideal mixture, i.e. , we have i iH H
,
1 1
,
1
0
0, 1,
o o
T TNR No o
R r k p V k i pikk iT T
NRo
i i R ki V k
k
V H c dT r F c dT Q
F F V r i N
Molar and enthalpy
balances give
N+1 unknown
variables T, Fi
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N+1 unknown variables in N+1 non linear algebraic equations
1 2, , ,.... 0, 1, 1i NG T F F F i N
Issues:
• multiple solutions
• slow convergence (divergence)
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Example Adiabatic CSTR with 1 reaction, constant heat capacities
1
1
1
1
( ) ( )
0
( )
N No o o o o o o
R r p V i pi i pi
i i
o
j j R j V
o Nj jo o o o
r p i pi
o
j j
V
R j
o o
r j jo
No o
j
ij
i pi p j j
i
o
j j
F
V H c T T r F c T T F T T y c
F X V r
y XH c
Xr
V
H y XT T
y c c y X
T X
T T T T y c
cf. adiabatic const.-volume BATCH
2 nonlinear algebraic equations for
unknown T and Xj
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0
0.2
0.4
0.6
0.8
1
310 330 350 370 390
Xj
T (K)
f2(T)
f1(T)
Steady state 1
Steady state 2
Steady state 3
Multiple steady states of adiabatic CSTR (exothermal reaction)
1 + ( , ) =0 (T) o
j j j V j R jF X r X T V X f
2 ( )o j j jT T X X f T
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11 10.001exp[ ( )] min
298
10 kcal/mol
V CO
a
a
r kc
Ek
R T
E
Example
You are to consider an irreversible gas-phase reaction in an adiabatic CSTR at constant pressure (101 kPa).
The gas phase reaction is:
CO(g) + 3 H2(g) CH4(g) + H2O(g)
The feed to the CSTR consists of CO and H2 at the following (stoichiometric) concentrations:
CCO(in) = 0.0102 mol/liter CH2(in) = 0.0306 mol/liter
The heat of reaction at 298 K is equal to –49.0 kcal/mol.
The heat capacities of CO, H2, CH4 and H2O are all constant and are equal to 7 cal/mol/K.
The temperature of the feed stream is equal to 298 K, the pressure is equal to 101 kPa, the volumetric flow
rate of feed is 8 liter/min and volume of reactor is 0.5 l. The gas mixture behaves as ideal gas.
A. Use the energy and molar balance to calculate the CO conversion and temperature of the effluent stream
from the adiabatic CSTR.
B. Calculate the composition in molar fraction of the outlet stream.
C. Calculate the volume of the adiabatic CSTR required to achieve the desired CO fractional conversion equal
to 0.99.
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 500 1000 1500 2000 2500 3000 3500 4000 4500
X1(T)
T/K
1
11
1 1
Adiabatic enthalpy balance
( )
( )
No o
i pi
i
o o o o
r p
y c T T
XH y c y T T
1 1
1 1
2
1
Molar balance of CO
(2 )( )( )
2 1
(2 ) 4
2
R
o
V P X Xk Tg T
RT F X
g gX
3 steady states
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Remarks:
Unrealistic temperature of the 3rd steady state backward
reaction will occur
The 2nd steady state is unstable carefull temperature control
has to be used
The dynamic behavior of reactor should be studied
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Homework 11 Determine steady states of adiabatic CSTR in which the exothermal liquid state reaction takes place
A1 A2 Reaction rate:
rV = A.exp(-E/RT).cA1 (kmol/m3/s) Data
A = 5.1017 s-1
E = 132.3 kJ/mol
VR = 2 m3
To = 310 K
= 800 kg.m-3
3.33 l/soV
3
1 2 /o
Ac kmol m
,298 100 /rH kJ mol
-1 -14.19 kJ.kg . Ko
pC
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z + z
VR + VR
z
VR
1R
N
i i
i V
F H
1
R R
N
i i
i V V
F H
v
R RV S z
Balance of enthalpy on Plug Flow Reactor (PFR)
1
1
1
1 1
in the steady state
0=
R R R R
R R R
N N
R i i i i
i iV V V V
N N
i i i i
i i
N
i i
iRV V RV
H dPV F H F H Q
t dt
dF H F H Q
dQF H
dV dV
RV
H
t
Q
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,
1 1 1
,
1 1
,
1 1
ideal mixture ( , , ) ( )
( ) ( )( ) ( )
i i
N N NRi i i
i i ki V k i i
i i kR R R
NR N
r k V k i pi
k iR R
NR Ni
ki V k ki i
k iR
H T P composition h T
dF dh T dh T dTh T F r h T F
dV dV dT dV
dT dQH r Fc
dV dV
dFr h
dV
, 21
1
2
,
1
1
( )
1 4( )
4
4( )
4
tr k e
R R
NRt R
r k V k eNk RR R R
i pi
i
NRR
r k V k eNk R
i pi
i
dSdQH T T
dV dV
dS d dzdTH r T T
ddV d dVFc dz
ddTH r T T
dz dFc
Circular cross
section
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2
2 2
4( )
4
. .( , ) ( , )
4 4
0, , 0
Rr V eo o
pM R
jj jR RV j V jo o
j j
o j
ddTH r T T
dz F c d
dX d dr X T r X T
dz F F
z T T X
One reaction, constant heat capacity of reaction mixture. Profiles of conversion and temperature are given by following
equations:
Limiting cases 1.Isothermal reactor
2.Adiabatic reactor
4 4
0 ( ) ( )r V e r V e
R
o
R
dTH r T T H r T T
dz
T T
d d
2 (
( )
)( )
4
o
j r
o jo
j p
o
j r jRr Vo o
M
o opM j pM
F H dXddTH r
dz F c dzF
T T
c
y HX
c
, ,
1 1
,
1
N N
i p i i p i
i i
No o o o
i p i pM
i
Fc Fy c
F y c F c
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2
,
1
2 2
4( )
4
. .( , ) ( , )
4 4
0, , 0
Rr V e
oNRjo o
i p i p j
i j
jj jR RV j V jo o
j j
o j
ddTH r T T
dz dyF y c c X
dX d dr X T r X T
dz F F
z T T X
One reaction, constant heat capacity of species. Profiles of conversion and temperature are given by following
equations:
Limiting cases 1.Isothermal reactor
4 4
0 ( ) ( )r V e r V e
R
o
R
dTH r T T H r T T
dz
T T
d d
, ,
1 1
,
1
ji ii V
R j R
o
j j
j o
j
o o oi ii i j j j j
j j
N No oi
i p i p i i j j
i i j
oNjo o
i p i p j
i j
dFdFr
dV dV
F FX
F
F F F F F X
Fc c F F X
yF y c c X
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2
, ,
1 1
,
1
,
1
( )( )
4
( )
( )o
o
j r jRr V
o oN Nj jo o o o
i p i p j j i p i p j
i ij j
o o o
j r p j
No o
j i p i j p j
i
Tj
No o ooj r pT
j i p i j
i
F H dXddTH r
dz dzy yF y c c X F y c c X
y H c T T dX
dzy c y c X
dXdT
y H c T Ty c y
,
1
0
( )
j
o o
j r jo
No o
j i
X
o
p
p
j
i j p j
i
y H XT T
y c y c X
c X
2.Adiabatic reactor
Cf. adiabatic BATCH and CSTR
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0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
100
200
300
400
500
600
700
0.0 0.2 0.4 0.6 0.8 1.0
XSO2 [-] T [oC]
V [m3]
Conversion of SO2
Temperature
„hot“ point
Exercise: reactor for oxidation of SO2 to SO3
Numerical method: • Euler method „Stiff“ solvers MATLAB www.netlib.org www.athenavisual.com http://wxmaxima.sourceforge.net/wiki/index.php/Main_Page
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Balance of mechanical energy in PFR Profile of overall pressure (P(z))
z=0
VR=0
z
VR
dR
(0)P ( )P z
2
2
(0) ( )
f R
f
R
P P z zv
d
dPv
dz d
Bernoulli equation
density of fluid(kg/m3) friction coefficient(-)
fluid mean velocity
f
(Re, / )w Rf d
v
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Catalytic PFR Profile of pressure is calculated using Ergun equation:
22 2
1 22 3 3
1 1150 1.75
f fb o o o obf f f f f f
p b p b
dPv v A v A v
dz d d
f - fluid dynamic viscosity (Pa.s)
f - fluid density (kg/m3)
o
fv - superficial fluid mean velocity (m/s)
b - bed porosity (-)
pd - catalyst particle diameter (m)
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2
2 2
2
1 2
4( )
4
. .( , ) ( , )
4 4
0, , 0,
Rr V eo o
pM R
jj jR RV j V jo o
j j
o o
f f f f
o j o
ddTH r T T
dz F c d
dX d dr X T r X T
dz F F
dPA v A v
dz
z T T X P P
PFR model for one reaction with constant heat capacity of reaction mixture
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Example
A gas phase reaction between butadiene and ethylene is conducted in a PFR, producing cyclohexene:
C4H6(g) + C2H4(g) C6H10(g)
A1 + A2 A3
The feed contains equimolar amounts of each reactant at 525 oC and the total pressure of 101 kPa.
The enthalpy of reaction at inlet temperature is -115 kJ/mol and reaction is second-order:
Assuming the process is adiabatic and isobaric, determine the volume of reactor and the residence time
for 25 % conversion of butadiene.
Data:
Mean heat capacities of components are as follows (supposing that heat capacities are constant
in given range of temperature)
cp1 = 150 J.mol-1.K-1, cp2 = 80 J.mol-1.K-1, cp3 = 250 J.mol-1.K-1
1 2
4 1 3 1
( )
115148.9( ) 3.2 10 exp (mol )
Vr k T c c
k T m sRT
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Project 15
A gas phase reaction between butadiene and ethylene is conducted in a PFR, producing cyclohexene:
C4H6(g) + C2H4(g) C6H10(g)
A1 + A2 A3
The feed contains equimolar amounts of each reactant at 525 oC and the total pressure of 101 kPa.
The enthalpy of reaction at inlet temperature is -115 kJ/mol and reaction is second-order:
1. Calculate temperature and conversion profiles in adiabatic PFR.
2. Assuming the process is adiabatic and isobaric, determine the volume of reactor and the residence time
for 25 % conversion of butadiene.
Data:
Heat capacities of components will be taken from open resources [1,2]
1. http://webbook.nist.gov/chemistry/
2. B. E. Poling, J.M.Prausnitz, J.P.O’Connell, The Properties of Gases and Liquids, Fifth Edition,
McGraw-Hill, N.Y. 2001.
1 2
4 1 3 1
( )
115148.9( ) 3.2 10 exp (mol )
Vr k T c c
k T m sRT