Ref: Seider et al, Product and process design principles, 3rd ed., Wiley, 2010.
1
Reactor Models
 Chemical reactors, particularly for continuous processes, are
contains: multiple phases, different geometries, and various
regimes of momentum, heat, and mass transfer. There are so
many configurations, involving different combinations of these
attributes, that attempts to develop generalized reactor models
have met with limited success.
 Most of the process simulators provide four kinds of reactor
models. These ideal models are used in the early stages of
process synthesis, when the details of the reactor designs are less
important, but reactor effluents and heat duties are needed.
2
Reactor Models
The common ideal reactor models are:
Stoichiometric or conversion model that permits the
specification of reactant conversions or extents of reaction for
one or more specified reactions.
2. Two models for multiple phases (vapor, liquid, and solid) in
chemical equilibrium, one for known reactions and the other
one for unknown reactions (Gibbs model).
3. A kinetic model for a continuous-stirred-tank reactor (CSTR)
that assumes perfect mixing of homogeneous phases (liquid or
vapor).
4. A kinetic model for a plug-flow tubular reactor (PFR), for
homogeneous phases (liquid or vapor) and assuming no
backmixing (dispersion).
1.
3
Reaction Stoichiometry
For most of the reactor models in the flowsheet simulators, it is
necessary to provide R chemical reactions involving C chemical
species:
C

ij
A j  0 , i  1 , , R
j 1
where Aj is the chemical formula for species j and νij is the
stoichiometric coefficient for species j in reaction i (negative for
reactants, positive for products). As an example, for the
manufacture of methanol, the reaction can be written as -2H2 - CO
+CH3OH = 0, and the stoichiometric coefficient matrix is:
 ij
 2 H 2


 1
CO


 1  CH 3 OH
4
Extent of Reaction
Consider a single reaction. In the stoichiometric reactor models,
one can specify the fractional conversion, Xk , of key reactant k :
Xk 
n k , in  n k , out
,
n k , in
0  Xk 1
or the extent (number of moles extent) of reaction:
 
n j , out  n j , in

, j  1 , , C
j
Based on the above equations, the molar flow rates of the
components in the reactor effluent can be calculated by:
n j , out  n j , in
 j
 n k , in X k 
 k

 , j  1 , , C


OR
n j , out  n j , in  
j
, j  1 , , C
5
Extent of Reaction
For multiple reactions, the reactions must be specified as series or
parallel.
The series reactions are equivalent to having reactors in series with
the feed to each reactor, except the first, being the product from the
previous reactor. Each reaction can have a different key reactant.
For parallel reactions, it is preferable to specify the extent of
reaction for each reaction, which results in:
R
n j , out  n j , in 

i
ij
, j  1 , , C
i 1
6
Chemical Equilibrium
A chemical reaction can be written as a general stoichiometric
equation, in terms of reactants A, B, etc., and products R, S, etc.,
aA  bB    rR  sS  
A reaction is characterized by two important thermodynamic
quantities, namely the heat of reaction and the Gibbs (free) energy
of reaction. Furthermore, these two quantities are functions of
temperature and pressure.
For many reactions, the effect of temperature on the heat of
reaction is relatively small. By contrast, the effect of temperature
on the Gibbs energy of reaction can be very large.
7
Chemical Equilibrium
For example, the reaction of CO and H2 to form gaseous methanol,
CO  2H
2
 CH 3 OH
o
has an standard heat of reaction,  H rxn
, at 25°C of -90,400 kJ/kmol
of methanol, while at 800°C, the heat of reaction, is -103,800
kJ/kmol, a relatively small change for such a large change in
temperature.
For the same methanol formation reaction, the standard Gibbs
o
energy of reaction,  G rxn
, is -25,200 kJ/kmol at 25°C. Already, at
500°C, the Gibbs energy of reaction, has undergone a drastic
change to +88,000 kJ/kmol.
8
Chemical Equilibrium
For a specified feed composition and final temperature and
pressure, the product composition at chemical equilibrium can be
computed by either of two methods:
1. Chemical equilibrium constants (K-values) computed from the
Gibbs energy of reaction combined with material balance
equations for a set of independent reactions. This method is
applicable when the stoichiometry can be specified for all
reactions being considered.
2. The minimization of the Gibbs energy of the reacting system.
This method requires only a list of the possible products.
9
Chemical Equilibrium
For the first method, a chemical equilibrium constant, K, is
computed for each independent stoichiometric reaction in the set,
using the equation,
r
s
o
  G rxn
K  a b
 exp  

RT
a AaB 

aRaS 




where ai, is the component activity.
For a gas solution, the activity is given by a i   i y i P   i Pi , where  i , is the
fugacity coefficient of component i in the gas mixture. It is common to replace
the activity in the equation for K with the above equation to give:
r
s
r
s
yR yS   R S 

K  a b
a
b
y A y B    A  B 
where
K   1 .0
r
s
r
s
 r  s  a b 
yR yS 
P
P

r  s   a b 
R
S
P
 a b
K P
 a b
K

yA yB 
P A PB 

for low to moderate pressures.
10
Chemical Equilibrium
For a liquid solution, the activity is given by
V
s 
a i  x i  i exp  i ( P  Pi ) 
 RT

where γi is the activity coefficient of component i in the liquid mixture and is
equal to 1.0 for an ideal liquid solution, V i is the partial molar volume of
s
component i, and Pi is the vapor pressure of component i.
It is common to replace the activity in the equation for K to give:
r
s
r
s
r
s
xR xS    R S  
xR xS 


K  a b
K
a
b
a
b


x A x B    A B  
xA xB 
where K 
 1 .0
for ideal liquid solutions at low to moderate pressures.
11
Chemical Equilibrium
Example 1: Steam-Methane Reforming
Example 1: Calculate the effluent component mole flow rates of a steammethane reforming reactor at 1050 K and 5 bar, if the feed contains 125 kmol
H2O/hr and 50 kmol CH4/hr.
Solution: The main two equilibrium reactions of the SMR are:
 H 2 O  CO  3H
Reaction
(1) : CH
Reaction
(2) : CO  H 2 O  CO
4
2
 H2
2
,
K 1 ( at 1050 K)  94 . 2 atm
,
K 2 ( at 1050 K)  1.2
2
At equilibrium, one can write (with ideal gas assumption):
3
K1 
y CO y H 2
y CH 4 y H 2 O
P
2
,
K2 
y CO 2 y H 2
y CO y H 2 O
The mole fraction of each component in the product (at equilibrium condition)
can be related to the extent of reactions (ξi), as follows:
n H 2 O , out  125   1   2 , n CH 4 , out  50   1 , n CO , out   1   2
n CO 2 , out   2 , n H 2 , out  3 1   2 , n total , out  175  2  1
12
Chemical Equilibrium
Example 1: Steam-Methane Reforming
After substitution the above equations in the Ki equations, the following set of
nonlinear equations will result:
 25 ( 1   2 ) ( 3 1   2 ) 3  94 . 2 ( 50   1 )( 125   1   2 ) (175  2  1 ) 2  0

 ( 2 ) ( 3 1   2 )  1 . 2 ( 1   2 ) (125   1   2 )  0
By solving the above set of equations (for example
),
the extant of each reaction and component mole flow rates are resulted as:
 1  44 . 42

 2  15 . 33

 n H O , out  65 . 25
2

 n CH 4 , out  5 . 58

 n CO , out  29 . 09

n
 15 . 33
 CO 2 , out
n
 H 2 , out  148 . 60
kmol/hr
"
"
"
"
13
Kinetics
Power Law
For homogeneous non-catalytic reactions, power-law expressions
are commonly used for regression of laboratory kinetic data. The
general power-law kinetic equation for key component (the
component that has the stoichiometric coefficient of +1) is:
rk  k 0 T 
n
E  C


exp  
  C i i , r j  v j rk
 RT  i 1
where rj is the rate of production of component j (unit in Aspen Plus: kmol/m3-s
or kmol/kg cat-s), Ci is the concentration of component i (unit in Aspen plus:
molarity, kmol/m3; mole fraction; mass fraction; partial pressure, Pa; mass
concentration, kg/m3), k0 is the pre-exponential factor, αi, is the order of reaction
with respect to component i, E is the activation energy, C is the number of
components, and T is the reaction temperature (unit in Aspen plus: Kelvin).
14
Kinetics
LHHW
For reactions that are catalyzed by solid porous catalyst particles,
the sequence of elementary steps may include adsorption on the
catalyst surface of one or more reactants and/or desorption of one
or more products. In that case, a Langmuir-Hinshelwood-HougenWatson (LHHW) kinetic equation is often found to fit the
experimental kinetic data more accurately than the powerlaw
expression.
The LHHW rate expression for key component can be written as:
rk 
(kinetic
factor)(dr iving force expression
(adsorptio n expression
)
)
, r j  v j rk
15
Kinetics
LHHW
where,
Kinetic
Driving
factor  k 0 T
n
E 

exp  

 RT 
force expression
 C

 k 1   C i i
 i 1


  K l   C

 l 1
 j 1
M
Adsorption
expression
C
 C


  k 2   C j j


 j 1
l, j
j



 




m
k1, k2 and Kls are constant or a function of temperature. The
definition of other parameters are the same as power-law
expression.
16
Ideal Kinetic Reaction Models
CSTR
The simplest kinetic reactor model is the CSTR, in which the
contents are assumed to be perfectly mixed. Thus, the composition
and the temperature are assumed to be uniform throughout the
reactor volume and equal to the composition and temperature of
the reactor effluent. A perfectly mixed reactor is used often for
homogeneous liquid-phase reactions.
In the CSTR model, the reaction may be takes place under
adiabatic or isothermal conditions.
Although calculations only involve algebraic equations, they may
be nonlinear. Accordingly, a possible complication that must be
considered is the existence of multiple solutions, two or more of
which may be stable.
17
Ideal Kinetic Reaction Models
CSTR
Consider the case of non-adiabatic operation with R chemical
reactions. the mole and energy balance, can be written as:
 w 


  C
 in 
j , in
 w 
C
 

  out 
j , out
 R

   v i , j rk i  V r  0
 i 1

,
j  1,  , C
w ( H in  H out )  Q  0
where, w is the mass flow rate, ρ is the mixture density, Cj is the molarity of
component j, vi,j is the stoichiometric coefficient of component j in reaction i, rk
is the rate of production of key component, Vr is the volume of reacting mixture,
H is the mixture enthalpy (energy/mass), and Q is the heat duty (energy/time)
added to the reactor.
In process simulators, the standard enthalpy of formation of each component are
considered in calculating of mixture enthalpy. Therefore, the heat of reaction is
handled automatically.
18
Ideal Kinetic Reaction Models
Example 2: Hydrolysis of Propylene Oxide
Example 2: Propylene glycol (PG, normal boiling point is 460.7 K) is produced
from propylene oxide (PO) by liquid-phase hydrolysis with excess water under
adiabatic and near-ambient conditions, in the presence of a small amount of
soluble sulfuric acid as a homogeneous catalyst:
C 3H 6O  H 2O  C 3H 8O 2
Because the exothermic heat of reaction is appreciable, excess water is used.
Furthermore, because PO is not completely soluble in water, methanol is added
to the feed, which enters the reactor at 23.9 oC, 3 bar, and with the following
flow rates (kmol/hr): PO = 18.712, Water = 160 to 500, Methanol = 32.73
It is proposed to consider the use of an existing agitated reactor vessel, which
can be operated adiabatically at 3 bar, with a liquid volume of 1.1356 m3. The
powerlaw kinetic equation of the reaction is:
 rPO  9 . 15  10
where
22
 1 . 556  10 5
exp  
RT

rPO  kmol/(m
3
 2
kJ
 C PO , R  8 . 314

kmol - K

- s) , C PO  kmol/m
3
, and
T  K
19
Ideal Kinetic Reaction Models
Example 2: Hydrolysis of Propylene Oxide
Carry out a sensitivity analysis to investigate the effect of the water feed
rate on the operating temperature and the PO conversion.
Solution: The reactor was simulated by Aspen Plus. The WILSON method and
RCSTR are used in the simulation.
After converging the simulation, a SENSITIVITY ANALYSIS was added to the
simulation with water flow rate as manipulated (independent) variable and
temperature and PO conversion as defined (dependent) variables.
The Sensitivity was run in two cases, one with varying the water flow rate in the
range of 160 to 500 kmol/hr with step size of 1.0, and the other with varying the
water flow rate in the range of 500 to 160 with step size of -1.0.
The resulted temperature and PO conversion in two cases as a function of water
flow rate are shown in the following FIGURES.
20
Ideal Kinetic Reaction Models
390
1
380
0.9
370
0.8
PO Conversion
Reactor Temperature (K)
Example 2: Hydrolysis of Propylene Oxide
360
350
340
330
320
0.7
0.6
0.5
0.4
0.3
0.2
310
0.1
300
0
290
160
210
260
310
360
410
Water flow rate (kmol/hr)
460
160
210
260
310
360
410
460
water flow rate (kmol/hr)
Analysis of this process shows the possibility of multiple steady states. For
example, at a water flow rate of 365 kmol/hr, the following steady states are
obtained: (1) XPO = 91.5% , T = 68.5°C, (2) XPO = 27.8%, T = 37.3°C, and (3)
XPO = 5.0%, T = 26°C. The intermediate steady state at 27.8% conversion is
unstable, while the other two steady states are stable.
21
Ideal Kinetic Reaction Models
PFR
All simulators provide one-dimensional, plug-flow models that
neglect axial dispersion. Thus, there are no radial gradients of
temperature, composition, or pressure; and mass diffusion and heat
conduction do not occur in the axial direction.
Operation of the reactor can be adiabatic, isothermal, or nonadiabatic, non-isothermal. For the latter, heat transfer to or from
the reacting mixture occurs along the length of the reactor.
At high flow rates (high Reynolds numbers) in a long tubular
reactor, the PFR model is generally valid because turbulent flow
may approximate plug flow without appreciable axial mass and
heat transfer. At Reynolds numbers below 2100 (laminar flow), the
PFR model is not valid because of the parabolic velocity profile.
22
Ideal Kinetic Reaction Models
PFR
Consider the case of adiabatic operation with R chemical reactions.
the mole and energy balance, can be written as:
w d C
St
j
dx

 R

   v i , j rk i  , B.C.) at x  0 , C j  C j 0 , for j  1,  , C
 i 1

H in  H out  0
where, w is the mass flow rate, ρ is the mixture density, Cj is the molarity of
component j, vi,j is the stoichiometric coefficient of component j in reaction i, rk
is the rate of production of key component, St is the cross section area of reactor
and H is the mixture enthalpy (energy/mass).
In process simulators, the standard enthalpy of formation of each component are
considered in calculating of mixture enthalpy. Therefore, the heat of reaction is
handled automatically.
23
Ideal Kinetic Reaction Models
CSTR and PFR
In general, for a gas-phase reaction, a CSTR model is not used,
because of the difficulty in obtaining perfect mixing.
For a single reaction at isothermal conditions, the volume of a PFR
is smaller than that of a CSTR for the same conversion and
temperature. However, for (1) auto-catalytic reactions, where the
reaction rate depends on the concentration of a product, or (2)
auto-thermal reactions, where the feed is preheated by product, and
the reaction is highly exothermic, the volume of a CSTR can be
smaller than a PFR.
For non-catalytic homogeneous reactions, a PFR is widely used
because it can handle liquid or vapor feeds, with or without phase
change in the reactor.
24
Ideal Kinetic Reaction Models
CSTR and PFR
For fixed-bed catalytic reactors, a PFR model with a pseudohomogeneous kinetic equation is usually adequate and is referred
to as a l-D (one-dimensional) model.
However, if the reactor is non-adiabatic with heat transfer to or
from the wall, the PFR model is not usually adequate and a 2-D
model, involving the solution of partial differential equations for
variations in temperature and composition in both the axial and
radial directions, is necessary.
Models for fluidized-bed catalytic reactors are the most complex
and cannot be adequately modeled with either the CSTR or PFR
models. Because some of the gas passing through the fluidized bed
can bypass the suspended catalyst, the conversion in a fluidized
bed can be less than that predicted by the CSTR model.
25
function SMR_ER
x0=[10 10]; %initial guess (extant of reactions)
x=fsolve(@fun1,x0)
% Component mole flow in reactor effluent
nH2O=125-x(1)-x(2)
nCH4=50-x(1)
nCO=x(1)-x(2)
nCO2=x(2)
nH2=3*x(1)+x(2)
y2=[nH2O nCH4 nCO nCO2 nH2];
end
function y=fun1(x)
y(1)=25*(x(1)-x(2))*(3*x(1)+x(2))^3-94.2*(50-x(1))*...
(125-x(1)-x(2))*(175+2*x(1))^2;
y(2)=x(2)*(3*x(1)+x(2))-1.2*(x(1)-x(2))*(125-x(1)-...
x(2));
end
26