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Computer-aided
chemical reaction engineering
CACRE
Forms of work:
Lectures
Demonstrations /case studies
Computational exercises
Final exam
Contents
1 Introduction
2 Stoichiometry and kinetics
3 Homogeneous reactors
4 Catalytic two-phase reactors
5 Catalytic three-phase reactors
6 Fluid-fluid reactors
7 Reactors with a reactive solid phase
8 Laboratory reactors and parameter estimation
Approach- procedure
1 Generalized models for chemical reactors (mass and energy balances)
2 Identification of the mathematical structure of the model (NLE, ODE, PDE…)
3 Modularization of the model
4 Selection of numerical strategy and methods
5 Selection of software
6 Model implementation
7 Test simulations
8 Final simulations
Bilagor 1-8
Chemical process in general
Physicaltreatment steps
Chemical treatmentsteps
Physicaltreatment steps
Products
Rawmaterials
Recycle
A chemical reactor
Transforms Raw material Products
Can be batchwise, semicontinuous or continuous
Can be stationary or non-stationary
Classification often basen on the number of phases
gas, liquid, solid, catalyst
The process chemistry determines very much the reactor selection
Reactor out In
Why is modelling and computation needed
It cannot be done in this way !
Mathematical
model
Reactor design in nutshell
Reactor ready
Idea
Experiment
Parameter
estimation
Optimization
Principles of reactor modelling
Kinetic
model
Mass and
heat transfer
model
Flow
model
REACTOR MODEL
Ingredients in the model
Stoichiometry
Kinetics and
termodynamics
Reaction & diffusion
Reactor model
Stoichiometry and och kinetics Desired reactions
Non-desired reactions (side reactions)
Often multiple reactions
Example: Methanol synthesis
CO + 2H2 W CH3OH (desired reaction)
CO2 + H2 W CO + H2O (side reaction)
Parallel reaction with respect to hydrogen
Consecutive reaction with respect to CO
Stoichiometric matrix
Reactants -
Products +
01
N
i
iia
0aT
Stoichiometric matrix - example CO + 2H2 = CH3OH
CO2 + H2 = CO + H2O
-1 CO - 2 H2 + 1CH3OH = 0
- 1CO2 -1 H2 + 1CO + 1H2O = 0
OH
CO
OHCH
H
CO
a
2
2
3
2
T
11011
00121
(1)
(2)
Reaction kinetics
Reaction rate R (mol/s m3) gives how many moles of substance is generated /time/volume
Important to know the difference between elementary and non-elementary reactions
Elementary reaction reflects directly the events, collisions on the molecular level
Rate expressions 2A + B = 2C
If the reaction is elementary:
22
CBA ckcckR
The construction of rate expressions for elementary reactions is
straightforward and can be done automatically by a computer; just the
stoichiometric matrix is needed as an input
Component generation rate ri
Rr ii
For methanol synthesis reaction
rH2 = - 2 R och rCH3OH = +1 R
Systems with many reactions and components
rH2 = -2R1 - 1R2
rCH3OH = +1 R1 + 0 R2
S
j
jiji Rr1
Generation rate
22
CBA ckcckR
2A + B = 2C
rA=-2R
rB=-R
rC=2R
Homogeneous reactors
Only one phase present (incl. a
homogeneous catalyst)
Tubular reactor (plug flow reactor(PFR))
Tank reactor (CSTR, semibatch reactor,
batch reactor)
Homogeneous reactors
One phase
Gas or liquid
Tank reactors can be continuous (CSTR),
semibatch (SBR) och batch reactors (BR)
Tubular reactor with plug flow (PFR) or axial
dispersion (ADR, ADM)
CSTR
Batch reactors
Mixing in tank reactors
Tubular reactor (PFR)
Tubular reactor
Advanced reactor technology
-parallel tube reactors
Cooling systems for reactors
Definitions
n nii
N
1
n nii
N
1
n n ii
N
0 01
n n ii
N
0 01
Total molar flow
Total molar amount
Definitions: mole fraction
xn
ni
i
xn
ni
i
xn
ni
i
0
0
xn
ni
i
0
0
Definitions: concentration
cn
Vi
i
c
n
Vi
i
c xn
Vx ci i i c c
n
Vx ci i i
Concentration and mole fraction
For continuous systems and batch reactors
Definitions: volumetric flow rate,
mass flow, density
V
m
m n Mi ii
N
1
Vn M n
x Mi i
i
N
i ii
N
1 1
Volumetric flow rate
Mass flow
Combination gives
Conversion
Xn n
nk
k k
k
,
,
0
0
Xn n
nk
k k
k
0
0
,
,
dt
dn + n = V r + n
iiii0
V r + n - n = dt
dniii0
i
V r = dt
dni
i
STIRRED TANK REACTORS WITH COMPLETE BACKMIXING
General mass balance
Rearranged to
For batch reactor, all flows zero:
The initial condition is
0 = t , n = n i0i
CSTR at steady state: dni/dt = 0
V r = n - n ii0i
V r + n - n = dt
nd0
V r = dt
nd
With arrays we can write
Batch reactor (BR)
a) Isothermal liquid phase CSTR
r + c - c
= dt
cd 0
τ = V/V0, τ = space time
V V , 00
Special cases
b) Batch reactor, gas and liquid phases
dt
dc V =
dt
d(cV) =
dt
dn r =
dt
cd
Liquid-phase semibatch reactor (SBR)
dt V + V = V 0
t
00
VV/ = r + c- c
= dt
cdR
0
const) = V( V =dV/dt 00
dt
dV c + V /dt cd =/dt nd
V r + n = dt
nd0
0 = V r + n - n0
V/n = c , V/n = c 000
, n = n , RT n Z= V p i
n / n = x iii
P
...)xP,(T, ZRT n = V i
Gas-phase CSTR at steady state
dt
dn + A
dl
dc D- + n = V r + A
dl
dc D- + n
ii
out
outi,ii
in
ini,
Denote
A
dl
dc D = A
dl
dc D - A
dl
dc D ,n - n = n
ii
in
i
out
ini,outi,i
which imply
dt
nd + n = V r + A
dl
dc D i
iii
An infinitesimal volume element ΔV is considered:
Tubular reactors: Plug flow (PFR) and axial dispersion model (ADM)
DADM
dt
cd V =
dt
V) c( d =
dt
dn iii
The volume element ΔV:
l A = V
dt
cd +
l
n
A
1 = r +
l
/dl)dc( D ii
ii
Plug flow (PFR) and axial dispersion model (ADM)
The accumulation term
General mass balance
dt
dc l A + n = l A r +
dl
dc A D i
iii
Let Δl 0:
dt
dc +
A
1
dl
nd = r +
dl
cd D ii
i2
i2
r + dl
cd D +
dl
nd
A
1 - =
dt
dci2
i2
ii where V c = n ii
For the plug flow model, the eqn. is reduced to (Adl = dV):
r + dV
nd - =
dt
dci
ii
At steady state conditions the time derivative of the concentration vanishes: 0 = dt
dci
We obtain
r + dl
cd D =
dV
ndi2
i2
i
r = dV
ndi
i
ADM
PFR
ADM
PFR
where w0 = the superficial velocity at the reaction inlet,
r + dl
cd D +
dl
dc w - =
dt
dci2
i2
i0
i
ADM
Liquid phase reactions:
dl
dcA w =
dl
dc V =
dl
)V c( d =
dl
nd i0
i
0
0ii
r + dl
dc w - =
dt
dci
i0
iPFR
r c
1 + V
dV
dc
c
1 - =
dV
Vdi
Vdz = dV , V/V = 0
A differential equation is obtained for δ:
0 = c
r -
dz
dc
c
1 +
dz
d0
i
For isothermal cases dc/dz is 0, and δ is easily obtained by integration (c = c0)
dz r c
= d i
1
0
0
0
1
dz r c
+ 1 = i
z
0
0
0
Gas-phase reactions
For both isothermal and steady state conditions, dc/dt = 0 and we
get:
c = c | r + dV
Vd c - V
dV
dc - =
dt
dcii
PFR
At steady state conditions the natural choice of variable is ni and
we get for the plug flow model:
r = dV
ndi
i
The concentrations which are needed in the rate lows are obtained from
n = n , V/n = c iii
The initial condition is
0 = Vat n = n i0i
The volumetric flow rate is updated by the formula
P
...)xP,(T, ZRT n = V i
Energy balances general considerations
A general energy balance for a volume element can be written as
dU/dt + Q + H = H0
The use of molar enthalpies:
n H = H = n H - n H = H - H imi
i
0i0mi
i
imi
i
0
The difference is split to two terms,
n H + n H = n H imi
i
imi
i
imi
What is Σ Δhmini? This becomes clear, when the molar heat capacity is introduced:
T c n = n H pmiiimi
is valid, provided that the pressure effect is neglected.
What is Σ HmiΔni? - The mass balance give
dt
dn - V r = n
iii
R = r jij
j
i
The generation rate
V c = n ii
dt
dc V =
dt
dn ii
V
dt
dc - V R H = n H
ijij
j
mi
i
imi
j
V dt
dc H - V R H = i
mijmiij
ji
where the sum Σi υij Hmi is de facto the reaction enthalpy, Δ Hrj.
and the molar amount being present in the volume element is
The term Σ Hmi Δni becomes
dt
dn U + n
dt
dU =
dt
)nU( d =
dt
dU =
dt
dU imi
i
imi
i
imi
i
i
j
V dt
dc U + c
dt
dU = i
mi
i
imi
i
V c dt
dU + V
dt
dc H - V R H + T c n i
mi
i
imi
i
jrj
j
pmii
i
0 = Q + V dt
dc U + i
mi
i
We get
c = dT
Hdpmi
mi c = dT
dUvmi
mi
dt
dT
dt
dU =
dT
Ud-1
mimi
dt
dT c =
dt
dUvmi
mi
VP + U = H mimimi
vmi is the molar volume.
V dt
dc VP = V
dt
dc )U - H(
imi
imimi
The energy balance becomes
V c dt
dT c + V
dt
dc V P - V R H + T c n ivmi
i
imi
i
jrj
j
pmii
i
0 = Q +
0=dV
Qd + R H +
dV
dT c n +
dt
dc V P -
dt
dT cc jrj
j
pmii
i
imi
i
ivmi
i
ΔV 0
Tank reactors
Because of the homogeneous contents, the integration over the entire tank volume can be carried out:
dV dt
dc V P - dV
dt
dT c c
V
0
imi
i
V
0ivmi
Qd + dV R H + dT c n +Q
0
v
0jrj
j
pmii
T
T0
dt
dc V P+
V
Q-R )H(- dT+ )c n (
V
1-
c c
1=
dt
dT imi
i
jrj
j
pmii
i
T
Tivmi
i
0
V
Q - R )H (-
c n
1 =
V
T - Tjrj
jpmii
0
c n = c n pipmii
V
Q - R )H (-
c n
1 =
V
T - Tjrj
jpi
0
For liquid phase reactions the term .ignoredoften is dt
dc V P i
mi
V
Q - R )H (- = dT c n
V
1jrj
j
pmii
T
T0
Steady state
)T - (T S U= Q cHeat transfer from/to the reactor
Tubular plug flow reactor
dt
dc V P -
dt
dT c c
imi
i
ivmi
i
0 = dV
Qd + R H +
dV
dT c n + jrj
j
pmii
Steady state
dV
Qd - R )H (-
c n
1 =
dV
dTjrj
jpmii
dV
Qd - R )H (-
c m
1 =
dV
dTjrj
jpi
)T - (T
V
S U=
dV
dS )T - (T U=
dV
Qdj
R
j
S is the total heat transfer area. For cylindrical tubes
D/4 =
L R
L R2 = VS/ T
T2T
TTR
where RT, DT and LT denote the radius, the diameter and the length of the tubes. The temperature outside the reactor is
T = T cj
Batch reactor
0 = dt
dU + Q R =
dt
dnjij
j
i
0 = dt
n U d
+ Q
imi
i
Reactor volume is constant, differentiation leads to
0 = dt
dc V U + V c
dt
dU + Q i
miimi
where
V R = dt
dc V =
dt
dnjij
j
ii
and
dt
dT c =
dt
dT
dT
dU =
dt
dUvmi
mimi
V
Q - R )U (- =
dt
dT c c jrj
j
ivmi
V
Q - R )H (-
c c
1 =
dt
dTjrj
jvmii
0 = V U R + dt
dT V c c + Q mi
i
jij
j
ivmi
Batch reactors are very frequently used for liquid-phase processes, for which ΔUrj ΔHij.
c c = V
c m =
V
c n = c c p0v0
R
0
R
vmiivmii
i
where the product ρ0 cp is the heat capacity of the reacting liquid.
V
Q - R H (-
c
1 =
dt
dTjrj
jp0
)T - (T S U= Q c
The mathematical structures of homogeneous reactor models
MODEL PROBLEM STRUCTURE
Steady state CSTR f (y) = 0 (N)LE
Dynamic CSTR dy/dt = f (y) ODE (IVP)
Steady state PFR dy/dz = f (y) ODE (IVP)
Dynamic PFR dy/dt = -A dy/dz + f (y) PDE (IVP)
Batch reactor (BR)
Semi batch reactor (BR)
dy/dt= f (y) ODE
Steady state axial dispersion model (ADM) A d2y/dz2 + Bdy/dz + f (y) = 0 ODE (PVP)
Dynamic axial dispersion model (ADM) dy/dt = A d2y/dz2 + B dz/dz + f (y) = 0 PDE
(N)LE = (non)linear equations
ODE = ordinary differential equations initial value problem
PD = partial differential equations, hyperbolic type
IVP = initial value problem
BVP = boundary value problem
• input of chemical engineering data (initial concentrations, temperatures, reactor
dimensions, kinetic, thermodynamic as well as mass and heat transfer parameters)
• input of data for steering of the numerical solution (numerical methods, system
step-length selection, convergence criteria e.t.c.
• definition of the kinetic and thermodynamic models (reaction rates, calculation of
rate and equilibrium constants)
• definition of the mass and heat transfer models (e.g. correlations for diffusion and
dispersion coefficients, heat and mass transfer coefficients and areas)
• definition of mass and energy balances along with equations for pressure drop
• definition of partial derivatives needed for the model solution
• numerical solver for the differential and/or algebraic equations involved in the model
• output routines, which tell not only the results (e.g. concentration, temperature and
pressure profiles) but also give information about the success or failure of the
numerical solution process
Software build-up; tasks
MAIN PROGRAM INPUT
c0, T0
OUTPUT
c, T
SOLVER
-mathematical library
routine for NLE or
ODES
RATE
-routine for
calculation of
reaction rates
MODEL (FCN)
-routine for the
model eqs, e.g.
dy/dt = f(y)
CORRE
-routine for
calculation of
correlations for
mass/heat transfer THERMO
-routine for
calculation of
thermodynamics
properties
MODEL DERIVATIVES (FCNJ)
-Jacobian routine J, contains fi/y
Simulation programme modules
Program flowsheet
y
)y( f - )y + y( f =
y
f
y
f
j
jijji
j
i
j
i
where k is the iteration index, f(k), is the function vector and J-1 is the inverted Jacobian matrix
Solution of algebraic equations
N1... = i , < |y
y - y|
i(k)
1)i(k-i(k)
0 = )y(x,f
f J - y = y 1)-(k-1
1)-(kk
Newton-Raphson algorithm
N
N
yy
yyyy
J
/f...... y/f y/f /f
. . . .
. . . .
. . . .
/f........../f /f /f
N3N2N1N
13121 11
End criterion
One unknown case
1)-(k
)1(k
1)-(k (k)f
f - y y
x),y( f = dx
yd
x =at x y = y 00
Solution of ordinary differential equations
kb + y = y ii
q
1=i
1-nn
)ka + y ,x( fh = k lil
i
1=l
nni
If the coefficient ail = 0 for l = i the method is explicit; if ail 0 for l = i the method is implicit.
)hka + y(f = k h)aJ - I( lil
1-i
1=l
niii
where I denotes the identify matrix:
1...00
. .
. .
. 1.
0...01
= I
:
where yn and yn-1 give the solution of the differential equation at x and x-Δx.
Runge-Kutta methods
where h=Δx.
and J is the Jacobian matrix
N
N
yy
yyyy
J
/f...... y/f y/f /f
. . . .
. . . .
. . . .
/f........../f /f /f
N3N2N1N
13121 11
y
)y(f - )y + y(f =
y
f
j
jijji
j
i
Semi-implicit Runge-Kutta method
kc +)h ka + y(f = kh)aJ - (I lil
1-i
1=l
1il
1-i
1=l
niii
An extension of semi-implicit Runge-Kutta methods is the Rosenbrock-Wanner method (ROW)
f h + y = y i-ni
K
0=i
i-ni
K
1=i
n
21
Adams-Moulton (AM) and backward difference-methods;
(AM) 1 - q = K 1 = K 1 = 211
(BD) 0 = K q = K 21
Linear multistep methods
(AM) f h + y = y i-ni
1-q
0=i
1-nn
(BD) fh + y = y n0i-ni
q
1=i
n
Backward difference-method:
Adams-Moulton method:
Temperature and concentration
Temperature dependence of the
rate constant
Jacobus Henricus van’t Hoff and
Svante Arrhenius:
RTEAAek/
RTEb AeTAk/
'
Transition state theory
Activation energy
Liquid-phase systems
Equilibrium constant often determined experimentally
Gas-phase system
Equilibrium constant can be calculated theoretically,
one knows
Reaction thermodynamics
0
rS 0
rH
0
0.2
0.4
0.6
0.8
1
1.2
0 2 4 6 8 10
Reaction thermodynamics
Temperature dependence
2
0)ln(
RT
U
dT
Kdrc
Often approximately
R
S
RT
UK rr
c
00
)ln(
0
rU
0
rS
Change in internal energy
Change in entropy
Reaction thermodynamics
Equilibrium constant Kp
2
0)ln(
RT
H
dT
Kdrp
integration gives
dTRT
HTKTK
T
T
rpp
0
2
0
0 )(ln)(ln
Reaction enthalpies Reaction enthalpy
0
rH
At reference temperature T0 (often 298 K)
From enthalpies of formations 0
0
0 )( fi
i
ir HTH
See for example in
Reid,Prausnitz, Poling,
The Properties of Gases and Liquids
Reaction enthalpy from
i
T
T
pmiirr dTCTHTH
0
000
Molar heat capacities exist as temperature (Cpmi) functions
CSTR- steady state multiplicity
Catalytic two-phase reactors
From reaction mechanism
to reactor design
20 m
Catalyst materials
20 m
200 m
Cu/SiO2
SAPO-5
Elektron microscopy (SEM) reveals catalyst morhpology
Isobutene i 10 MR H–FER pores
Fibre catalysts
4 nm
TEM image of the 5 wt.% Pt/Al2O3
(Strem Chemicals) catalyst SEM image of the 5 wt.% Pt/SiO2
fiber catalyst
1 mm
125-90 m particles
D = 27%
dPt= 4 nm
D = 40%
dPt= 2.5 nm
Catalysts in micro- and nanoscale
5 m
TEM-bild SEM-bild
4 nm 5 m
• 5wt.% Pt/SF (Silikafiber) 5wt.% Pt/Al2O3 (Strem)
4 nm
New products and
processes from renewable
sources
Chemicals from biomaterial, particularly wood
Catalytic production of biodiesel
From wood to food
Isomerization of linoleic
acid was first time
carried out on a
heterogeneous catalyst
Catalytic two-phase reactors
Heterogeneous catalytic reactor
Solid catalyst accelerates reactions
Gas or liquid present in the reactor
Molecular path to the catalyst
diffusion to the outer
surface of the catalyst
particle
diffusion through the
catalyst pores
Molecules come to the
active sites on the
surface
Molecular path to the catalyst
molecules adsorb
on the active sites
and react with each
other
Product molecules
desorb and diffuse
out
Concentration and temperature
profiles in catalyst particles
Catalytic Reactors
Packed bed
Fluidized bed
Packed bed: traditional design
Gauze-reactor
Oxidation av ammonia
High temperature, 890C
Network of Pt catalyst
Monolith reactor
Multibed reactor
Catalyst beds in series (often adiabatic)
Heat exchangers between the beds
Multibed reactor, SO2 to SO3
Multitubular reactor
Models for packed beds
Pseudo-homogeneous model
concentration and temperature in the catalyst particle on
the same level as in the fluid bulk
neither concentration- nor temperature gradients in the
catalyst particle
Diffusion resistance negligible in the catalyst particlen
Pore diffusion can be included with the aid of an
effectiveness factor
Models for packed beds
Heterogeneous model
Separate balance equations for bulk phase
and catalyst particles
Catalyst bulk density
B
cat
R
catalystm
V
kg
m 3
rmol
s mi B
reactor volume
3
rmol
skg catalysti ( )( )
One-dimensional pseudo-
homogeneous model - stationary
, ,n r V ni in i B i out
, ,n n ni i out i in
[in] + [generated i] =
[out] + [accumulated]
One-dimensional pseudo-
homogeneous model
RdV
ndB
.
The diffusion of the molecules through the fluid film around
the catalyst particle and their diffusion through the pores
Influences the reaction rate
The real reaction rate becomes
)('
Bjejj cRR
cB = concentration in the bulk phase
ρB = mass of catalyst / reactor volume
Effectiveness factor
Real diffusion flow/Intrinsic kinetics
diffej
diff
N
N
i
i
sR
i
b s
s r r dr
r R
10
1c
=1 if diffusion resistance negligible
Diffusion i porous particle
Catalyst surface
Diffusion in pores
Fick’s law
dr
dcDN i
eii
Ni diffusion flow mol/(time surface)
Dei effective diffusion coefficient
Mass balance for a catalyst
particle – steady state
01
pi
siei
sr
dr
rdr
dcDd
r
If diffusion coefficient is constant:
ei
ipii
D
r
dr
dc
r
s
dr
cd
2
2
Form factor – shape factor (a=s+1)
R
s
V
A
p
p 1
R characteristic dimension of the particle
Ap outer surface of the particle
Vp particle volume
Form factor
S=0
slab S=1
cylinder
S=2
sphere
Biot number
ei
GiM
D
RkBi
Relation between the diffusion resistance in the fluid film
and the catalyst particle
Biot number is usually >>1 for porous particles
Thiele modulus:
first order reaction
22 RD
k
ei
pi
Reaction rate / diffusion rate
Effectiveness factor
Asymptotic effectiveness factor
Semianalytical expressions for arbitrary kinetics
Good approximation for positive reaction orders
Erroneous if the reaktion order is negative, reaction is accelererated with decreasing concentration
Heat effects in catalyst particles
Fourier’s law (heat conduction)
Temperature gradient inside the particle typically
small
Temperature gradient can exist in the fluid film
around the particle
The film is thin
Heat effects in catalyst particle
Energy and mass balances for catalyst particle are
coupled via reaction rate and should be solved
numerically
Effectivity factor can be >1 for strongly exothermic
reactions, rate constant increases with
temperature !
Steady state multiplicity possible
Steady state multiplicity
Lactose hydrogenation - diffusion
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
x
c (
mol/l)
0.03 mm
0.3 mm
0.3 mm
0.03 mm1.0 mm
1.0 mm
3.0 mm
3.0 mm
Concentration profiles
Inside the particle
-various particle sizes
Yta Center
lactose + H2 =
lactitol
0 200
400 600
0
0.5
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
0 200
400 600
0
.5
1
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Concentration profiles of lactose in the
particle
No deactivation Deactivation
center x x
Two-dimensional model
Large heat effect induce a fradial
temperature gradient, which leads to
concentration gradients
Reaction rates vary in radial direction
Concentrations gradients created
Catalytic hydrogenation
Hot spot Hydrogenation
of av toluene
Oxidation of o-xylene
Dependence of Hot spot on the the coolant and inlet temperature
Two-dimensional model:
temperature profile
Two-dimensional model
Mass balance
[in plug flow] + [in radial disp.] + [generated] = [out
plug flow] + [out radial disp.]
iBii
mr
i rd
dc
d
cd
Pe
a
dz
wcd
w
1)(12
2
0
Two-dimensional model
Energy balance
j
rjB
p
HRjd
dT
d
Td
Rcdz
dT)(
12
2
2
0
Numerical solution
Parabolic PDEs converted to ODEs
Finite difference + (RK, Adams Moulton,
Backward difference)
Orthogonal collocation + (RK, Adams
Moulton, Backward difference)
Fluidised bed
Fluidisation
Solid particles in vertical bed
Gas blown from the bottom
Particles remain stagnant at low gas velocities
At higher gas velocities the particles fluidise
The bed is expanded and the particles become
dispersed in the gas phase
Minimum fluidisation velocity
Fluidised bed
With increasing gas velocity a bubble phase
is formed – rich in gas
Most part of catalyst in the emulsion phase
Fluidised bed resembles boiling liquid
Fluidised bed
If the gas velocity is increased even
more, the bubbles become equal to the
bed diameter (slug flow)
Limit velocity for slug flow = ws
Fluidised bed and pressure drop
Fluidisation is recognised by measuring the
pressure drop
Pressure drop increases monotonically with
the gas velocity (Ergun equation)
At minimum fluidisation, the increase of
pressure drop stops
Fluidiserad bed:
hydrodynamics
Bubble phase
Emulsion phase
Wake
Cloud
Reactions proceed in all phases !
Phase structure in detail
Fluidised bed
Matematical model - principles
Separate balance considerations for each
phase
Catalyst particles are very small (in
micrometer scale) so internal diffusion can
be ignored
Vigorous turbulence implies that the bed is
isothermal
Fluidised bed
Matematical model
Plug flow model
Unrealistic but gives the maximum limit
Backmixing model
Sometimes tanks-in-series model is tried
iRiBi nVrn.
0
.
Fluidised bed
Matematical model
Kunii-Levenspiel model
Realistic description
Bubble phase in plug flow
Gas flow in the emulsion phase is negligible
Cloud and wakephases have the same
concentrations
Kunii-Levenspiel model
Transport of a reacting gas from bubble
phase to cloud-wake phase and further to
emulsion phase
Three parts in the volume element
ecb VVVV
Kunii-Levenspiel model
Bubble phase
Cloud-wake phase
Emulsion phase
Kunii-Levenspiel model, mass
balances
b
eeBe
b
ccBcbBb
b
b
V
VR
V
VRR
d
dc
0b
cBcceccecbbc
V
VRccKccK
0b
eBeeecce
V
VRccK
Kunii-Levenspiel model
structure
3 * N mass balances (N= number of
components)
1 * N ODEs
2*N algebraic equations
Numerical solution with DASSL
Kunii-Levenspiel
model parameters
Volume fractions Vc/Vb och Ve/Vb
Kbc och Kbe from correlations
Mean residence time of bubbles
b
bw
L
Diffusion coefficients
Depend on molecular structure – in
general dependent on concentrations, too
Fick’s law gives a simple relation between
the diffusion flux and the concentration
gradient
dx
dcDN i
eii
Effective diffusion coefficients
Effektive diffusion coefficient in a porous
particle
Di molekular diffusion coefficient
p porosity < 1
p tortuosity or labyrinth factor,
> 1
i
p
p
ei DD
Effective diffusion coefficient
Molecular diffusion
Intermolecular diffusion
collisions between molecules
Knudsen diffusion
Collisions between molecules and pore walls
kimii DDD
111
Fuller-Schettler-Giddings equation
T temperature
M molar mass
v volyme contribution
Diffusion coefficient
Gas phase
23/13/1
27
75.1
/10//
ki
ki
ik
vvatm
P
smM
molg
M
molg
K
T
D
Volyme contributions Diffusion volumes of simple molecules
He 2.67 CO 18.0
Ne 5.98 CO2 26.9
Ar 16.2 N2O 35.9
Kr 24.5 NH3 20.7
Xe 32.7 H2O 13.1
H2 6.12 SF6 71.3
D2 6.84 Cl2 38.4
N2 18.5 Br2 69.0
O2 16.3 SO2 41.8
Air 19.7
Atomic and Structural Diffusion Volume Increments
C 15.9 F 14.7
H 2.31 Cl 21.0
O 6.11 Br 21.9
N 4.54 I 29.8
S 22.9
Aromatic ring -18.3
Heterocyclic ring -18.3
Diffusion coefficient
Gases
Knudsen’s diffusion coefficient
Sg specific surface area,
which can be determined by nitrogen adsorption, BET (Brunauer-Emmett-Teller)-theory
ipg
p
kiM
RT
SD
2
3
8
Diffusion coefficient
Liquids
Not as well developed theory as for gases
General theory for calculation of binary
diffusion coefficients in liquids is missing
Correlations typically describe a solute in a
solvent
Correlations exist for neutral molecules and
ions
Diffusion coefficient
Liquids
Stokes-Einstein equation
Molecule radius RA is the bottleneck !
AB
ABR
RTD
6
Diffusion coefficient
Liquid
Wilke-Chang equation
VA solute molar volume at normal boiling
point
B solvent viscosity
sm
VcP
K
T
molg
M
D
AB
B
AB //
104.72
6.0
12
Molar volumes
Methane 37.7
Propane 74.5
Heptane 162
Cyclohexane 117
Ethylene 49.4
Benzene 96.5
Fluorobenzene 102
Bromobenzene 120
Chlorobenzene 115
Iodobenzene 130
Methanol 42.5
n-Propylalcohol 81.8
Dimethyl ether 63.8
Ethyl propyl ether 129
Acetone 77.5
Acetic acid 64.1
Isobutyric acid 109
Methyl formate 62.8
Ethyl acetate 106
Diethyl amine 109
Acetonitrile 57.4
Methyl chloride 50.6
Carbon tetrachloride 102
Dichlorodifluoromethane 80.7
Ethyl mercaptan 75.5
Diethyl sulfide 118
Phosgene 69.5
Ammonia 25
Chlorine 45.5
Water 18.7
Hydrochloric Acid 30.6
Sulfur dioxide 43.8
Atomic increments for estimation of VA Increment, cm3/mol (Le Bas)
Carbon 14.8
Hydrogen 3.7
Oxygen (except as noted below) 7.4
In methyl esters and ethers 9.1
In ethyl esters and ethers 9.9
In higher esters and ethers 11.0
In acids 12.0
Joined to S, P or N 8.3
Nitrogen
Doubly bonded 15.6
In primary amines 10.5
In secondary amines 12.0
Bromine 27
Chlorine 24.6
Fluorine 8.7
Iodine 37
Sulphur 25.6
Ring, Three-membered -6.0
Four-membered -8.5
Five-membered -11.5
Six-membered -15.0
Naphtalene -30.0
Anthracene -47.5
Double bond between carbon atoms -
Triple bond between carbon atoms -
Diffusion coefficient
Liquids
Wilke-Chang equation has been extended to
solvent mixtures
Association factor
water 2.6
methanol 1.9
ethanol 1.5
non-polar solvents 1.0
Viscosity
Use experimental data if available
Correlation equations
A, B, C och D from databanks
2/ln TDTCTBA
