κίνησις "kinesis", movement or to move
APPLIED REACTION KINETICS Friday A01 09:00-11:00 Friday BS9 13:00-15:00
(Assoc. prof. Bohumil Bernauer, A27,
[email protected], +420220444017 ) (Dr. Milan Bernauer, H08,
[email protected], +420220444287)
Resources and references • Notes from lectures • Internet web.vscht.cz/bernauem/ • Textbooks Fogler Scott H.: Elements of Chemical Reaction Engineering, 4th Edition, Prentice Hall, 2006. (http://www.engin.umich.edu/~cre/) Missen R.W., Mims C.A., Saville B.A., Introduction to Chemical Reaction Engineering and Kinetics, J. Wiley&Sons, N.Y. 1999. • Journals (on-line) • Software MS Excel, (Matlab, Octave, Athena Visual Studio, FORTRAN, Maple….)
Fischer-Tropsch (SASOL, RSA )
N2O decomposition (IPC AS, CZ)
WGS (BASF, FRG)
Chemical reactor(s) heart of chemical process
Methane aromatization (ICTP, CZ)
Raw material separation reaction separation product
r ( com position , T , P , ..., param s )
Summary of the 1st lecture
• • • • • • •
Stoichiometry Extent of reaction Fractional conversion of key component Stoichiometric matrix Balance of chemical elements Selectivity, Yield Reaction rate definition
" " the m aterial elem ent (P lato)
Stoichiometry
" " the count, the quantity
2NO N2 + O2
closed (batch) system t =0
Atoms of oxygen
n N O 2 nO2
Atoms of nitrogen
n NO 2 n N 2
o
t>0
o
o
n N O 2 nO2
o
o
n NO 2 n N 2
o
o
n N O 2 nO2 n N O 2 nO2 o
o
n N O n N O 2( n O 2 n O 2 )
2
Symbols for species Stoichiometric coefficients
-2NO + NO = A1 1 = -2
o
o
n N O n N O 2( n N 2 n N 2 ) o
o
n NO n NO
o
n NO 2 n N 2 n NO 2 n N 2
( nO2 nO2 )
O2 + O2 = A2 2 = 1
1
o
N2 = 0 N2 = A 3 3 = 1
i 0
p ro d u cts
i 0
in erts
3
i
Ai 0
(nN2 nN2 )
i 1
i 0
reac tan ts
1
Molar extent of the reaction o
o
n NO n NO 2
( nO2 nO2 ) 1
[ksi:] o
(nN2 nN2 ) 1
o
ni ni
i
From the definition of the reaction extent follows: 1. The reaction extent has the dimension of moles (number of molecules) 2. The reaction extent value depends on stoichiometry of reaction 3. The reaction extent is an extensive variable
Reaction extent for a single reaction in closed (batch) system t=0
t>0
o
ni
ni
Closed system
i 0
p ro d u cts
i 0
in erts
N
i
Ai 0
i 1
i 0 o
ni ni
i
ni ni i o
reac tan ts
Example
2NO N2 + O2
A1 N O , A2 N 2 , A3 O 2
1 2, 2 1, 1 1 ν 2 T
ν A 2 T
A1 1 1 , A A2 A 3 A1 1 1 A2 0 A 3
Matrix notation
Fractional conversion of key component, j
X
j
j
m ax *
max
X
o
nj nj
Number of moles of key component in limits (chemical equilibrium or 0)
j
o
*
o
nj 0
X
j
max
ni ni
i o
nj
Xj
i j
nj nj
o i
n ni
j
i
n
o j
X j (0,1)
o
nj
o
X
j
j
100
nj nj n
X j (0,100)
o j
o
X j m ax X
nj j
j
o i
ni n
i
j
o
njX
j
Stoichiometric matrix in the case of several reactions N
ki
Ai 0
k 1, N R
i 1
reaction
component
Stoechimetric matrix has NR rows and N columns
11 21 ν . N R ,1
12
...
1N
22
...
2N
N R ,2
... N R , N
νA 0
,A
A1 A2 . AN
Number of moles of i-th component consumed or created in k-th reaction : o
Molar extent of k-th reaction :
k
n ki n ki
ki
NR
Number of moles of i-th components:
o i
ni n
ki
k
k 1
Matrix notation
o
T
nn ν ξ n1 n2 n .. nN
n1o 1 o 2 n o 2 ,n ,ξ .. .. no NR N
o
n ki n ki
Problem 1.1 Oxidation of ammonia on Pt-Rh catalyst NH3 + 1.25 O2 NO + 1.5H2O NH3 + O2 0.5 N2O + 1.5H2O NH3 + 0.75O2 0.5 N2 + 1.5H2O
(1) (2) (3)
Task: To write down the stoichiometric matrix.
A1
A2
A3
A4
A5
A6
Reaction
NH3
O2
NO
N2O
N2
H2O
(1)
-1
-1.25
1
0
0
1.5
(2)
-1
-1
0
0.5
0
1.5
(3)
-1
-0.75
0
0
0.5
1.5
Molar balance table of component in closed (batch) system Component NH3 O2 NO N2O N2 H2O
t=0
t >0
o
n1 n1 ( 1 2 3 )
o
n 2 n 2 1.25 1 2 0.75 3
o
n3 n3 1
n4
o
n 4 n 4 0.5 2
o
n 5 n 5 0.5 3
o
n 6 n 6 1.5( 1 2 3 )
o
n1
o
n2
o
n3
o
o
n5
o
n6
6
6
n i 1
o i
n
o i
0.25 1 3
i 1
e.g .
m o lar fractio n o f N H 3 n1 ( 1 2 3 ) o
y1
6
i 1
n i 0 .2 5 1 3 o
Independent reactions Set of NR reactions in reaction network is independent if
Rank()=NR or number of independent reactions = Rank()
Problem 1.2 NH3 + 1.25 O2 NO + 1.5H2O NH3 + O2 0.5 N2O + 1.5H2O NH3 + 0.75O2 0.5 N2 + 1.5H2O 2NO N2 + O2
(1) (2) (3) (4)
Task: To calculate the number of independent reactions. We determine the rank of stoichiometric matrix by Gaussian elimination:
1 1 1 0
1.25
1
0
0
1
0
0.5
0
0.75
0
0
0.5
1
2
0
1
1 0 0 0
1.5 1 1.5 0 0 1.5 0 0
1.25
1
0
0
0.25
1
0.5
0
0
1
1
0.5
0
2
2
1
Rank()=3 only 3 reactions are independent
1.25
1
0
0
0.25
1
0.5
0
0.5
1
0
0.5
1
2
0
1
1.5 1 0 0 0 0 0 0
1.5 0 0 0
1.25
1
0
0
0.25
1
0.5
0
0
1
1
0.5
0
0
0
0
1.5 0 0 0
Extent of the reaction in a flow system
Fi
o
Fi
Inlet molar flow rates [mole of i-th species/s]
REACTOR
Outlet molar flow rates [mole of i-th species/s]
N
i 1
ki
Ai 0 NR
Fi Fi ki &k o
k 1, N R Open (flow) system in a steady state
k & k
k 1
o
Fi , Fi -o u tlet, in let m o lar flo w rates o f i-th sp ecies [m o le/s]
&k ex ten t o f k -th reactio n p er u n it o f tim e [m o le/s] m ean resid en ce tim e [s] o T F = F + ν ξ&
F1 F2 F .. FN
F1o o , F o F2 , ξ& .. o FN
& 1 & 2 .. & NR
A reaction is at steady-state if the concentration of all species in each element of the reaction space (i.e. volume in the case of homogeneous reaction or surface of catalyst in the case of catalytic heterogeneous reaction) does not change in time.
Balance of chemical elements
o j
REACTOR
j
N
ki
Ai 0
k 1, N R
Open (flow) system at steady state
i 1
j
o j
o
j , j -outlet, inlet m olar flow s of j-th chem ical elem ent [m ole/s] Φ =Φ
o
1 2 Φ .. NEL
1o o ,Φo 2 .. o NEL
NEL - Number of chemical elements
Formula matrix E
NEL elements NEL=3 N components N=6 NH3
N
H
O
1
3
0
O2
0
0
2
NO
1
0
1
N2O
2
0
1
N2
2
0
0
H2O
0
2
1
Formula vector of NH3
There are no creation or annihilation of chemical elements in chemical reactions:
νE = 0
11 21 . N R ,1
12
...
1 N E 11
E 12
...
22
...
2N
...
...
E N ,2
...
N R ,2
... N R , N
E 21 . E N ,1
E 1, N E L 0 E 2, NEL 0 . E N , N E L 0
0
...
0
...
0
...
NEL
0 0 0
NR
Molar* weight (relative molecular mass) of i-th species: NEL
Mi
E
ij
mj
j 1
Atomic weights (relative atomic mass) of j-th element
M = Em M1 M2 M .. MN
m1 m2 ,m .. m NEL
and we have for molar weights of species
νM = νE m 0 because νE 0 *The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12. Avogadro constant = 6.022 141 29(27) × 1023 mol−1 (http://www.nist.gov)
Balances of atoms in batch and flow systems
C lo sed (b atch ) system : o
T
n =n +ν ξ T
E n = E n + E ν ξ E n + νE ξ E n T
T
o
T
T
T
o
T
o
O p en (flo w ) system at stead y state: o T F = F + ν ξ& T T o T o & E F = E F + E ν ξ E F + νE ξ& E F T
T
o
T
T
b ecau se νE 0 Finally
E
T
E
T
n n E n F F = E F o
o
T
T
o
n 0 o
F 0
These equations are used in data reconciliation tasks around chemical reactors. The last equation (flow system) is valid only at steady state.
Problem 1.3 Selective reduction of NOx by C3H8
Fi
o
Fi
NO 4.563 2.2905
Fi [ m ole / m in] Fi [ m ole / m in] o
Steady state, unknown stoichiometry of reactions
NO2 0.1845 2.3355
C3H8
CO 0 0
CO2 0 x
2.943 2.898
H2O 0 x
O2 90 x
N2 0 x
Calculate the missing outlet molar flow rates
E
N O C H
T
F
o
- F = E F 0 T
1-NO
2-NO2
3-CO
4-C3H8
5-CO2
6-H2O
7-O2
8-N2
1 1 0 0
1 2 0 0
0 1 1 0
0 0 3 8
0 2 1 0
0 1 0 2
0 2 0 0
2 0 0 0
T
E2
T
E1 T
T
E 1 F K N O W N E 2 FU N K N O W N 0 T
T
E 2 FU N K N O W N E 1 F K N O W N
FU N K N O W N E
T 2
1
T
E 1 FK N O W N Q FK N O W N
T
Q E2
F1 F2 F3 F4 F 5 F6 F 7 1 T E 1 F8
known
Fi
unknown
Fi
We obtain using Excel (homework 1)
0 0 Q 0.5 0.5
and taking
F2 E 2
Fi [ m ole / m in] Fi [ m ole / m in]
NO 4.563 2.2905
o
T
1
T
0
1
0
0
1
0.5
0.5
0
E 1 F1 Q F1
NO2 0.1845 2.3355
3 4 5 0
we have CO 0 0
C3H8 2.943 2.898
CO2 0 0.135
H2O 0 0.18
O2 90 88.76
N2 0 0.061
Selectivity Moles of a particular product generated per mole of key reactant consumed
Yield Moles of a particular product generated per one initial mole of key reactant
Ammonia oxidation
Component NH3 O2 NO N2O N2 H2O
t=0
t >0
o
n1 n1 ( 1 2 3 )
o
n 2 n 2 1.25 1 2 0.75 3
o
n3 n3 1
n4
o
n 4 n 4 0.5 2
o
n 5 n 5 0.5 3
o
n 6 n 6 1.5( 1 2 3 )
o
n1
o
n2
o
n3
o
o
n5
o
n6
n3 1 o
S NO NH 3 YNO NH 3
( 1 2 3 ) n n 1 o 3
o 1
n
o 3
0
1 o
n1
1 ( 1 2 3 )
Reaction rate (IUPAC Gold Book = rate of conversion )
m ax
r
d dt
d dt
i dt
m ole.s
1
dk
rk 0; 0
1 d ni
Closed system of uniform pressure, temperature and composition.
dt
t
Rate of generation (consumption) of component Ai
R Ai
d ni dt NR
R Ai
i.
ki . k 1
d dt
dk dt
i .r NR
ki .rk k 1
o n e reactio n
R = νr
several reactio n s
T
R = ν r
We measure usually the rates of generation (consumption) of components R and we want to calculate r
R
exp
T
ν r
O bjective function
r R
exp
T
ν r
R T
exp
T
ν r
m inim ization of r = least squares solutio n of overdefined system (N > N R )
r νν
T
1
νR
exp
Problem 1.4 Steam reforming of methane (5 species, N=5, 2 reactions, NR=2) 1 2 3 4 CH4 + H2O CO + 3H2 (1) 5 CO + H2O CO2 + H2 (2)
Measured R : (-0.97572, -2.88778, -1.02127, 4.832804, 2.078537)T (mol/s) 1 ν 0
1
1
3
1
1
1
0 1
r1 0.94619 r2 1.99546
12 T νν 3
3 4
νν T
1
ν
-0.10256 -0.02564 0.179487 0.230769 -0.07692 0.076923 -0.23077 -0.38462 0.076923 0.307692
Specific reaction rate d
The reaction rate dt is, like , an extensive property of the system, a specific rate (intensive property) is obtained by dividing d by the total volume, mass, surface of the system: dt
R eaction rate per vo l um e rV
1
r
V
rV , k
1 V
1 d V dt
rk
1 1 dn i V i dt
3
m ole.m . s
1
1 dk V
dt
R eaction rate per m ass rM
1 m
rM , k
r 1
m
1 d m dt
rk
1 dk m dt
1 1 dn i m i dt
1
m ole.kg . s
1
R eaction rate per surface rS
1
r
S
1
rS , k
1 d
S
S dt
rk
1 1 dn i S i dt
m ole.m
2
.s
1
1 dk S dt
R eaction rate per active center (turn over num be r) rR S
1 n RS
rR S , k
r
1 n RS
1 d n R S dt
rk
1 dk n R S dt
1 1 dn i n R S i dt
s
1
C entral proble m of A P P LIE D C H E M IC A L K IN E T IC S r function T , c1 , c 2 , . ..c N , P , catalytic activity, transport param e ters,... .
Rule 1: The rate function r at constant temperature generally decreases in monotonic fashion with time (or extent or conversion). Rule 2: The rate of irreversible reaction can be written as r k (T ) g c1 , c 2 , ...c N
Rule 3: The rate constant k depends on temperature (Svante Arrhenius, 1889): k (T ) A e
E
RT
Rule 4: The function g is independent of temperature: g c1 , c 2 , ...c N
c1
1
2
N
c 2 ...c N
N
i
ci
i 1
Rule 5: When a reaction is reversible: r r f rb k f (T ) g f c1 , c 2 , ...c N k b (T ) g b c1 , c 2 , ...c N
Problem 1.5 (homework 2) In flow catalytic reactor the synthesis of methanol is carried out CO(g) + 2H2(g) CH3OH(g) The inlet mass flow rate of CO is 1000 kg.h-1 of CO and the inlet flow rate of hydrogen is supplied so that the inlet molar ratio H2:CO is equal to 2:1. 1200 kg of the catalyst is placed in the reactor. The outlet mass flow of CO is 860 kg.h-1. To determine: 1. Mean reaction rate per mass of catalyst in mol.kg-1.s-1 . 2. If specific internal surface of catalyst is 55 m2.g-1, calculate mean reaction rate per surface of catalyst in mol.m-2.s-1. 3. If per 1 m2 of catalyst contains 1019 active sites, calculate mean reaction rate per active site in s-1. 4. Calculate inlet and outlet gas mixture composition in molar fractions. Data: MCO= 28.010 kg.kmol-1 NA=6.0221413x1023 mol-1 (Avogadro number)
C O A1 , H 2 A2 , C H 3 O H A3 1 1 ν 2 , R 2 r 1 1 NR
o
Fi Fi
Fractional conversion of key com ponent 1 key com ponent X1
&k Fi Fi i & o
ki
& &M A X
k 1
rM
& m CAT
Fi Fi
o
, rS
i m CAT
& S
Fi Fi
, rR S
iS
inlet 1-CO
F1
2-H2
2 F1
F
i 1, & rM rR S
o
o
o
y2
o
1.157 10
o o
3 F1 o 2 F1 3 F1
o
1
3
1
3 2
55 10 1200 10
i n RS
o
y1 y2
o F3 F3 1 & y 3
F F 2 & o
1.38839 m ol.s
3
55 10 1200
23 3
s
1
o F1 1 & o 3 F1 2 & o 2 F 2 & 1
3 F1 2 & & o
F1 F1
o
1
X1
o
o
F2 2 F1 2 F1 X 1 o
F3 F1 X 1
3 F1 2 & o
o
o
F 3 F1 2 F1 X 1
1
1
1.38839
1
1.267 10
yi
F2 F2 2 &
3
m ol.kg . s , rS
19
o
o F1 F1 1 &
1
1.38839 6.0221413 10 3
F1
0
3 F1
Fi Fi
outlet
o
y1
o
1
1200
o
860 1000 / 0.02801 / 3600
1.38839
n RS
yi
0
3-CH3OH
&
o
F Fi o F1o i o o i F1 X 1 / Fi Fi 1 i 1
8
2.104 10 m ol.m
2
.s
1
X1
1000 860 1000
0.14
inlet 1-CO
y1
2-H2
y2
F1
o
o
outlet
o
yi o
3 F1
o
2 F1 3 F1
o
o
1 3 2 3
y1
y2
yi
F1
o
1
X1
o
o
3 F1 2 F1 X 1 o
o
o
o
2 F1 2 F1 X 1 3 F1 2 F1 X 1
o
3-CH3OH
0
1
y3
F1 X 1 o
o
3 F1 2 F1 X 1
1
1 X1 3 2X1
2 1 X 1 3 2X1 X1 3 2X1
0.3162
0.6324
0.05147