Lecture 1: Reaction networks and critical effects

• Sudden change in behavior (bifurcation): bistability, periodic behavior, explosions

• When and why do we need a mathematical model?

• Model-based understanding and control• Examples at different scales

Reactor Safety

JOURNAL OF LOSS PREVENTION IN THE PROCESS INDUSTRIES, (2002)    15(3)   p. 213-222 

Acrylic reactor runaway and explosion accident analysis

On May 18, 2001 a destructive fire and explosion accident occurred in an acrylic resin manufacturing plant located in the northern part of Taiwan. More than 100 people were injured and totally 46 plants including 16 high-tech companies nearby were severely damaged. The resulting blast wave destroyed and shattered many large and small windows of residences within half-a-kilometer. The immediate cause turned out to be a vapor cloud explosion and the blast mass was estimated to be equivalent to 1000 kg of TNT. However, the original cause was found to be a runaway reaction of a 6-ton reactor that contained methyl acrylate, methyl alcohol, acrylonitrile, isopropyl alcohol, acrylic acid, methacrylic acid, and benzoyl peroxide. The investigation and experimental runaway results revealed that during the runaway, the temperature had risen rapidly from 60degreesC to about 170-210degreesC and the maximum temperature rising rate could reach 192 K/ min. Since the final temperature of the process was much higher than the boiling points of all the reactants, vapors generated inside the reactor were released to the atmosphere.

CO+1/2O2 ->CO2

CO+NO -> 1/2N2+CO2.

Negative order kinetics

21

: OnCO

PModel I r k

P

22

2 2

:(1 )

CO O

CO CO O O

P PModel II r k

K P K P

Langmuir, Hinshelwood, Ertl

200719561932

Langmuir-Hinshelwood Mechanism

Single crystal catalyst, Pd(111)

Temperature series of measured hysteresis loops between 370 and 490  K. The CO2 rate is shown as a function of the CO fraction Y in the gas flux to the surface.

The Journal of Chemical Physics, 7 February 2009J. Chem. Phys. 130, 054706 (2009)

Analysis of catalytic reactions by PEEM

Photoemission Electron Microscopes (PEEM's) record electrons emitted from a sample in response to the absoprtion of ionizing radiation. The electrons are accelerated by a strong electric field between the sample and the outer electrode of the objective lens, and the image is magnified hundred- or thousand-fold by a series magnetic or electrostatic electron lenses. An electron-sensitive detector records the electron emission.

UVe-

PEEM images the surface work function, which depends on the nature of the adsorbed species and coverage

2

1 1 3 1 1

22 3 2 2

1 1 3

22 3

1

1 1

Steady state analysis:

CO coverage: (1 ) '

O coverage: (1 ) '

CO: 0 (1 )

O: 0 (1 )

(1 )From the CO equation:

CO

O

dxk x y k x k xy k k p

dtdy

k x y k xy k k pdt

k x y k x k xy

k x y k xy

k yxk k

22 1 3 1 3 1 1

1 3

1

3

3

( )& (1 )

Substitute into O balance:

(1 ) (1 )( ) ( )( ) 0k y k k y k k y k

x k k yx

k

yk y

y y

k

k

1 2 2

3

_1 _ 2

' '

s 2 s 20

CO ; 2 2 O ; 2CO Ok P k P

ks s

k k

Model III

CO s O s CO O CO s

Parametric analysis of reaction rate as a function of k1

22 1 3 1 3 1 1 3

1

22 2 1 3

1 1 13

1

3

3

1

Cubic equation in :

(1 )( ) ( ) 0

Rewrite as a quadratic equation in :

(1 )( )( ) 0

The roots are of opposite signs. Need a positive roo :

1

t

y

k y k k y k k y k k k y

k

k y k k yk k

k k

k k y

y

k y

k

2

3

4 (1 )1

2

k yk y

Steady state analysis

k1

Limit PointsBistability can be analysed graphically, or numerically or formallyFor negligible desorption (k-1=0) we find one solution at y=0 and

two more solutions obtained from

2 3 1 3 12 (1 ) ( )k k y y k k y k- = +

Find the extremum of k1 by dk1/dy= 0. 2k2k3(1-2y)=k3+(2k1+k3) dk1/dy

This yields ylp and the boundary by substituting back into eq above

1 2 21 ' / 2 ' / 2lp CO Oy k P k P

3 3 3 31 2 2 1 3' / 2 ' 1 2 4 (1 ); ' /CO O COlp

k P k P K K K K k P k

The other boundary (wout proof) at k1=0, ylp=0

1 2 2' / 2 ' 0CO O lpk P k P

klp k1

y

ylp

ylp

Limit points (2)

Single crystal catalyst, Pd(111)

The Journal of Chemical Physics, 7 February 2009J. Chem. Phys. 130, 054706 (2009)

Dynamics

(0,1) unstable saddle point

(0.0542, 0.4513) stable node

(0.5220, 0.0372) unstable saddle point

(0.8838, 0.0015) stable node

11 2 30.00. 5, 1,5, 10k k kk

Summary

• Critical effects in simple physicochemical systems

• Sudden changes in reaction rate• Same operating conditions lead to multiple

regimes• Different regimes exhibit different parametric

dependence• This leads to critical effects• What about the stability of different steady

states? We will find out next time…