10.34, Numerical Methods Applied to Chemical Engineering
Professor William H. Green
Lecture #32: Kinetic Monte Carlo and Turbulence Modeling.
Thermal Relaxation
pi(E,t)
C+D
B
A
Figure 1. Energy diagram.
Branching Fraction:
One approach:
φ
B
C+D
(t ; T , P, p A (ε , t = 0)) ⇒
∫p
∫p
i=B
(ε , t )dε
i =C + D
(ε , t )dε
d
p i (ε ) = ... {track entire distribution}
dt
Second approach: Kinetic Monte Carlo (Gillespie) {track individual molecule}
initial conditions: i = A; ε, t
τ=
1
k isom ( E ) + k disc ( E ) + Z coll
t new = t − τ ln(rand ) = t − Δt
Pick rand2: if rand2 < kisom(E)τ
inew = B
Enew = E
else if rand2 < (kisom(E)+kdisc(E))τ
inew = C+D
else inew = iold
random ΔE Enew Å E + ΔE
Dilute in A so we can assume A does not interact with other molecules
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
Problem Set 11
water
organic
vitE
r• Æ vitE Æ vitC ÆÆ
vitC
Figure 2. A droplet of organic solvent containing
Vitamin E.
When a radical forms, Vitamin E picks up the radical. This radical is not as reactive. Some
diffuses to the surface where Vitamin C picks up the radical. Enzymes then remove the
radical.
Simplified model
RH
ROO•
O2
R· Æ ROO•
RH + O2 (+ROO·) Æ ROOH (+ROO·)
ROOH
inert Å 2ROO•
d[ROO·]
RO•
/dt = k[ ]
d[ROOH]
vitC
/dt = k…[ROO·]
Figure 3. A model of the reactions inside the droplet.
Track one droplet using Gillespie
NRH
N O 2 T Æ are constant
t NROO· NROOH
When one has a small droplet, some droplets have no ROO·. How small is small? When will
the continuum equation converge with the Gillespie algorithm? Your challenge is to write
down the equations.
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 32
Page 2 of 4
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
τ=
k RH + O
2
1
( N ROO ) + k ROOH ⎯⎯→ 2 ROO ( N ROOH ) + k 2 ROO ⎯⎯→ inert ( N ROO ) + k ROO ⎯⎯→ bye −bye ( N ROO )
(
⎯
⎯→ ROOH
The model: R· reacts with O2 to make ROO·. The ROOH makes 2ROO· over several steps.
Reactions are autocatalytic. The oxidation of the organic molecules leads to arteriosclerosis.
With small droplets, there may not be many radicals so one needs to keep track of single
molecules instead of simply using the mean value equations. This is where Gillespie’s
Algorithm applies.
A similar phenomenon occurs in emulsion polymerization where there may be only 1 radical
in a droplet.
Equation for Reacting Flow
probability density function (pdf)
f(V, Ψ; x, t)
V = velocity. Ψ = all other state variables
p(Ψ )
=
N solvent
G
∂f
∂ < p > ⎞ ∂f
∂
⎛
[ p(Ψ ) S n (Ψ ) f ] =
+ p(Ψ )V ⋅ ∇f + ∑ ⎜ p(Ψ ) g α −
+ ∑
⎟
∂t
∂α ⎠ ∂Vα
n =1 ∂Ψт
α = x, y , z ⎝
∂
∑
α = x , y , z ∂Vα
⎛
⎜
⎜
⎝
∑
β = x, y, z
∂τ βx
∂β
+
⎞
∂ ( p − < p >)
∂
V,Ψ f ⎟ + ∑
⎟ n ∂Ψт
∂α
⎠
⎛
⎜
⎜
⎝
⎞
∂J αn
V,Ψ f ⎟
∑
⎟
α ∂α
⎠
S.B. Pope, Prog. Energy Combust. Science 11(2) 119-192 (1985)
[Eq. 3.109]
Terms
∂f
⇒ Time dependence of probability density
∂t
G
p(Ψ )V ⋅ ∇f ⇒ Convection
p(Ψ )
∂ < p > ⎞ ∂f
⎛
⇒ Buoyancy and Pressure-driven flow
⎜ p(Ψ ) g α −
⎟
∂α ⎠ ∂Vα
= x, y , z ⎝
∑
α
N solvent
∑
n =1
∂
[ p(Ψ )S n (Ψ ) f ] ⇒ Chemical reaction source term
∂Ψт
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 32
Page 3 of 4
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].
∂
= x , y , z ∂Vα
∑
α
⎛
⎜
⎜
⎝
∑
β
= x, y, z
∂τ βx
∂β
+
⎞
∂ ( p − < p >)
V , Ψ f ⎟ ⇒ Reynold’s stress tensor and pressure
⎟
∂α
⎠
fluctuation: expectation value for turbulence
∂
∑n ∂Ψ
т
⎛
⎜
⎜
⎝
⎞
∂J αn
⎟ ⇒ Diffusive mixing
Ψ
V
,
f
∑
⎟
∂
α
α
⎠
10.34, Numerical Methods Applied to Chemical Engineering
Prof. William Green
Lecture 32
Page 4 of 4
Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall
2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD
Month YYYY].