Page 378 - Numerical Methods for Chemical Engineering

Problems 367

iterations are terminated, as it is assumed to be very unlikely that the particle will find its
way back to stick to the aggregate. No site is then filled, but another particle insertion at R 1
is performed. After each insertion, compute the radius of gyration of the aggregate,

2
m,n ϕ mn (m − x c ) + (n − y c ) 2 x c = ( m,n ϕ mn m)/( m,n ϕ mn )
2
R = (7.274)
g
m,n ϕ mn y c = ( m,n ϕ mn n)/( m,n ϕ mn )
and stop the simulation when R g exceeds R end . Using spy, make a plot of the fractal geometry
of the resulting aggregate for the parameters
R 0 = 1 R 1 = 30 R 2 = 60 R end = 15 (7.275)
7.B.4. Consider an ideal polymer chain of N freely-jointed segments of length l. We wish to
simulate the evolution of the polymer chain under flow without accounting for any entan-
glements between chains (this is valid as long as the molecular weight is less than ∼1000).
We coarse-grain the geometry of the chain by treating it as a collection of N b “beads,” each
subsequent pair of beads being connected by a “strand” of Z = N/N b segments. The set
of SDEs that describe the motion of the chain under flow in a velocity field v(x)is
+
1 2k b T

T (c) (v)
d R v = v(0) + (∇v) · R v + F v dt + dW t (7.276)
ζ ζ
(v)
dW t is an independent vector of three Wiener processes for each bead. ζ is the drag
constant for each bead, and the total conservative spring force acting on bead ν ∈ [0, N]is
(c)
F = F(R v − R v+1 ) + F(R v − R v−1 ) (7.277)
v
where we use the finitely-extensible (FENE) spring law
−K Q 3k b T
F(Q) = 2 2 K = 2 Q max = Zl (7.278)
1 −|Q| /Q Zl
max
When the chain is stretched only a little, this force law agrees with the Gaussian model
of Problem 7.A.5. As the spring is stretched more, it becomes stiffer so that the chain can
never be extended beyond its contour length.
Write a MATLAB program that simulates and animates the motion of a polymer chain in
[1]
a shear velocity field v(x) = ˙γ x 2 e . For clarity in the animation, shift the center of mass
to the origin after each time step. Perform a number of simulations at varying shear rates
˙ γ> 0 with the parameters

N = 100 l = 1 N b = 10 k b T = 1 ζ = 1 (7.279)
Describe the nature of the motion and how the shape of the chain is altered.

7.B.5. Another approach to global minimization is the particle swarm optimization (PSO)
method (Kennedy & Eberhart, 1995). Let F(x) be the cost function surface. We simulate a
population of N particles using an algorithm that is meant to mimic the motion of a swarm of
animals in a search for food (www.swarmintelligence.org). Each particle α has a position
x [α] and velocity v [α] that are updated at each iteration. Let p [α] be α’s “personal best,” i.e.
the coordinates with the lowest F(x) that have been visited to date by particle α. Let g be