Commit: 576c2b8dd7db3ed275c39a6ddfdec5a813fc998e - myprojects-hg-reborn (hg)

Commit MetaInfo

• Revision: 576c2b8dd7db3ed275c39a6ddfdec5a813fc998e (tree)
• Time: 2008-10-01 19:42:33
• Author: iselllo
• Commiter: iselllo

Log Message

I simply removed a few typos from the manuscript.

Change Summary

• modified: latex-documents/intro_smoluchowski.tex

Incremental Difference

diff -r 323ab84bc8b3 -r 576c2b8dd7db latex-documents/intro_smoluchowski.tex
--- a/latex-documents/intro_smoluchowski.tex	Wed Oct 01 10:41:42 2008 +0000
+++ b/latex-documents/intro_smoluchowski.tex	Wed Oct 01 10:42:33 2008 +0000
@@ -142,7 +142,7 @@
 The code I emailed you is one of my research codes to solve Smoluchowski equation (a
 population-balance equation useful to investigate agglomeration
 problems).
-Consider a set of fundamental units, spherules hereafter called
+Consider a set of fundamental units, spherules, hereafter called
 monomers.
 A $k$-mer is an object formed by $k$ monomers. They can coalesce,
 giving rise to a sphere, or retain their identity giving rise to more
@@ -159,18 +159,21 @@
   \label{eq:volume_conservation}
   \nu_k=\nu_i+\nu_j,
 \end{equation}
-whereas the radius of the resulting sphere is given by:
+where $\nu_k=k\nu_1$ is the $k$-mer volume (of course $k$ times the
+one of a monomer).
+The radius of the resulting sphere is given by:
 \begin{equation}
   \label{eq:radius_final}
-  R_k=(R_i^3+R_k^3)^{1/3}.
+  R_k=(R_i^3+R_j^3)^{1/3}.
 \end{equation}
+
 As time progresses, the total volume is conserved, whereas the overall
 particle concentration, $N_\infty=\s_kn_k$, inevitably decays with time. 
 An initially monodisperse (i.e. consisting of  monomers only)
 distribution will evolve, as time progresses, and spread covering
 larger sizes ($k\gg 1$).
 The ``direct'' approach consists in binning the initial distribution
-with a uniform bin structure (i.e. where the $k$-th
+with a linear (i.e. uniformly-spaced) bin structure (i.e. where the $k$-th
 bin contains $k$-mers only)
 and solving Eq. \ref{eq:smoluchowski} in each bin.
 This  quickly
@@ -182,9 +185,9 @@
   \label{eq:smolu_garrick}
   \f{dn_k}{dt}=\f{1}{2}\s_{ij}\chi_{ijk}\beta_{ij}n_in_j-n_k\s_i\beta_{ik}n_k, 
 \end{equation}
-where $\chi_ijk$ is the so-called splitting operator re-distributing
+where $\chi_{ijk}$ is the so-called splitting operator re-distributing
 the newly-created particles into different bins.
-This allows the use of non-uniform (typically logarithmically spaced)
+This allows the use of a non-uniformly-spaced (typically logarithmically-spaced)
 bin structure, thus saving a lot of computational time.