修訂 | 576c2b8dd7db3ed275c39a6ddfdec5a813fc998e (tree) |
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時間 | 2008-10-01 19:42:33 |
作者 | iselllo |
Commiter | iselllo |
I simply removed a few typos from the manuscript.
@@ -142,7 +142,7 @@ | ||
142 | 142 | The code I emailed you is one of my research codes to solve Smoluchowski equation (a |
143 | 143 | population-balance equation useful to investigate agglomeration |
144 | 144 | problems). |
145 | -Consider a set of fundamental units, spherules hereafter called | |
145 | +Consider a set of fundamental units, spherules, hereafter called | |
146 | 146 | monomers. |
147 | 147 | A $k$-mer is an object formed by $k$ monomers. They can coalesce, |
148 | 148 | giving rise to a sphere, or retain their identity giving rise to more |
@@ -159,18 +159,21 @@ | ||
159 | 159 | \label{eq:volume_conservation} |
160 | 160 | \nu_k=\nu_i+\nu_j, |
161 | 161 | \end{equation} |
162 | -whereas the radius of the resulting sphere is given by: | |
162 | +where $\nu_k=k\nu_1$ is the $k$-mer volume (of course $k$ times the | |
163 | +one of a monomer). | |
164 | +The radius of the resulting sphere is given by: | |
163 | 165 | \begin{equation} |
164 | 166 | \label{eq:radius_final} |
165 | - R_k=(R_i^3+R_k^3)^{1/3}. | |
167 | + R_k=(R_i^3+R_j^3)^{1/3}. | |
166 | 168 | \end{equation} |
169 | + | |
167 | 170 | As time progresses, the total volume is conserved, whereas the overall |
168 | 171 | particle concentration, $N_\infty=\s_kn_k$, inevitably decays with time. |
169 | 172 | An initially monodisperse (i.e. consisting of monomers only) |
170 | 173 | distribution will evolve, as time progresses, and spread covering |
171 | 174 | larger sizes ($k\gg 1$). |
172 | 175 | The ``direct'' approach consists in binning the initial distribution |
173 | -with a uniform bin structure (i.e. where the $k$-th | |
176 | +with a linear (i.e. uniformly-spaced) bin structure (i.e. where the $k$-th | |
174 | 177 | bin contains $k$-mers only) |
175 | 178 | and solving Eq. \ref{eq:smoluchowski} in each bin. |
176 | 179 | This quickly |
@@ -182,9 +185,9 @@ | ||
182 | 185 | \label{eq:smolu_garrick} |
183 | 186 | \f{dn_k}{dt}=\f{1}{2}\s_{ij}\chi_{ijk}\beta_{ij}n_in_j-n_k\s_i\beta_{ik}n_k, |
184 | 187 | \end{equation} |
185 | -where $\chi_ijk$ is the so-called splitting operator re-distributing | |
188 | +where $\chi_{ijk}$ is the so-called splitting operator re-distributing | |
186 | 189 | the newly-created particles into different bins. |
187 | -This allows the use of non-uniform (typically logarithmically spaced) | |
190 | +This allows the use of a non-uniformly-spaced (typically logarithmically-spaced) | |
188 | 191 | bin structure, thus saving a lot of computational time. |
189 | 192 | |
190 | 193 |