Conservation of Mass

Conservation of Mass

”Let us consider a system consisting of \(n\) components amongst which \(r\) chemical reactions are possible. The rate of change of the mass of component \(k\) within a given volume \(V\) is:

\begin{equation} \label{dGM1}\frac{dM_{k}(t)}{dt}=\frac{d}{dt}\int^{V}\rho_{k}dV=\int^{V}\frac{\partial\rho_{k}}{\partial t}\,dV\\ \end{equation}

where \(M_{k}\) is the mass of the constituent \(k\), \(\rho_{K}\) is the density (mass per unit of volume) of \(k\). This quantity is equal to the sum of the material flow of component \(k\) into the volume \(V\) though its surface \(\vec{\Omega}\) and the total production of \(k\) in chemical reactions which occor inside \(V\)

\begin{equation} \label{dGM2}\int^{V}\frac{\partial\rho_{k}}{\partial t}\,dV=-\int^{\vec{\Omega}}\rho_{k}\,v_{k}\cdot d\vec{\Omega}+\sum_{j=1}^{r}\int^{V}\nu_{ki}J_{i}\,dV\\ \end{equation}

where \(d\vec{\Omega}\) is a vector with magnitude \(d\Omega\) normal to the surface and counted positive from the inside to the outside. Furthermore \(v_{k}\) is the velocity of \(k\) and \(\nu_{kj}J_{j}\) the production of \(k\) per unit volume in the \(j^{th}\) chemical reaction. The quantity \(\nu_{kj}\) divided by the molecular mass \(M\) of the component \(k\) is proportional to the stoichiometric coefficient with which \(k\) appears in the chemical reaction \(j\). The coefficients \(\nu_{kj}\) are counted positive when the components \(k\) appear in the second, negative when they appear in the first member of the reaction equation. The quantity \(J_{j}\) is called the chemical reaction rate of reaction \(j\). It represents a mass per unit volume and per unit time. the quantities \(\rho_{k}\), \(v_{k}\) and \(J_{j}\) occurring in (\ref{dGM2}) are all functions of time and space coordinates.

Fluxes in the r.h.s member of (\ref{dGM2}) can be, obbiously seen also at the integrated scale, where we have then:

\begin{equation} \frac{dM_{k}}{dt}=Q_{k}^{in}-Q_{k}^{out}+\sum_{j=1^{r}}\langle\nu\rangle_{ki}\langle J_{i}\rangle\\ \end{equation}

where \(Q^{in}\) are the mass fluxes entring the control volume and \(Q^{out}\) the fluxes exiting, \(\langle\nu\rangle_{ki}\) represent a spatially averaged value of \(\nu_{ki}\) and \(\langle J_{i}\rangle\) a spatially average reaction rate. The identification

\begin{equation} -\int^{\vec{\Omega}}\rho_{k}\,v_{k}\cdot d\vec{\Omega}=Q_{k}^{in}-Q_{k}^{out}\\ \end{equation}

and

\begin{equation} \langle\nu\rangle_{ki}\langle J_{i}\rangle=\sum_{j=1}^{r}\int^{V}\nu_{ki}J_{i}\,dV\\ \end{equation}

is quite obvious, and matter of convenient representation at the coarse grained scale.

A further specialisation of the mass budget is obtained when flows happens through networks. In this case

\begin{equation} \frac{dM_{k}^{(l)}}{dt}=A^{(lm)}_{k}Q_{k}^{(m)}+\sum_{j=1^{r}}\langle\nu\rangle_{ki}^{(l)}\langle J_{i}\rangle^{(l)}\\ \end{equation}

where the indexes \(l\) and \(m\) indicates two arcs (links) of the network, and \(A^{(lm)}_{k}\) is an adjacency matrix with entries \(a^{(lm)}=1\) when the matter flux of link \(m\) enters in \(l\), \(a^{(lm)}=-1\) when the matter flux of link \(l\) enters in \(m\), and \(a^{(lm)}=0\) otherwise.