# 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:

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

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 $$V$$

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

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:

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

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

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

and

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

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

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

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.