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Obviously from the picture it seems the variation of concentration in the junction itself might be considerable that lead the authors to consider a spherical control volume at the junction. When you ignore the volume of the junction and consider it as a point and write the mass balance equation as an algebraic equation of balance of incoming and outcoming ...
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Mass is conserved always. That's the known fact. To show this holds true in convection-diffusion equation, I need to introduce material derivative to you. The material derivative of a scalar quantity as $C(\mathbf{r},t)$ is defined as:
$$\frac{D C(\mathbf{r},t)}{D t} = \frac{\partial C}{\partial t} + \mathbf{v} \cdot \nabla C$$
Where $\mathbf{v}$ is the ...
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