I'd like to request for a reference in which I can look up for derivation of advection-diffusion equation that is applicable in a variable cross-sectional area.
$$\frac{\partial AC}{\partial t} = \frac{\partial}{\partial x}D\frac{\partial C}{\partial x} - \frac{\partial (vAC)}{\partial x}$$
C is the concentration, A is area, v is velocity and D is diffusion
The advection-diffusion equation in $n$ dimensions is $$ \frac{\partial C}{\partial t} + \nabla \cdot (\mathbf{v} C) = \nabla \cdot (D \nabla C). $$ You can find the derivation of this equation pretty much anywhere, but it comes from the balance of the fluxes in an infinitesimal volume.
If we integrate this equation in the volume between two sections of the duct (I hope you can visualize that), we have $$ \int_V \frac{\partial C}{\partial t} \mathrm{d}V + \int_V \nabla \cdot (\mathbf{v} C) \, \mathrm{d}V = \int_V \nabla \cdot (D \nabla C)\, \mathrm{d}V. $$ Let's look at the advective term. Using the divergence theorem, we write $$ \int_V \nabla \cdot (\mathbf{v} C) \, \mathrm{d}V = \int_{\partial V} \mathbf{v} C \cdot \mathbf{n} \, \mathrm{d}A = \int_{S_1} \mathbf{v} C \cdot \mathbf{n} \, \mathrm{d}A + \int_{S_2} \mathbf{v} C \cdot \mathbf{n} \, \mathrm{d}A + \int_{S_3} \mathbf{v} C \cdot \mathbf{n} \, \mathrm{d}A, $$ in which $S_1$ is one section, $S_2$ is the other section and $S_3$ is the wall of the duct. Look at what happened: we got rid of the divergence operator and expanded the surface integral into three sections.
If the walls of the duct are impermeable, there is no flow in the direction normal to the wall, and therefore the third term is $0$. Now we assume that $C$ and $\mathbf{v}$ are uniform along each section, and the integrals simplify to $$ \int_V \nabla \cdot (\mathbf{v} C) \, \mathrm{d}V = \left[ v C A \right]_2 - \left[ v C A \right]_1. $$ Now, if the sections that define the volume are close, we can write this difference as a derivative, $$ \int_V \nabla \cdot (\mathbf{v} C) \, \mathrm{d}V = \frac{\partial}{\partial x} (v C A) \Delta x, $$ in which $\Delta x$ is the distance between the sections. See that this derivative is similar to the divergence operator in the original equation. We transformed the equation to the integral form, but now we are back to the differential form.
If we do the same thing to the diffusion term, assuming that there is no flux of $C$ to the wall (e.g., if $C$ is temperature then the wall is adiabatic), we have $$ \int_V \nabla \cdot (D \nabla C)\, \mathrm{d}V = \frac{\partial}{\partial x} \left(D \frac{\partial C}{\partial x} A \right) \Delta x. $$ This hypothesis (of no flux of $C$ to the wall) is not always valid. If you want to relax this assumption you will need to include a term associated to the flux through $S_3$.
Finally, the transient term is $$ \int_V \frac{\partial C}{\partial t} \mathrm{d}V = \frac{\partial}{\partial t} \int_V C\, \mathrm{d}V. $$ If this volume is very thin, $C$ is pratically constant inside it, and the volume integral is $C A\Delta x$. Therefore, $$ \int_V \frac{\partial C}{\partial t} \mathrm{d}V = \frac{\partial}{\partial t} (AC) \Delta x. $$
Therefore, the equation is $$ \frac{\partial}{\partial t} (AC) + \frac{\partial}{\partial x} (v C A) = \frac{\partial}{\partial x} \left(A D \frac{\partial C}{\partial x} \right). $$
I find it unlikely that the duct shape will be varying with time. In that case, you can take $A$ outside the time derivative: $$ A \frac{\partial C}{\partial t} + \frac{\partial}{\partial x} (v C A) = \frac{\partial}{\partial x} \left(A D \frac{\partial C}{\partial x} \right). $$
As a final comment, notice that the equation in the question lacks an $A$ in the diffusive term. The equation derived here corrects this.