I need some help to understand how to use the continuity equation for the solution of a system where the density and the velocity changes all the time.

Im working on the simulation of a Downdraft gasifier. My partial differential equations include the mass balance per component in a solid fase and a gas fase, the energy balance and the momentum equation in two dimensions.

In my algorithm the idea is to solve the velocity profile first, then calculate the concentration and temperature using multigrid and then correct the velocity again.

The issue is that I don't know where to include the continuity equation.

Any ideas??

This is what actually CFD does and maybe you should first try to read some book before post such a question.

A simple, maybe not working, idea could be that of explicitly integrate every equation and so the continuity will give you an equation for the density. In a multiphase flow, with n components, n equations (one per phase) must be included. But there are a lot of issue connected with the time and mesh size and with your case.

Actually, in general, in compressible solvers the continuity equation is used to compute the density; in incompressible solvers it's used to derive a pressure correction equation that will force the velocity to a divergence free one

And how do you change the continuity equation to compute the density?

Hi, I did not understand what exactly u want but if u see at the continuity equation it is partial difference equation which means that the quantities are the change with respect to the parameters even in the discretized form so when u write are solve an pde it essentially means that u are solving the changed quantities

hope u understand