This actually could be kind of complicated.
From how the problem is stated, I am assuming there is a liquid input, a solids input, a solids output, a liquids output, and a 2-phase solid-liquid mix inside the reactor.
For a given chemical component i:
dN_i/dt = x_i*L_in + y_i*S_in - r_i*V_mixture - z_i*L_out - w_i*S_out
where N_i is the total moles of i
x_i, y_i, z_i, and w_i are mole fractions
L_in is the molar flow rate of liquid into reactor (mol/s)
S_in is molar flow rate of solid into reactor (mol/s)
r_i is the reaction rate of component i (mol L^-1 s^-1)
L_out and S_out are analogous to their aforementioned counterparts.
V_mixture is the volume of the reacting medium. Presumably, it can be determined from simple level measurements in the tank. It is a function of L_in, S_in, L_out, and S_out.
N_i = rho_mixture*V_mixture*MW_i
where rho_mixture is the density of the mixture, which is a function of L_in, S_in, L_out, and S_out. I assume the reaction does little to change the density.
MW_i is the molecular weight of component i.
rho_mixture and V_mixture however, are function of the amount of liquid and solid in their content.
The best thing I can think of is to do some simple experiments in the lab with different amounts of liquid and solid to make a chart of the density as a function of liquid or solid mass %. Then you can take the density change into account and get the greater precision you seek.
Hope this works pal!