Analytical approximation to the fully developed concentration profile of diffusive aerosol particles in laminar flow in a circular tube

An analytical approximation is presented for the dimensionless concentration profile n(r,x) of aerosol particles of diffusion coefficient D   flowing with mean axial velocity u¯ in a cylindrical tube of length L and radius R,n(r,x)=nmax(x)f(r)=Cexp(−λβx)[1−111(18r2−9r4+2r6)]wherein C=1.477 and λ  =3.659 are fit constants to the numerical solution of the convection–diffusion equation for βx=DLx/u¯R2≥0.2, i.e. for fully developed concentration profile.