The process that I need to describe is a water atomization by using injectors of micrometric diameter. Starting from a fixed volumetric water flow rate, I have to spray the droplets inside an expanded liquid (supercritical CO2 + ethanol). I have to take into account the gravity, the buoyancy force and the dissipative attrition force. Are there any valid correlations to calculate the terminal velocity, in this thermodynamic conditions? Thanks a lor for help
Typically you have to write the equation of motion. In this case maybe you could consider the droplet as a sphere, so the dissipative force could be calculated as
-6 pi eta R v, being eta the viscosity of the surrounding fluid, R the radius of the droplet and v its velocity. The equation of movement could be written as
m dv/dt = Fg - FB - 6 pi eta R v,
where Fg id the gravity force and FB the buoyancy force.
Given that there will be, for enough large times, a terminal velocity, you can consider that the change of velocity is zero, this is dv/dt = 0. Now you can solve the equation for finding v.
Thank you for your really interesting answer. Actually I had thought to write a balance of forces, but I was thinking of avoiding the solution of a PDE with a boundary condition, but as I see it is indispensable to have a profile of velocity. Anyway I still have a question: are you sure that the acceleration of the droplet has the same sign of the dissipative force? If you put to the first member the "-6 pi eta R v" term, it becomes positive, as the acceleration of the droplet of water. Are you sure this is the sign?
No, you do not need to solve the differential equation, because when the terminal velocity is zero, then dv/dt=0. The resulting equation to solve is not differential.
About your second question, -6 pi eta R v is in the second member of the equation (right hand side), indicating that is opposite to the acceleration. I am pretty sure about thar sign.
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