Solids Transportation Calcs 1

[samlukeben]

Can anyone help?
I'm trying to determine the minimum velocity at which sand will begin to flow with the carrying fluid. I have access to all the necessary data e.g particle size distribution, densities, viscosity of carrying media, temperatures, pressures etc etc...and fluid velocity and flow regime.The pipe will be in the vertical position.

can anyone direct me to a general formula?

Many thanks in advance.....

 

[m777182]

There are different ways to get the minimum transport velocity.I shall describe the one that I know: it starts considering the minimum fluidization velocity. When you know it then you are quite near to your solution: you have to expand your bed by increasing velocity.
One of the ways to get the minimum fluidization velocity is based on graphs of settling Reynolds number and Archimedes number, Re_settling=f (Ar) for a given particle and particle and fluid properties (densities, viscosity, paricle_dia, ...).From Re_settling you can calculate the velocity at equilibrium, when a particle does not sink nor move upwards. With increasing velocity it will become fluidized.
There is as well quite a number of empirical equations describing the minimum velocity needed for fluidization; some of them are in implicit form:

Re=Ar/[1400+5,22*sqrt(Ar)]
or
Ar=1406*Re_min_fluid+27,3*(Re_min_fluid)^2
or similar
Ar=1650*Re_min_fluid+24,5*(Re_min_fluid)^2
the other one claims
Re_min_fluid=33,7*[sqrt(1+3,59E-5*Ar)-1]
or
Ar=150*(1-epsilon)*epsilon^2*Re_min_fluid + 1,75/epsilon^2*Re_min_fluid
or again in terms of epsilon(=porosity of the bed):

Re_min_fluid=sqrt[1830+0,571*epsilon^3*Ar/epsilon^2]-42,8

Fluidization occurs at epsilon=0,6....0,8, at epsilon =1 you will get complete transport.With a certain particle size distribution you have to calculate with a sort of average particle diameter, sometimes called equivalent dia.It is calculated as

equivalent_dia=1/summa(w_i/dia_i), where w_i is a weight fraction of particles at diameter_i)

From all these equations you should explicitely get the minimum fluidization velocity, u_min_fluid. In some equations it is already explicitely given:

u_min_fluid=420 *particle_density*particle_dia^2
or
u_min_fluid=fluid_kinem_viscosity/(fluid_density*particle_dia)*[sqrt(1135,7+0,0408*Ar)-33,7]

There will be significant differences among calculations by one or the other equation that describe fluidization velocity. And there will be a range of velocities for particular particle distributions.
Here I have two equations that describe the entrainment velocity:

Re=Ar/(18+0,61*sqrt(Ar)) and

Re= - 42,8 + sqrt[1830+0,572*epsilon^3 /(1-epsilon)^2 * g* particle_dia^3/ fluid_kinem_viscosity*(particle_density-fluid_density)/fluid_density

I am sure there are many more, but somewhere you have to start.
Regrettfully I cannot give you literature citations at the moment.If needed I can manage it later.
regards
m777182