Calculate Flow Rate of Sand in Inclined System

[pterodox]

Hi. First of all, I should mention that I'm quite a novice in both physics and engineering.

Let's consider an hourglass (or a silo) that is inclined at a constant angle. I'm unsure of how to calculate the necessary diameter of an orifice for a given mass or volume of granular matter to discharge in a given time, considering the angle at which the system is held.

Searching the internet, I've found a paper describing an equation referred to as "Beverloo's law", which dictates that as long as the particle diameter is small enough in relation to the orifice, (Wikipedia claims that for an ideal flow, the diameter of the average particle (d_p) should be between 1/12 and 1/2 of the orifice's diameter - d_o ) the discharge rate is equal to
Cρ_b√g(d_o-kd_p)^5/2
...where C and k are described as "empirical coefficients of discharge and shape", and ρ_b is the matter's density.

Thanks to online engineering tables, I've found that the density of sand for example tends to revolve around 1.3 or 1.4 g/cm³ but I'm uncertain about finding values for C and k, and how significant they are from a practical point of view.
I'm still studying the effect of the bulb's inclination.

Any ideas/advice to put me on a right track? Thank you for the time.

PS: How do I use Latex on this board?

 

[Puma]

First the information about d of the sand particle being up to 1/2 of the orifice is not correct. You get bridges forming - like stonework bridges which I've noticed can be about 5 or 6 grains wide.

Second it really depends on the roughness of the sand, not all sand is the same and further more the humidity if there are hydroscopic salts within the sand will affect the results.

With tilting you get laminar flow so that the second layer tends to speed x, and the third layer tends to 2x approx - both moving at the same speed relative to the layer beneath them (excluding air resistance).

I would have thought physical testing would be the way to go here?