# Computation of forces of liquid sloshing inside a cylindrical tank

Hello,
I am trying to develop a formula to calculate the force of water sloshing against the inside of a cylinder.
While there has been a lot of development of baffles and structures designed to minimize the effect of sloshing liquids in tanks, we are doing some studies of liquids in small tanks without baffles.

Our tank is approximately 3’ long by 1’ diameter. It is placed horizontally. It contains 5 gallons of water. The tank is angled downward 3 deg. At the lower end there is a small cylindrical structure on top for a vent tube, baffle and filter.
Our test is to prove that the sloshing of the liquid in the tank will not reach the vent filter.

We will accelerate the tank along the horizontal at 15 ft/sec^2 for a time of 0.5-1.0 sec.

Now through simple V=a*t and D=v*t equations we know that the tank will move to a speed of 15 ft/sec and 15ft distance in 1 sec.

But how much FORCE will be required to accel (15 ft/sec^2) the tank held at 3 degrees with 5 gallons of water horizontally?
Assume for now no friction. We will be using very low drag linear bearings to support the tank in the track.
Dry weight of the tank is 24 lb
5 gallons water is 42 lb

F=m*a
a is known constant. However, “m”, the mass of the tank with the liquid is NOT constant. Your not pushing a 66 lb static block…

The tank is being accel’d out from under the liquid. The tank moves at first instant acceleration and the liquid shears to the end of the tank, it then “sloshes”at the tank end and a complex interaction of forces take place with momentum and the viscosity of the liquid, playing a large part.

The mass dynamically shifts…
Is there a formula derived or empirical to give me a estimate at least of the force required to push the tank at a constant accel? Taking into account the physical properties of the liquid?

This really isn’t a textbook problem. It’s a real world testing design problem.
I work for an aerospace company, and our tank is a lavatory waste tank, filled with, waste..
The 15 ft/sec^2 is an estimate of the decel of the plane as it lands, with a full tank and the pilot is surging (pumping) the brakes to slow down the plane.
The liquid in the tank sloshes forward, but the waste liquid cannot slosh so much that the vent filter on top of the tank is contaminated.