### Horizontally Placed Drums and Real-Time Flow Calculations

Drums and tanks are an integral part of any manufacturing industry. Textile, Dairy, Cement, Salt, Petroleum or fertilizers, they are used almost everywhere. Technology undergoes a change every day but our need for storing things or maintaining buffer quantities in industries will hardly be replaced. When we install a storage vessel anywhere, there is attached with it the problem of quantification. How much quantity will it hold? What will be the change in quantity with every delta change in its level? How will be the different flow and level measurements done etc?
While calculating the amount in vertical tanks and drums is often quite easy. Multiply the cross-section area with height or level of the liquid inside and there you go, you get the quantity inside it. All this is quite simple because the vessels are usually simple cylindrical bodies and so the quantity inside a vertically placed drum is a simple function of its height or level of liquid multiplied by a constant (cross-sectional area here). What makes things really complicated is when we place these cylindrical bodies horizontally. The volume at any point here is also equal to the cross-sectional area multiplied by length, but the question is which cross-sectional area to consider, as it is different at different levels. The below diagram may help to visualize the problem.  ## Volume Calculations

Suppose a drum (horizontal) has a level filled up to height x, shown in the diagram below. The radius of Cross-sectional circular area is considered as R and length of the drum is considered as L. The below diagram may make it clear (excuse the sketching skills :p) The volume of the liquid, filled here can be calculated by multiplying the cross-sectional area occupied by the liquid and the length of the drum. The area under consideration can be calculated in several ways, one of them being integration. We need to express the segment AB as a function of x and R and then integrate the function covering the variable length of AB from 0 to x. But that will be quite tedious and boring. Let's adopt a simpler way. The area occupied by the liquid can be thought of as an Area under sector OACB less the area of triangle OAB. Not going much into the detailed calculations, if we calculate the area of the sector, it would be equal to: The Area under the triangle ABC will be equal to : Now that we have areas of both the figures, the area occupied by the liquid can be easily calculated, given below: And the volume of the liquid, filled up to level x would be then: This looks quite an ugly equation but it correctly finds out the volume of the liquid, filled in a horizontally placed drum up to any level (x)
The above equation becomes more useful when in any industry we need to find out real-time outgoing or incoming flow and when the only instrument installed in the drum is meant to give us the level readings only.
The equation derived above can be programmed and by taking the real-time level readings, the flow incoming or outgoing can be calculated.
Also, the above equation is valid for the lower semicircular shape of the side view. It can be generalized by making certain adjustments or programmed in that way.
Thanks :)