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This question is part of a larger investigation into specific machinery process rates.

I know the dimensions of the hopper, and thus the volume of the hopper (inverted frustum of a pyramid). However, I would like to know how to calculate the volume of a liquid in the hopper based on the level of the liquid within the hopper.

Hopper e.g.

  1. When the level of a liquid in the hopper is $75.1$ cm ($50\%$ in terms of distance between the top and the bottom of the hopper), what is the volume of the liquid?

  2. When the level of a liquid in the hopper is $95\%$ ($142.69$ cm in terms of distance from the bottom of the hopper), what is the volume of the liquid?

I would like to be able to input a formula into an excel spreadsheet to calculate this volume, and ultimately further integrate this data into the control system.

FYI By my calculations the volume of the hopper is $2.04198802$ $\text{m}^3$.

Shubham Johri
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J. Tucker
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1 Answers1

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For the side 208.1cm we have the side increased to at a height of h:

$$35.2 + \frac{208.1 - 35.2}{150.2}h = 35.2 + 1.1511h$$

Similarly for the other side :

$$34.2+ \frac{159.8 - 34.2}{150.2}h = 34.2 + 0.8362h$$

Hence at a height of H the volume is

$$\int_0^H(35.2 + 1.1511h)(34.2 + 0.8362h)dh$$

$$= 1203.84H + 34.4H^2 + 0.32H^3$$

To verify if H is 150.2 we have volume 2.041 $m^3$

KY Tang
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