Control Engineering

Charlie Masi

Is the energy reaching a gamma-ray level sensor really inversely proportional to the level?
September 3, 2007

This question came up internally at Control Engineering during the preparation of an article on gamma-ray level sensors for the September 2007 issue, which states: “As the process level rises in the vessel, energy reaching the [gamma-ray] detector will decrease in an inversely proportional relationship to the level.”

Strictly speaking, the relationship is not inversely proportional, but an inverse, or decreasing, exponential. When looking through most fluids, however, it is correct to a very good approximation. To see why, we need to start with what happens when gamma rays—or any electromagnetic radiation for that matter—travels through a semi-transparent medium.

First, recognize that the article points out that the gamma-ray beam used in such level measurement applications is collimated. That means all of the rays are, to a very good approximation, parallel. Beam divergence (or spread) is negligible over the distance involved.

If you think about it, you’ll realize that the amount of energy (more properly power, but we’ll let that slide in the interest of not being persnickety) pulled out of the beam as it travels a distance Δx has to depend on the beam’s intensity I:

Doing all the usual math things to get a derivative, then solving for the constant gives:

A little Calculus 102 gives

where t is the distance the beam has to travel through the material.

Integrating and solving for I_t gives

So, strictly speaking, the function is a decreasing exponential.

Assuming that it is small compared to one, we expand it in a Maclaurin series (which is just a Taylor series expanded around zero) and keep only the first and second terms:

This is a linear function, so the gamma-ray intensity at the detector is inversely proportional to the fill level to within the linearization error.

Posted by Charlie Masi on September 3, 2007 | Comments (0)