21.5.2Distance considerations

Placing a sensor at three integer distances (x, 2x, and 3x) from a radiation point-source results in relative power levels of 100%, 25% (one-quarter), and 11.1% (one-ninth) falling upon the sensor at those locations, respectively:

2x

3x

This is a basic physical principle for all kinds of radiation, grounded in simple geometry. If we examine the radiation flux emanating from a point-source, we find that it must spread out as it travels in straight lines, and that the spreading-out happens at a rate defined by the square of the distance. An analogy for this phenomenon is to imagine a spherical latex balloon expanding as air is blown into it. The surface area of the balloon is proportional to the square of its radius. Likewise, the radiation flux emanating from a point-source spreads out in straight lines, in all directions, reaching a total area proportional to the square of the distance from the point (center). The total flux measured as a sphere will be the same no matter what the distance from the point-source, but the area it is divided over increases with the square of the distance, and so any object of fixed area backing away from a point-source of radiation encounters a smaller and smaller fraction of that flux.

If non-contact pyrometers really were “looking” at a point-source of infrared radiation, their signals would indeed decrease with distance. The saving grace here is that non-contact pyrometers are focused-optic devices, with a definite field of view, and that field of view should always be completely filled by the target object (assumed to be at a uniform temperature). As distance between the pyrometer and the target object changes, the cone-shaped field of view covers a surface area on that object proportional to the square of the distance26 Backing the pyrometer away to twice the distance increases the viewing area on the target object by a factor of four; backing away to three times the distance increases the viewing area nine times:

Target object (e.g. furnace wall)

3x

So, even though the inverse square law correctly declares that radiation emanating from the hot wall (which may be thought of as a collection of point-sources) decreases in intensity with the square of the distance, this attenuation is perfectly balanced by an increased viewing area of the pyrometer. Doubling the separation distance does result in the flux from any given point on the wall spreading out by a factor of four, but the pyrometer now sees four times as many similar points on the wall as it did previously. So long as all the points within the field of view are uniform in temperature, the result is a perfect cancellation with the pyrometer providing the exact same temperature measurement at any distance from the target.

If the sensor’s field of view expands far enough to capture objects other than the one whose temperature we intend to measure, measurement errors will result. The sensor will now yield a weighted average of all objects within its field of view, and so it is important to ensure that field is limited to cover just the object we intend to measure.

26The field of view (a circle where the viewing “cone” intercepts the flat surface of the object) increases linearly in diameter with increases in distance between the sensor and the object. However, since the area of a circle is

proportional to the square of its diameter (A = πD2 or A = πr2), we may say that the viewing area increases with

4

the square of the distance between the sensor and object.

Non-contact sensor fields-of-view are typically specified either as an angle, as a distance ratio, or both. For example, the following illustration shows a non-contact temperature sensor with a 5:1 (approximately 11o) field of view:

Cable

The mathematical relationship between viewing angle (θ) and distance ratio (D/d) follows the tangent function:

A sampling of common field-of-view distance ratios and approximate viewing angles appears in this table: