Optical Liquid-Level Sensing Eases Industrial Applications

Sensors Insights by Jonathan D. Weiss and Joseph S. Accetta

Laboratory measurements and optical modeling suggest that extremely simple optical sensors can be used to measure liquid level in an industrial environment. Two such sensors are discussed here: a hollow-tube sensor and a solid-rod sensor.

Introduction

The detection of liquid level is essential in many industrial environments, some of which are potentially explosive and electrically noisy. The potentially explosive nature of the environment may be due to vapors emanating from the liquid whose level is to be measured in the presence of oxygen. An aircraft tank is a good example. Unintended background electric and magnetic fields may be generated by nearby equipment or even atmospheric phenomena.

Optical sensors, such as those discussed here, have the advantage over their electrical counterpart of not producing sparks and of being immune to electromagnetic interference. These particular optical sensors have the further advantage of simplicity and the absence of any moving parts.

Additionally, the hollow-tube sensor is also uninfluenced by the refractive index of the liquid, for virtually all liquids likely to be encountered in practice. Although the solid-rod sensor does not possess this last advantage, in liquids of a certain range of refractive index, it does possess extreme sensitivity over a narrow range of liquid level. The discussion that follows relies heavily on ray-trace calculations* and limited laboratory measurements.

Basic Description of the Sensor

Four optical configurations of the sensor are shown in Fig. 1, with both modeling information and experimental data obtained on the first. The second and third were only modeled, and the fourth is a proposed generic single-ended configuration that would be more convenient in practice to deploy than any of the double-ended versions.

Fig.1: Four schematics of the optical liquid-level sensor, with simplified ray paths showing various reflections and transmissions. A: Hollow tube with input and output optical fibers inserted into the opening. B: Same as A, but with the output fiber covering the entire end face to collect any light remaining in the wall. C: Solid rod. D: Generic single-ended configuration, for ease of deployment.
Fig.1: Four schematics of the optical liquid-level sensor, with simplified ray paths showing various reflections and transmissions. A: Hollow tube with input and output optical fibers inserted into the opening. B: Same as A, but with the output fiber covering the entire end face to collect any light remaining in the wall. C: Solid rod. D: Generic single-ended configuration, for ease of deployment.

Because of the round-trip path of the light, the response of the double-ended configuration is equivalent to that of the others over twice their length and the height of the liquid level. The material of the sensor has been assumed to be nominally pure silica glass and the one that was fabricated was made of that material, but it could have been made of another clear material.

Regarding the second embodiment, it is the same as the first, but the output fiber, leading to a photodetector covers the entire output face of the tube, to transmit any light that emerges from the annulus at that end. The third is a solid rod, which means that optical guidance is possible, unlike the hollow tube for which only Fresnel reflections can occur. Alluded to in the introduction, we shall see that the response of the solid rod is quite different from that of the hollow tube.

Varying certain parameters: hollow-tube sensor

What are the likely modifiable influences on the response of the hollow-tube sensor to liquid level? The inner and outer diameters appear to be two because they influence the number of (lossy) reflections that a light ray undergoes from one end to the other. The refractive index of the surrounding liquid is expected to be another because it affects the transmission of light out of the sensor at the outer diameter. The refractive index of the glass or plastic material affects the strength of reflections at both the inner and outer interface. Finally, the launch condition of the light from the input fiber, in the form of the angular distribution of power is another potential influence. This launch condition is discussed in Fig. 2.

Fig. 2: Meaning of the angular distribution of power possible from each point at the input fiber to the sensor, and various such distributions modeled here. A result of the rotational symmetry of the fiber, these angular distributions of power are independent of the azimuth, φ.
Fig. 2: Meaning of the angular distribution of power possible from each point at the input fiber to the sensor, and various such distributions modeled here. A result of the rotational symmetry of the fiber, these angular distributions of power are independent of the azimuth, φ.

As shown in Fig. 2, the angular distribution of power is the power per solid angle per area emitted from the face of the input fiber. In general, it could be a function of θ, the angle to the normal to that face, the azimuthal angle φ, and position across the fiber face. In the case of a highly multimode fiber, with a substantial fraction of the possible bound modes excited within it, averaging takes place such that the angular distribution of power is essentially independent of azimuth and position. Since the fibers envisioned here have tens of millions of possible bound modes, this assumption of uniformity has been made in the modeling calculations.

The first distribution, F1(θ), is peaked at θ = 0 and gradually tapers off to zero at the numerical aperture (NA) of the fiber, assumed to be 0.5. This NA corresponds to a maximum output angle of π/6 or 30°, which is typical of plastic fibers. This distribution is easiest to obtain with an LED as the optical source and, based on previous experiments, is well-approximated by cos2 (3θ). The next two, F2(θ) and F3(θ), can be approximately achieved by more or less collimation of the optical source than the standard output of an LED. The second distribution, F2(θ) involves only rays emerging at angles up to half the limit imposed by the numerical aperture of the fiber.

The last one, F4(θ), can be approximately achieved by angling the source to the axis of the fiber, resulting in, ideally, no rays emerging at small angles to the normal. All of these functions result in different distributions of angles of incidence at the interfaces and thus different reflectivities at those interfaces. Those, in turn, influence the power propagated to the far end of the sensor.

Experimental Data

The laboratory data obtained on this sensor are shown in Figure 3, along with the experimental arrangement. The legend refers to the refractive index (at the sodium D-line) of the liquids whose level was measured.

Fig. 3: Experimental configuration and data for liquids of various indexes of refraction. The first angular distribution of power in Figure 2 should closely approximate the actual one from the input fiber.
Fig. 3: Experimental configuration and data for liquids of various indexes of refraction. The first angular distribution of power in Figure 2 should closely approximate the actual one from the input fiber.

The indexes were obtained by adding various amounts of sugar to water, starting at zero. The relationship between the sugar content of water and its refractive index is well known and it was used to calculate the index, which covered most of the range for common liquids. In addition, because the input fiber was coupled to the maximum output of a yellow LED, the angular distribution of power out of it was very close to F1(θ).

We note three characteristics of the data. The first is that the signal drops monotonically as any liquid level increases from zero to the length of the tube. This is the result of optical energy that is lost to the liquid, caused by the improved index matching between the material of the tube and the surrounding medium. The second is that the response curve is independent of the liquid index, although the curve for n = 1.39 does deviate from the others by a few percent. Since there is no fundamental reason why it should, it is assumed to have been caused by some unknown irregularity in the experimental procedure.

As to why there is little or no dependence of the response on index, the index matching between the silica and the liquid is good enough that the reflected energy at the outer surface is essentially zero for all of the liquids considered. Thus, only multiple reflections at the inner interface are relevant. However, since the index matching between air and the silica is not as good, those smaller-angle rays (referring to θ) which first impinged upon the inner surface above the liquid level undergo multiple reflections at the outer interface and contributed more to the signal. As the level rises, the ray angle at which this takes place gradually diminishes.

For liquid indexes much below that of water (~ 1.333), a dependence of the response on index must become discernible because the index matching diminishes. However, such liquids are not easy to come by. Cryogenic liquids such as oxygen (n = 1.22), nitrogen (n = 1.21), hydrogen (n = 1.10) and helium (n = 1.024) qualify, as does a high-pressure liquid like carbon dioxide (n = 1.20). A dependence on index will also appear for relatively high-index liquids starting at, say, 1.63, for the same reason. Carbon disulfide is an example.

A potentially useful consequence of the index match between the silica glass and liquids in the mid-range of index and the poorer match between the glass and air is that the presence of bubbles in the liquid will cause noise in the output signal if they attach and detach themselves from the outer surface of the sensor in a random manner. Noise, though at a lower level, will also be caused by optical reflection off of moving bubbles that are reasonably close to the outer surface. The noise may be a useful diagnostic concerning boiling or other phase transitions, cavitation, etc. in the liquid.

A third visible characteristic of the data is that the response is independent of whether the light originates from the top or bottom of the sensor. This occurs because the signal is basically a product of many reflection coefficients and such a product does not depend on their order.

Ray-trace calculations
Optical modeling allows us to examine how the response of the sensor depends on the various parameters mentioned under "varying certain parameters: hollow-tube sensor", without performing time-consuming laboratory experiments. Selected cases are presented here. In every such case, as was true experimentally, the tube length is equal to the maximum liquid level considered. In addition, for comparison purposes among the cases, the total power emerging from the input fiber is assigned a fixed value, regardless of F(θ).

Effect of Liquid refractive index
Fig. 4, for example, displays the results of a modeling calculation for a tube sensor similar to the real one in Figure 3. The data and calculation display a response that is independent of the liquid index and the two are of the similar shape.

Fig. 4: Model calculations of the response of an 18 cm (7") tube for several liquid refractive indexes. Independence of refractive index is clearly shown.
Fig. 4: Model calculations of the response of an 18 cm (7") tube for several liquid refractive indexes. Independence of refractive index is clearly shown.

Effect of wall thickness

The computer-generated sensor has a wall thickness of 0.5 mm, while the real one has a wall thickness of 1.0 mm. How important is wall thickness, for a given I.D.? Fig. 5 presents the calculated results for various wall thicknesses (0.25, 0.5, 1.0, 2.0 mm), at a fixed I.D. of 2.0 mm.

Fig. 5: Model calculations of the response of an 18 cm (7") tube for a liquid refractive index of 1.33, F<SUB>1</SUB>(&theta;), and several outer diameters. Inner diameter = 2 mm. Only the last curve ("full covering") refers to an output fiber covering the entire face of the tube, as opposed to just the core of the tube. Independence of wall thickness is clearly shown for a fiber inserted in the tube.
Fig. 5: Model calculations of the response of an 18 cm (7") tube for a liquid refractive index of 1.33, F1(θ), and several outer diameters. Inner diameter = 2 mm. Only the last curve ("full covering") refers to an output fiber covering the entire face of the tube, as opposed to just the core of the tube. Independence of wall thickness is clearly shown for a fiber inserted in the tube.

Aside from small undulations in the response, which are just numerical noise in the calculation, they tell us that the response is essentially independent of wall thickness over this range. Suppose that the output fiber covers the entire face of the tube, rather than just the hollow core? Will the results deviate significantly? The last curve in Figure 5 displays those results for the tube with the largest wall thickness. Some increase in the signal is seen. At zero liquid level, full covering results in a 7% increase in signal, while it monotonically drops to 2.7% at full liquid level. This is true, even though the wall area is eight times that of the core. Thus, the fraction of the total optical power carried in the wall of a reasonable thickness is very small and can be ignored.

Scaling of Dimensions

Fig. 6 is a comparison between sensor (2, 3 mm) and (6, 9 mm), although as we have just seen, it is the I.D. that is of primary importance.

Fig. 6: Model calculations of the sensor response for tubes of I.D., O.D. = 2, 3 and 6, 9 mm, and of three different lengths in cm (inch). F(&theta;) = F<SUB>1</SUB>(&theta;).
Fig. 6: Model calculations of the sensor response for tubes of I.D., O.D. = 2, 3 and 6, 9 mm, and of three different lengths in cm (inch). F(θ) = F1(θ).

Three lengths (17.8, 35.6, 53.3 cm) have been used for each sensor. In other words, the transverse dimensions have been scaled by three, while the sensor length scales by no more than that. The liquid is water and, as in the previous calculated examples, F(θ) = F1(θ).

For a given length and liquid level, the signal will be smaller for the sensor with the larger number of lossy reflections required for light to travel from one end to the other. Since the longitudinal advance of a light ray between reflections scales with the diameter, the larger signals are associated with the larger sensor. The largest changes in signal (but not the largest fractional change) with changes in liquid level are also associated with the larger sensor, presumably because there is more power associated with each ray that is lost to the liquid. This result suggests that the sensor be made as wide as possible, consistent with other requirements of a particular application.

As special case occurs, when the length of the larger sensor is three times the length of the smaller one, the effects of the length and diameter cancel and the two responses are expected to be the same when normalized signals are plotted against normalized liquid levels. This is shown to be exactly true in Figure 7, though the two calculations were performed completely independently of one another.

Fig. 7: Graph showing the perfect scaling between 2, 3, 17.8 and 6, 9, 53.3 in Figure 6. The normalized signal is the ratio of the signal to the signal at zero liquid level and the normalized liquid level is the ratio of the liquid level to the maximum liquid level.
Fig. 7: Graph showing the perfect scaling between 2, 3, 17.8 and 6, 9, 53.3 in Figure 6. The normalized signal is the ratio of the signal to the signal at zero liquid level and the normalized liquid level is the ratio of the liquid level to the maximum liquid level.

The last parameter to vary is the angular distribution of power from the input fiber to the sensor. That subject is covered in Figure 8 for a (2, 3 mm) sensor immersed in water.

Fig. 8: Model calculation of the response of 2, 3, 17.8 for all four angular distributions of power. The response with F4(&theta;), which eliminates all small-angle rays, is qualitatively different from the other three.
Fig. 8: Model calculation of the response of 2, 3, 17.8 for all four angular distributions of power. The response with F4(θ), which eliminates all small-angle rays, is qualitatively different from the other three.

Although the fractional change in signal is about the same for F1(θ) → F3(θ), the absolute value of the signal is largest for F2(θ), second for F1(θ), and smallest for F3(θ). This is also the order in which small-angle rays (here, θ ≤ π/12 or 15°) are represented in F(θ) as a fraction of the integrated angular distribution of power.

The small-angle rays generally make contact with the inner surface further along the tube, have a larger angle of incidence when they do, and thus suffer less loss as they progress toward the far end. The fractional change in signal is about 0.42. The signal associated with F4(θ) (see inset) is qualitatively and quantitatively different from the other three, even for a much shorter tube. Since small-angle rays are completely excluded from this distribution, the opposite argument applies here. They contribute almost nothing to the signal, but they contribute to much of the integrated angular distribution of power in F3(θ).

Solid Rod

As stated earlier, this configuration behaves very differently from the tube because, unlike the tube, it is a waveguide for liquid indexes below that of the rod. Its response to liquid level depends strongly on the index over that range. For an input fiber of NA = 0.5 and a rod having an index = 1.458, there is no response to liquid level for liquid indexes below about 1.37. This occurs because the NA of the rod/liquid waveguide has a higher NA than that of the input fiber. Thus, all of the light from the input fiber is guided, even when the rod is fully immersed in the liquid.

For indexes above 1.37, some of the larger-angle rays injected into the rod are fully refracted into the liquid after only a few reflections. The remainder of them continues to be guided. Thus, the response to liquid level is a rapid drop in signal followed by a constant signal. As the index of the liquid increases, the drop becomes more rapid and the constant signal diminishes. For a liquid index of 1.458, equal to that of the core, the constant signal is only that caused by those few rays reaching the far end of the rod without any contact with its wall. Curiously, the return of Fresnel reflections causes the signal to start up again for a liquid index above 1.458, but very slowly.

Fig. 9 illustrates the behavior of the rod sensor for two liquid indexes above 1.37. Of course, the size of the drop for n = 1.38 is so small that this sensor would not be practical, but it demonstrates the point.

Fig. 9: Response of solid rods (dia. = 2.5 mm, index = 1.458) and 17.8 cm long to liquids of index 1.42 and 1.38. F(&theta;) = F<SUB>1</SUB>(&theta;).
Fig. 9: Response of solid rods (dia. = 2.5 mm, index = 1.458) and 17.8 cm long to liquids of index 1.42 and 1.38. F(θ) = F1(θ).

For given liquid and rod indexes, the rapidity of the drop will diminish with the diameter of the rod. This sensor may be useful when high sensitivity is needed over a narrow range of liquid level. The parameters to vary here are the rod and liquid indexes, the rod diameter, and the NA of the input fiber.

Summary

Using some experimental data and optical modeling, we have demonstrated how a clear hollow tube can be used as an optical liquid level sensor. The dependence of its behavior on sensor dimensions, the refractive index of the liquid, and launch conditions of the light were investigated. The behavior of a solid rod as an optical liquid level sensor was also examined.

References:

*Peformed by Patricio Durazo of Breault Research Organization (Tucson, Arizona), using Breault's ASAP optics modeling code.

ABOUT THE AUTHOR
Jonathan D. Weiss, Ph.D. ([email protected]) is the Chief Technology Officer of JSA Photonics LLC. Joseph S. Accetta, Ph.D. ([email protected]) is the Chief Executive Officer of JSA Photonics LLC.

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