# It floats boats

## Buoyancy technology has a long record of accomplishment in industry and its influence on variables is clear and concise

### FAST FORWARD

• Buoyancy-technology level sensors mechanically convert liquid level into force.
• Modern digital transmitters compensate for many ill effects.
• Given existing techniques, it is possible to track any liquid/liquid interface.

Since Archimedes formally described the underlying principle in the third century BCE, buoyancy has been available as an instrumentation tool for a respectable range of measurement applications.

Let’s review the basic theory, address the sizing of sensor hardware for an interface measurement application, discuss the major sources of measurement error, and suggest compensation techniques for error reduction.

Intelligent instruments with auxiliary inputs can implement these techniques to provide a reliable and accurate process indication over changing operating conditions.

Buoyancy-technology level sensors are transducers that mechanically convert a liquid level into a force as the first transformation in the measurement. The buoyancy produced is proportional to the volume of fluid displaced, and the density of the fluid.

F = L × ρ × V

Where:
L = Fluid level normalized to length of displacer, with reference to bottom of displacer
ρ = Density of fluid
V = Volume of displacer

The displacing element must have a uniform cross-section over its design range (vertical length) in order to provide a linear change of force with level change.

Buoyant force and interface

Every “level” measurement is actually an interface level measurement, as the upper portion of the displacer is also experiencing a buoyant force from its environment.

If the upper phase in the measurement chamber were a pure vacuum, the density of the upper fluid and its buoyancy contribution would be identically zero. If the upper phase is a highly compressed gas with high molecular weight, its buoyancy contribution may become significant.

In practice, we can usually ignore the upper fluid until the ratio of its density to the density of the lower fluid approaches the accuracy requirements of the measurement.

The relationship between buoyant force and interface level, when considering the upper phase density, is:

F = (ρupper + L(ρlower - ρupper))V

Where:
L = Position of interface, normalized to length of displacer, referenced to bottom of displacer
ρupper  = Density of upper fluid
ρlower = Density of lower fluid
V = Volume of displacer

 Change lever-arm length

Sizing of sensor hardware

Since the force change is the next signal, we must choose the displacer dry weight and the force sensor configuration so the net load measurement does not violate the bounds on the linear input range of the force sensor.

Displacers for standard buoyancy sensors are usually sized (by weight and volume) to use as much of the useful range of the force sensor as possible for an air/water interface at room temperature and atmospheric pressure. Operating at or near the design span (100% proportional band condition) keeps resolution and accuracy high.

As can be seen from the interface buoyancy equation above, the actual span of buoyant force can become quite small as the difference between the densities of upper and lower fluids diminishes. In order to regain some of the lost gain in the mechanical path:

• Displacer cross-sectional area (and thus, volume) can be increased within the constraints of the enclosure
• A lever arm length within the force sensor linkage can be increased in some constructions
• Force sensor spring rates can be modified in some constructions

After adjusting the mechanical gain, the dry weight of the displacer will again require adjustment to remain within the linear range of the force sensor. The degree of sizing modification that we ultimately adopt will depend on the cost versus accuracy requirements for a given application.

Sources of measurement error

When the differential density between upper and lower fluids is small, any uncertainty in the knowledge of the density of each of the fluids can represent a large percentage of span.

For example, if an instrument is calibrated for water at SG = 0.99 and oil at SG = 0.85, the calibrated span is represented by the differential SG of 0.14. An error of only 0.01 SG in the knowledge of the density of the oil results in a 7.1% span output error at the input condition where oil completely covers the displacer (0% interface level).

That output error will be proportionately ramped to zero as the input-interface level increases to 100%.

If the force sensor uses some form of mechanical spring, the spring rate will vary as a function of temperature. This effect will change the base mechanical gain of the system.

If calibration happens at ambient conditions, the gain change accumulated at operating temperature will result in a larger net error at one end of span versus the other, providing a significant zero-shift in addition to the span change.

When this effect is large, the sizing becomes more critical. More of the force sensor’s available input span has to be reserved for potential zero shift and span growth, to prevent exceeding the linear input range.

Sediment or coating buildup on a displacer can modify its weight and/or volume, resulting in span and zero shifts. When operating at higher gain, sensitivity to these parameters will be proportionally higher. Excessive sedimentation can eventually restrict displacer or fluid motion, resulting in gross errors.

Applications that allow a double interface to occur on a displacer (e.g. air/oil/water layers) result in an indeterminate equation. There is no longer a “functional” relationship between input and output. A large number of possible input conditions could produce a given output. At this condition, we can only compute worst-case bounds for whatever possible input conditions.

 Density differences at interface are important.

 With mechanical spring, there’s temperature dependence.

Compensation techniques

Modern multiple-input digital transmitters are well suited to monitor influence variables and apply appropriate compensation for many of these effects.

For applications with varying density, the dynamic correction of calibration is possible by adjusting the density parameters in the instrumentation algorithm.

If the fluid densities come from independently instrumented sources for some reason, we can feed them to the algorithm digitally, at a rate appropriate to their rate of change in the vessel.

If the fluid densities vary in a deterministic way as a function of other measurable process states, as do water and steam density in a saturated system, we can apply a tabular compensation, using process temperature and/or pressure as an input.

Modulus changes over temperature are well understood and documented for most spring alloys. It is possible to normalize the modulus versus temperature relationship to the value at the calibration temperature, and build a table for gain correction as a function of spring temperature change.

A combination of information from the modulus and density changes can compensate for the zero shift.

Developing diagnostic algorithms to test displacer buoyant force signals against known input conditions and compare them to initial commissioned values is possible. This approach will provide a means to detect sediment or coating buildup. It can also identify a collapsed, punctured, or disconnected displacer.

Independent high-low limit switches or knowledge of tank geometry can help to detect the presence of, or even identify the location of a second interface on a displacer. These types of information can transmit directly into the instrument algorithm to allow validation of the process measurement, or provide an additional independent equation to remove the indeterminate condition.

Batch and other applications

From the buoyancy equation and discussion, it is apparent that accurate interface level measurement using buoyancy technology requires:

• Accurate knowledge of fluid density
• The ability to size the sensor for sufficient sensitivity to resolve the equivalent density change between 0 and 100% conditions
• Knowledge of the force transducer performance versus temperature
• Adequate control over the system boundaries to validate the measurement
• Given the compensation and diagnostic techniques discussed, the following interface categories are good candidates for buoyancy technology measurement.
• Saturated steam/water up to approximately 700°F
• Oil/water
• Water/denser chemicals
• Compressed gas/liquids
• Any liquid/liquid interface, provided the mechanical gain can be adjusted sufficiently to meet the transmitter’s minimum input span for a given accuracy requirement and the fluid density can be accurately tracked

Types of applications that present interesting challenges for buoyancy include batch applications where the density of the fluid changes as a function of the progress of the batch process rather than of an easily instrumented parameter like temperature.

The desalter application, where the differential density is often quite small and the process is removing salt from one fluid and adding it to the other, is an application that would require independent density measurement to attain high accuracy.

Processes that allow high rates of precipitation of solids onto the sensor hardware can become expensive to maintain for many technologies. Buoyancy technology allows signal quality to degrade in a gradual manner for processes that develop a foam, or “rag” layer between two liquids.

The force signal will indicate a level somewhere within the extents of the rag layer. The value of the signal will have an uncertainty of about half the depth of the rag layer.

Alarm limits can be set based on knowledge of the process.

Buoyancy technology has a long record of accomplishment in industry, and its influence on variables is clear and concise.

Applying compensation for these influences through modern multi-variable digital techniques allows accurate, reliable application of buoyancy to a wide range of interface measurement applications.

Thomas W. Kirner (Thomas.Kirner@EmersonProcess.com) is a senior engineer at for Fisher Products, Emerson Process Management in Iowa. Nicholas Sheble (nsheble@isa.org) is InTech senior technical editor.

### TERMINOLOGY

BCE is “Before Common Era,” corresponding to BC in Christian terminology.

Elastic modulus, or modulus of elasticity, is the mathematical description of an object or substance’s tendency to be deformed elastically (i.e., non-permanently) when a force is applied to it (as a spring).

Gain is the ratio of output (signal) magnitude to input (signal) magnitude.

Desalter is a process unit in an oil refinery that removes salt from the crude oil. The salt is in the water in the crude oil, not in the crude oil itself. Desalting is usually the first process in crude oil refining.