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Multivariable approach to liquid level

Density-compensated leveling eliminates nearly all error.

By Dudley Harrelson and Jonathan Rowe

Tank liquid level measurement is the second most common application for differential pressure (d/p) cell measurement technology. The first is fluid flow rate measurement.

The accuracy of either the flow or level measurement is highly dependent on knowing one or more actual fluid densities. Let's look at multivariable transmitters with integral density computation for improved measurement of liquid level.

Differential pressure transmitters are common in the measurement of liquid level. Their relatively low cost, ready availability, ease of installation, and wide variety of materials, process connections, and pressure ranges make them popular for liquid level applications.

However, since the hydrostatic head generated by a column of liquid is dependent on the density of the liquid, the accuracy of liquid level inferred from a hydrostatic head measurement is dependent on the accuracy of the density used in the level calculation.

For applications that are tolerant of some degree of liquid level error, assumptions of liquid density can work. But that liquid level measurement will have errors proportional to the variation in actual density from the assumed density values.

A level computing solution that independently measures differential pressure, absolute pressure, and several temperatures could compute all the density values required. In order to do this properly, however, we would need multiple measurement devices with all their field wiring and I/O channels, plus sophisticated fluid computation algorithms.

Alternately, a single multi-variable transmitter with integral computational capability could provide greatly improved accurate level measurement despite variations in density caused by variations in pressure and temperature. This approach is much simpler and more robust than multiple devices and all the resultant controller or systems calculations.

Rid computational resources

Despite continued improvements in other technologies, differential head flow meters and level transmitters continue to dominate in many areas of industrial measurement, including hydrocarbon production and refining, chemical production, and electric power generation.

However, differential head flow measurements are inaccurate if only differential pressure is measured and density is not determined. Therefore, knowledgeable users have applied flow computing techniques to head flow measurement for decades, particularly when measuring valuable compressible fluids such as natural gas, ammonia, and steam. This approach often requires separate measurement and transmission of both fluid pressure and temperature and calculation of fluid density in a system or flow computer.

Multivariable transmitters have rapidly gained in popularity over the past few years. This is due to both more cost effective measurements and flow computation software to greatly improve the accuracy of flow rate measurements.

Multivariable transmitters continuously measure differential pressure, absolute pressure, and process and device temperatures. These multivariable transmitters employ a d/p cell transmitter form factor.

To date, the most common applications have been density-compensated mass or standard volume flow rate measurements achieved by using the transmitters with primary head devices, such as orifice plates, venturi tubes, or multipoint averaging pitot elements.

Multivariable transmitters with onboard computing are being widely deployed in order to achieve better accuracy at the point of measurement. In addition, they also increase reliability and reduce cost by eliminating devices and their process connections; as well as minimizing field wiring and I/O channels and rack space. Plus they eliminate the need for sophisticated computational resources in the panel, PLC, or control system. Note that for optimum accuracy, the density effect goes far beyond simple ideal gas law corrections.

Furthermore, the reduced cost of this distributed capability makes it more cost effective and practical to apply compensation to measurements engineers previously had to ignore because improved process measurement and control was unjustified.

Today, their use is also expanding due to the expanded use of digital busses such as Hart, Foundation fieldbus H1, and Profibus PA, all of which can digitally transmit multiple measurements simultaneously.

Now with the advancements discussed, these multivariable transmitters can also work for more accurate liquid level measurement. Onboard liquid and vapor density calculations, based on measured pressures and temperatures, can compensate the differential pressure measurement for varying densities. Calculating the density of many process liquids is possible using the measurements of pressure and temperature, providing an economical solution for improved hydrostatic liquid level measurement.

Performance better density

For a given acceleration of gravity, the liquid head in a tank or vessel generates a force per unit area or pressure (P) that is directly proportional to the liquid level (L) above the measurement point times the average density (r) of the liquid in the column. This means that L = P/r.

A differential pressure transmitter with one side vented to atmosphere can determine the liquid level in a vented tank.

A vented tank

While this formula (L = P/r) is simple, its usage can be complicated. Virtually all applications using pressure transmitters for liquid level include one or more of the following issues:

• Transmitter is not located at the zero level point.
• Transmitter is remote from the tank, above or below the primary pressure connection.
• Transmitter is isolated from process fluid with a flange or seal system.
• Tank is closed and, hence, subject to pressure or vacuum above the liquid.
• Fluid above the liquid may be the vapor of the liquid itself or an outside sourced fluid, such as a nitrogen blanket.
• Tank pressure reference connection is filled with a vapor (dry leg).
• Tank pressure reference connection is filled with liquid (wet leg).
• External wet legs can exist on both high and low pressure sides of the transmitter.
• Environmental conditions can be different for each of these external legs.
• Environmental conditions are usually different than tank conditions, e.g. a wet leg temperature might be very different from the in-tank temperature.
• Plus, of course, changes in liquid and vapor densities.

A fully defined application is here.

Fully defined application

Here, we need four density values for the transmitter, which calculate using measured pressures and temperatures.

They include:

• Density of the tank liquid, r1 at temperature T1 with pressure M2 on the liquid.
• Density of the tank liquid in the bottom external leg, r2 at temperature T2 with M2 pressure on the liquid.
• Density of the vapor above the liquid, r3 at temperature T1 and pressure M2.
• Density of the liquid in the upper external leg, r4 at temperature T4 with M2 pressure on the liquid.

The temperatures of the tank liquid and tank vapor are about equal within the accuracy limits expected.

Also, the pressure in the tank, M2, is calculated from the pressure measured by the multivariable transmitter less the hydrostatic head of liquid in the column between the top of tank and the transmitter (H3)¡¤(r3).

Temperatures T1, T2, and T4 can be user-assigned to any one of several sources. The multivariable transmitter has one external resistance temperature detector (RTD) input. The RTD can measure any one of these temperatures.

Recognizing that the external liquid legs are often at an ambient temperature that is similar to the temperature of the transmitter itself, the transmitter can configure to continuously use measured internal electronics or sensor temperatures for the liquid leg temperatures.

Finally, user-entered constant temperature values can also work in lieu of RTD, electronics, or sensor temperatures, if desired.

While there are many possible variables and issues to resolve in determining an accurate liquid level, usually the key variable to better performance is better density data. Furthermore, when the pressure in the tank is significant, the density of the overhead vapor can also be significant. Therefore, we also need to know or calculate the vapor density.

For many applications we can find the liquid and overhead vapor densities directly from the measured temperatures and pressures. These density calculations are virtually identical to the density calculations used in flow computing products, where pressure and temperatures are measured and then computation of mass or standard volume of flow is possible.

A variety of computation methods and standards work including the ASME (American Society of Mechanical Engineers) tables for common fluids such as water and steam, API (American Petroleum Institute) equations for hydrocarbon fluids, and ANSI (American National Standards Institute) and ISO (International Standards Organization) equations for other fluids. Also the calculations should allow for mixtures and user-entered custom fluid properties.

Accordingly, the transmitter configuration tools utilize physical properties tables for many common fluids and provide a means for user entry of the data for custom fluids.

The following equation shows one possible level calculation (L), based on a differential pressure measurement (?P), a pressure measurement and calculation (M2), and the individual density calculations.

L = [DP - (r3)(H3-H1-H2) - (r1)(H1) - (r2)(H2) + (r4)(H3)] / (r1 - r3)

This equation will vary according to transmitter orientation and the various possible combinations of open and closed tanks and wet and dry legs.

Independent temperatures

Density-compensated level, using multivariable transmitters, eliminates most of the errors that occur when uncompensated, single variable transmitters are used. The multivariable transmitter provides for continuous, dynamic calculation for up to four separate densities (tank liquid, tank vapor, external leg liquid between the tank and transmitter, and either vapor or liquid in an external leg from the top of tank to the transmitter, known as dry or wet legs, respectively).

The level error remaining after dynamic compensation for density is minimal. With the density error gone, the resulting level accuracy is primarily a function of the inherent accuracy of the differential pressure transmitter itself. This means the resultant level error is approximately the same as the error that would occur when using a conventional single variable transmitter on a level application that had no density variations.

The exact value of the compensated accuracy will vary depending on several conditions, including tank height and the measurement spans of the differential pressure and pressure measurements within the multivariable transmitter.

Depending on the transmitter selected, the measurement span, and a few secondary factors, level accuracies on the order of 0.1% of span are obtainable de7spite wide variations in densities of both the liquid and vapor. This accuracy is in marked contrast to the uncompensated accuracies described above in both the propane and boiler drum applications.

The successful implementation of multivariable transmitters for density-compensated level measurements also requires that the installation and configuration account for both open and closed (pressurized) tanks, both wet and dry legs, vapor density correction for both the tank and the dry leg, liquid density correction for both the tank and wet leg, independent temperature assignments for liquids in the tank and in each external wet leg (high- and low-pressure sides), and in the dry leg, when used.

Compensated liquid level

Here are two application examples from the propane storage business (first) and the power generation industry (second).

A propane storage vessel is a twelve foot (~3.5 M) diameter horizontal cylinder. The storage cylinder is outside and subject to atmospheric conditions of 0 to 110¡ãF (-25 to 65¡ãC). Plus direct solar radiation might raise the temperature to 150¡ãF.

A horizontal cylinder (not a sphere) with level transmitter

The nominal accuracy of compensated liquid level using a multi-variable, computing transmitter is +/- 0.1% of level. To illustrate the benefits of using multivariable transmitters for density-compensated liquid level, we will first examine the errors in liquid level that would probably exist if traditional single variable transmitters were in use without density compensation.

Level measurement performance of a single variable d/p cell transmitter on this application would come from the direct measurement of differential pressure and transmitter (ambient) temperature. Assumptions would include: tank temperature, tank pressure, and fluid composition.

Level measurement precision (or uncertainty) prediction would proceed along this line of logic.

Let's assume that for this particular application at design conditions, 75¡ãF, and that level measurement precision would be ~0.1% or 1/8 inch. We can then examine the amount of level error that would probably occur due to liquid and vapor density variations.

The environment can vary from 0-150¡ãF so performance is therefore:

• At 10¡ãF, the density of the stored LPG would be 33.58 lbs/ft₃, giving us a secondary inaccuracy of ~10% or 12.7 inches.
• At 100¡ãF, the density of the stored LPG would be 29.52 lbs/ ft₃, giving us a secondary inaccuracy of 7% or 7.8 inches.
• At 150¡ãF, the density of the stored LPG would be 26.18 lbs/ ft₃, giving us a secondary inaccuracy of 31% or 37.6 inches.

The power generation example is a 75 (75 inch, 2 meter) boiler drum. The drum is at saturated conditions and normally operates at 600 psig. Boiler drum pressure can vary plus or minus 25 psig from the set value. In addition, during start-up the drum pressure starts at close to atmospheric.

Boiler drum and level transmitter at a power generation facility

Level measurement performance of a single variable d/p cell transmitter on this application would derive from the direct measurement of differential pressure and transmitter (ambient) temperature. Assumptions would include drum pressure and isolation wet-leg temperatures.

The level measurement precision (or uncertainty) calculation would work as follows:

• At reference conditions, 600 psig, level would be accurate to ≈0.1inch.
• However, at 575 or 625 psig, level would increase to 1.1 inch or >3%.
• Plus during start-up or shut-down—very critical times—100 psig, level would be 3.1 inch off or >9% in error.

Finally, level measurement using hydrostatic, force, or pressure, technology has been in wide-spread use for decades. Until now, the level measurement was either marginally accurate or expensive based on density variation. Now, multivariable transmitters with internal dynamic density compensation can provide a cost effective means of greatly improving this tried and true level measurement technology. DT

Behind the byline

Dudley Harrelson and Jonathan Rowe both work for Foxboro division of Invensys. Harrelson is a technical specialist, and Rowe works as senior marketing specialist.