# Understanding the Additional Measurement Error a Specific Monitoring Device Would Add when Performing Square Root Density Correction for Rotameters

Note: The below assumes it is clearly understood why rotameter monitoring is done and how to use the monitoring properly within the system.  The information contained below is also not a full uncertainty analysis but is typically sufficient enough get a better idea of the magnitude of potential influence.

Pressure and temperature affect the flow output of volumetric devices like rotameters as it changes the gas density. The formula for rotameter square root density correction is: The scale printed on rotameters represents what the flow would be if the rotameter was used at some previously defined pressure and temperature determined by the rotameter manufacturer, often referred to as the “design conditions”. Since these conditions will likely not be present during calibration each needs to be monitored directly at the rotameter and corrected to the design conditions. Atmospheric pressure varies significantly with location (altitude, longitude, latitude) so pressure will likely be much different. There might also be back pressure on the rotameter rather than exhausting to atmosphere. If not in a controlled environment the temperature could be very different. Any of these situations and several other factors cause significant density changes and is thus why they must be monitored rather than make assumptions. If conditions are not monitored and/or this correction is not applied significant flow errors will likely occur.

Once the need to monitor these conditions is understood the next question generally asked is, “How accurate of a pressure and temperature monitoring device do I need to purchase for the square root density correction”. The real question is, “How much total error is acceptable when combined with the reference error”.

We don’t want to simply add in the accuracy of the pressure and temperature monitoring devices selected. One reason for this is the output falls under the square root in the formula that is used. The other reason is when doing this type of analysis each component is generally root sum squared (RSS) as each will very likely not be at worse case at the same time.

To determine the error a specific pressure and temperature monitoring device might contribute, what we are actually trying to do is determine what the change in gas density would be if the monitoring device was reading perfectly vs. at either end of its spec limit. The easiest way to calculate this is to assume the design conditions are the exact same as the operating conditions.  This assumption is not typical of the true operating conditions and is only being used for math purposes so that we don’t need to calculate the density at another pressure and temperature. If we plug in the same design density as operating density into the formula above it becomes a multiplier of 1. Now we just need to calculate what the difference from 1 would be if the devices were operating at their spec limits using the below slightly modified formula: An easy to use tool to analyze the pressure and temperature affects on density is the Unit of Measure Converter. This comes packaged with COMPASS software.

First, we need to calculate design density. In this example we will use conditions of 14.7 psi and 70F in N2 as this is a common design condition. Design density calculation in this example is 1.16075 kg/m3.

For the pressure monitoring device in this example, we will analyze a Fluke Pressure Module with uncertainty of ±0.07% of reading and its affect on density. The monitoring device should be of the same range with absolute pressure output.

When we factor in the pressure uncertainty of the .07% of reading device at 14.7 psi it calculates out to ±0.01029 psi. So, worse case this device should measure this pressure as 14.71029 psi or 14.68971 psi at its specification limits. For math purposes, we will use the low side specification reading of 14.68971 psi to see what the change on density would be: The density changed from 1.16075 to 1.15993 kg/m3.

If we now plug these two densities into the formula we can see the potential pressure error affect on flow: This pressure monitoring device adds a potential ±0.0353% error on the final flow value.

For the temperature monitoring device in this example, we will analyze a Fluke 1523/1524 thermistor that has an accuracy of ±0.002C (0.0036F) and its effect on density. A thermistor is recommended in most cases as their temperature range is generally much closer to what rotameters typically see so they are more sensitive and have a faster response time than PRTs.

When we factor in the temperature uncertainty of 70F with ±0.0036F accuracy, worse case this device should measure this temperature as 70.0036F or 69.9964F at its specification limits. For math purposes, we will use the high side limit of 70.0036F to see what the change on density would be: The density changed from 1.16075 to 1.16074 kg/m3.

If we now plug these two densities into the same formula we can see the potential temperature error affect on flow: This temperature monitoring adds a potential 0.0004% error on the final flow value.

Our final step is to RSS (Root Sum Square) all three of the involved uncertainties on flow to give us our total potential error. The molbox/molbloc system has a standard uncertainty of ±0.2% reading, the rotameter pressure monitor is ±0.035% of reading, and rotameter temperature is ±0.0017% of reading. The following formula is used: Our final amount of measurement error in this example is about ±0.20309% of reading when all three factors are combined.

Note: If less additional uncertainty is desired consider a Fluke Calibration RPM4 for the pressure monitor at ±0.01% of reading which produces only a ±0.005% error on the final flow correction. When combined with the same Fluke 1523/1524 thermistor for temp monitoring, total RSS would be only ±0.20006% of reading.

Tip: It is highly recommended to use COMPASS for Flow when calibrating rotameters. Performing the square root density correction is critical for proper measurements and the proper way to do this is real time monitoring during each flow set point and measured directly at the rotameter. Using COMPASS for Flow is the only way have this correction applied to the data at each point and avoids having to run the data through calculation spreadsheets or similar