Understanding and Expressing Measurement Uncertainties
associated with Thermodynamic Metrology(A primer)
Edward Brown
Quality Systems Lab
Boca Raton, FL 33487

Introduction
In Thermodynamic Metrology all too often
personnel are assigned to perform tasks withlittle or no experience. The importance ofsound temperature metrology was of little
interest or was not given the same respect as
DC Low Frequency or RF microwave. Overthe past 20 years temperature realization has
become more important due to the effect it hason other metrology disciplines. There aremany broad and some minute aspects to thisdiscipline. This paper is intended as a primer
to assist the new technician in understanding
thermodynamic metrology, determination ofmeasurement uncertainty and expression ofuncertainty SI units.
Measurement of Temperature
A variety of instruments can be used tomeasure temperature; LIG (liquid in glass)thermometers, RTD’s (resistive temperaturedetectors), thermocouples, SPRT’s (standard
platinum resistance thermometers) and a few
others. Most of these with the exception of
LIG’s need an external device to read thesignal from the sensing element.
Sources of uncertainty in this type ofarrangement are the sensor, the indicating
device, and in many cases, the connection
method.
Heat Source
All heat sources introduce measurement errors
as a result of their thermodynamic properties.
This measurement error should be quantified todetermine the heat sources contribution to themeasurement uncertainty. All laboratories
providing temperature calibration services
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maintain a variety of standards for realizationof temperature depending upon the complexity
or level of calibration being performed. Dryblocks, liquid baths and fluidized sand baths
are all used in the comparison method ofcalibration, where a calibrated SPRT iscompared to an unknown sensor. This method
is typically used by secondary laboratories
calibrating commercial temperature elements
and is the most common type ofthermodynamic calibration.
In the commercial laboratory, a less common
method is the use of ITS-90 (InternationalTemperature Scale – 1990) fixed point
apparatus. These devices are based on fixed,
known temperature points which are derived
from physical constants of nature. Properly
designed fixed point maintenance apparatus
and the associated fixed point cell will
maintain a known temperature over many
hours and in some cases days. The mostcommon fixed point is the triple point of water(TPW), which, when properly prepared canmaintain 0.01°C with better than .0001°C of
uncertainty.
There are many sources of uncertainty in a heatsource; axial uniformity, radial uniformity,loading effect, temperature stability, stem
conduction error, reference probe, and manyothers. We will discuss each of these briefly
and how to arrive at a combined measurement
uncertainty and an expanded uncertainty (k=2).

Loading effect
The number of probes impacts the amount ofheat drawn away from or into a bath or well.
This effect can be minimized by allowing
sufficient time for equalization and saturationof the work load before taking measurements.

Figure 1. Diagram of comparison calibration
Uncertainties associated with comparison
calibrations
The major sources of uncertainty are
uniformity, stem conduction, loading, bathinstability, reference thermometer accuracy,
and unit under test accuracy.
Uniformity
The vertical gradient in a bath or dry well istermed “axial uniformity”. The surface of abath or dry well is exposed to ambientenvironment and to a controlled temperature
along a portion of the vertical length. This
axial uniformity can be minimized be aligning
the reference probe and UUT sensing centers.
The horizontal gradient in a bath or dry well istermed “radial uniformity”. The bath or dry
well construction as well as the distance
between sensors and the heating source can alllead to radial measurement uncertainty.
Minimize this type of uncertainty by using areference sensor of the same diameter as theUUT and placing the sensors as close together
as possible.
Stem conduction error
Stem conduction is the parasitic loss of heatalong the length of the thermometer stem. Thisaffects both the reference and UUT.
Following the recommended depth forminimum thermometer insertion (20 X probe
OD) will minimize stem conduction error
resulting in very little uncertainty.

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Other Considerations
Drift
Control sensor and reference sensor drift willvary depending on the care and frequency ofuse of any bath or well. This drift can be
determined with regular calibration andintermediate checks.
Control Sensor
When using a bath or well without an externalreference the control sensor becomes anothersource of uncertainty.
Minimize youruncertainty by using a reference sensor
whenever possible.
Hysteresis
Hysteresis is the difference in indicatedtemperature dependant on the direction thebath or well approaches set point.
Uncertainty Budget
Any uncertainty budget should be an aggregateof the most important contributors from thepreviously mentioned sources of error. The
most common method is to use the “Type B”
approach. In this approach, the uncertainties
published in the instrument manuals are used.
Some of the values needed for an uncertainty
budget can be found on the calibration reportfor the instruments used. If uncertainties are
required at points between those expressed on
the calibration report, then rigorous calculating
methods are needed to propagate uncertainties.
The calibration lab can use the law ofPropagation of Uncertainties as described in
the GUM to achieve these uncertainties.
However, a less complex method forestablishing
theuncertainties
betweencalibration points would be a technique calledLinear Interpolation. In this technique weassume that the uncertainty changes linearly
between the points of interest.
Combining uncertainties
We must first convert to standard uncertainty
in order to evaluate the total uncertainty. At thesame time we must ensure that all values are
expressed in the same unit of measure, see “SIUnits” below.
There are two types of uncertainties;
uncertainties based on known or assumed
probability distribution and those based onlimits of error. When uncertainties are statedwith a known distribution with a coveragefactor of k=2, it means a normal, or Gaussian,distribution has been assumed. To convert thistype of uncertainty to a standard uncertainty
we simply need to divide by two. For otheruncertainties types such as limits of error that
wereassigned
without
probability
distributions, a rectangular distribution isassumed. To convert a rectangular distribution
to a standard uncertainty, the value is divided
by the square root of three. The individual
Table 1
UncertaintySpecification (°C)
Contributors @ 0.0°C
Stability.002
Loading effect.001
Radial uniformity.002
Axial uniformity.004
Reference probe.004
calibration
Reference probe drift,.002
including hysteresis
Reference probe stem.001
conduction
Thermometer readout.001
accuracy
Combined Standard Uncertainty (RSS method)
Expanded Uncertainty (k=2)

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standard uncertainties are then combined
through a process called Root Sum Square orRSS. This means that each of the standard
uncertainties is squared before adding all of thesquared components together. The square rootof the result is taken as the total combined
standard uncertainty.

U=S12+ S22+ S32...

The expanded uncertainty is obtained bymultiplying the resulting value U by a factor oftwo (k=2). This expanded uncertainty isshown in the uncertainty statement of scope ofaccreditation.

ExampleIllustrated below is a theoretical uncertainty
table for a metrology bath with a standard
thermometer as the readout. This example is at
0°C and does not take the UUT into
consideration.
Please remember this istheroretical, the math is sound the uncertainties
are without basis.
Probability
Distribution
Rectangular
Rectangular
Rectangular
Rectangular
Rectangular

Uncertainty (°C)

Rectangular.0011

Rectangular.0005

Rectangular.0005

.0011
.0005
.0011
.0023
.0023

.003868.007736

RoundingRounding should only be performed once allcalculations have been performed. As seen inTable 1 the expanded uncertainty is reflected to6 places after the decimal. This number can
now be rounded to a more practical value
before being expressed on a scope ofaccreditation or customer certificate ofcalibration. The final expanded uncertainty
should reflect the resolution of the laboratories
indicating device. In the example above therounded expression would be, ±. 0.008°C for athree decimal place digital display.
SI Units
When starting the process of determining
uncertainty, always convert non-standard units
to SI units. The Kelvin and degree Celsius are
the accepted units for expressing uncertainty.
This conversion should be performed before
any calculations are done. It should be notedthat Kelvin is not expressed as °K, the Kelvinis an absolute value of temperature and should
always be expressed as mK or K.
Note: 0°C = 273.16K.
Summary
There are many factors to take into
consideration when expressing total andexpanded uncertainties in thermodynamic
metrology. This paper is intended to give thenew technician some insight into thecomplexity of thermodynamic metrology. This
paper does not, by any means, cover thediversity and complexity of Thermodynamic
Calibrations in today’s metrology laboratory.
However, more extensive information can begleaned from websites such as NCSLI.org orNIST.gov.

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