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NIST Technical Note 1606

Assessment of Uncertainties for the NIST 1016 mm Guarded-HotPlate Apparatus: Extended Analysis for Low-Density FibrousGlass Thermal Insulation

Robert R. Zarr

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2009

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National Institute of Standards and Technology • US. Department of Commerce

NIST Technical Note 1606

Assessment of Uncertainties for tlie NIST 1016 mm Guarded-HotPlate Apparatus: Extended Analysis for Low-Density FibrousGlass Thermal Insulation

Robert R. Zarr

February 2009

U.S. DEPARTMENT OF COMMERCE

Carlos M. Gutierrez, Secretary

NATIONAL INSTITUTE OF STANDARDS AND TECHNOLOGY

Patrick D. Gallagher, Deputy Director
Certain commercial entities, equipment, or materials may be identified in this document in order to describe an

experimental procedure or concept adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the entities, materials, or equipment are necessarily the best available for the purpose.

National Institute of Standards and Technology Technical Note 1606

Natl. Inst Stand. Technol. Tech. Note 1606, 48 pages (February 2009)

CODEN: NSPUE2
Mahn H. Hahn

In Memoriam

This technical note is dedicated to the memory ofthe late Dr. Mahn Hee ("Manny") Hahn

whose vision, enthusiasm, and commitment advanced innovative ideas on guarded-hotplate technology into use as a new generation of apparatus at the National Institute of

Standards and Technology (NIST). Mahn Hee Hahn received his Doctor of Engineering

from Catholic University in 1972. His dissertation, entitled "The Line Heat Source

Guarded Hot Plate for Measuring the Thermal Conductivity of Building and Insulation

Materials, " became the blueprint for the new plate designs. Under contract with the

National Bureau ofStandards (NBS), Dr. Hahn developed the prototype 305 mm line-heat- source guarded-hot-plate apparatusfrom a detailed mathematical analysis to a completed

engineering design. The apparatus was fabricated in the NBS machine shop under his

supervision. From 1979 to 1981, as a project leader on the NBS thick insulation project

tester. Dr. Hahn proceeded to develop and construct the second generation line-heat

guarded-hot-plate apparatus at NBS. The 1016 mm line-heat-source guarded-hot-plate

apparatus was commissioned in 1981 for the development of thick thermal iiisulation reference materials. The apparatus has been in continuous operation since 1981 mid has

subsequently been vital in the development of five thermal insulation NIST Standard

Reference Materials and over 100 individual calibrations of thermal ijisulation transfer

specimens. After several years ofeffort, the line-heat-source technology has been adopted

as a standardized practice in conjunction with the North America guarded-hot-plate

standard. In the late 1990s, Mannyjoined another NIST team to develop and construct a 500 mm guarded-hot-plate apparatus for extended temperatures and operation wider

controlled atmospheres. Manny 's continued involvement and contributions over the years

have been essential in supporting and advancing the guarded-hot-plate apparatus at NIST.

Ill
IV
Abstract

An assessment of uncertainties for the National Institute of Standards and Technology (NIST)

1016 mm Guarded-Hot-Plate apparatus is presented. The uncertainties are reported in a format

consistent with current NIST policy on the expression of measurement uncertainty. The report

describes a procedure for determination of component uncertainties for thermal conductivity and

thermal resistance for the apparatus under operation in either the double-sided or single-sided

mode of operation. An extensive example for computation of uncertainties for the single-sided

mode of operation is provided for a low-density fibrous-glass blanket thermal insulation. For this

material, the relative expanded uncertainty for thermal resistance increases from 1 % for a

thickness of 25.4 mm to 3 % for a thickness of 228.6 mm. Although these uncertainties have been

developed for a particular insulation material, the procedure and, to a lesser extent, the results are appUcable to other insulation materials measured at a mean temperature close to 297 K (23.9 °C,

75 °F). The analysis identifies dominant components of uncertainty and, thus, potential areas for

future improvement in the measurement process. For the NIST 1016 mm Guarded-Hot-Plate

apparatus, considerable improvement, especially at higher values of thermal resistance, may be

realized by developing better control strategies for guarding that include better measurement

techniques for the guard gap thermopile voltage and the temperature sensors.

Keywords

building technology; fibrous glass blanket; guarded hot plate; thermal conductivity; thermal

insulation; thermal resistance; uncertainty
VI
TABLE OF CONTENTS

PAGE

Memoriam iii Abstract v

List of Tables viii List of Figures viii L Introduction 1 2. Reference Material 1 3. Steady-State Thermal Transmission Properties 2

4. Measurement Principle 2

Double-Sided Mode 2

Single-Sided Mode 4

5. Apparatus 5

Guarded Hot Plate 6

Meter-Plate Electrical Power 6

Cold Plates 6

Environmental Chamber 7

Primary Temperature Sensors 7

Temperature Sensors in the Guard Gap 8

Temperature Control 9

6. Measurement Uncertainty Estimation 9

Terminology 10

Procedure 11

7. Mathematical Process Model 12

8. Sources of Uncertainty 12

9. Quantification of Uncertainty Components 14

Meter Area (A) 14

Thickness (L) 16

Temperature Difference (AT) 20

Heat Flow (0 22

10. Calculation of Combined Uncertainty 33

Combined Standard Uncertainty and Expanded Uncertainty for Xexp 33

Combined Standard Uncertainty and Expanded Uncertainty for R 36

11. Reporting Uncertainty 38

12. Discussion 39

Dominant Uncertainty Components 40

Dominant Contributory Sources 41

Comparison with Previous Error Analysis 42

13. Summary 43

Recommendations for Future Research 43

14. Acknowledgements 45

15. References 46

Appendix A 48

Vll
LIST OF TABLES

PAGE

Table 1 Steady-State One-Dimensional Thermal Transmission Property Equations 2

Table 2 List of Uncertainty Sources for X, for the NIST 1016 mm Guarded-Hot-Plate Apparatus 1

Table 3 Summary of Standard Uncertainty Components for Meter Area (A) 15

Table 4 Summary of Rephcation Statistics for Uncertainty Component u(L3,) 18

Table 5 Summary of Standard Uncertainty Components for 25.4 mm Thickness (L25.4) 20

Table 6 Combined Standard Uncertainty Uc (L) 20

Table 7 Summary of Standard Uncertainty Components for T 22

Table 8 Summary of Standard Uncertainty Components for Power Input {Qm) at L = 25.4 mm..25

Table 9 Predicted Values for s due to Edge Heat Transfer (Peavy and Rennex [221) 27

Table 10 Nominal Settings for Imbalance Study in Yates Order 28

Table 11 Imbalance Test Data in Yates Order 29

Table 12 Statistical Significance for Estimated Effects for Imbalance Study 29

Table 13 Estimates and Standard Deviations for a^ 02, and a^, in Equation (24) 30

Table 14 Comparison of Empirical and Theoretical Values of Edge Heat Flow Error (e) 32

Table 15 Estimates for Uc (AQ) 32

Table 16 Combined Standard Uncertainty Uc (Q) 33

Table 17 Combined Standard Uncertainty for Thermal Conductivity (kgxp) 34

Table 18 Combined Standard Uncertainty for Thermal Resistance (R) 37

Table 19 Typical Values of R and Ur for Low-Density Fibrous-Glass Blanket NIST CTS 38

Table 20 Percent Contribution of Individual Components for X^xp and R 40

Table 21 Comparison of Individual Components in (%) for Xexp 43

LIST OF FIGURES

Figure 1 Figure 2

Figure 3

Figure 4

Figure 5 a Figure 5b

Figure 5c

Figure 6

Figure 7

Figure 8

Figure 9

Figure 10

Figure

Figure

Figure

Figure

1112

13

12

Figure 13

Guarded-hot-plate schematic, double-sided mode of operation - vertical heat flow 3

Guarded-hot-plate schematic, single-sided mode of operation - heat flow up 4

NIST 1016 mm Guarded-Hot-Plate Apparatus 5

Electrical schematic for meter-plate power measurement 7

Location of cold plate PRT (top view) 8

Location of hot plate PRT in guard gap on meter side of guard gap (top view) 8

Cross-section view of PRT in guard gap (guard plate not shown) 8

Angular locations of Type E thermopile junctions in the guard gap (not to scale) 9

Cause-and-effect chart for Xexp (2 levels of contributory effects) 13

Control chart for 0.1 Q. standard resistor, S/N 21736 24

Full-factorial experimental design for 3 factors, 2 levels 27

Combined standard uncertainty and individual components for X, 35

Thermal conductivity measurements of Fibrous-Glass Blanket NIST CTS 36

Combined standard uncertainty and individual components for/? 38

Thermal resistance measurements of Fibrous-Glass Blanket NIST CTS 39

Relative expanded uncertainties {k=2) versus thermal resistance of FibrousGlass Blanket NIST CTS 40

Cause-and-effect chart for X^xp with dominant contributory sources identified 41

vni
1. Introduction

In October 1992, NIST officially adopted a new policy [1] for the expression of measurement

uncertainty consistent with international practices. The NIST policy is based on recommendations

by the Comite International des Poids et Mesures (CIPM) given in the Guide to the Expression of

Uncertainty in Measurement [2] hereafter, called the GUM. This report assesses the uncertainties

for the NIST 1016 mm Guarded-Hot-Plate apparatus and expresses the uncertainties in a manner

consistent with NIST policy. The uncertainty assessment presented herein elaborates on a previous

effort [3] presented in 1997 for the production of NIST Standard Reference Material (SRM) 1450c

and supersedes the previous error analysis prepared by Rennex in 1983 [4]. Technical details of

the apparatus design and fabrication have been described previously [5-6] and, therefore, are only

briefly presented here.

The guarded-hot-plate method was standardized in 1945 after many years of effort and designated

ASTM Test Method C 177 [7]. Essentially, the method establishes steady-state heat flow through

flat homogeneous slabs - the surfaces of which are in contact with adjoining parallel boundaries

(i.e., plates) maintained at constant temperatures. The method is considered an absolute

measurement procedure because the resulting heat transmission coefficients are directly

determined. That is, the test results are not determined by ratio of quantities. In principle, the

method can be used over a range of temperatures but, in this report, the mean temperature is limited primarily to 297 K (23.9 °C, 75 °F). This report discusses the measurement principle and

presents a procedure for the assessment of uncertainties for a particular lot of low-density fibrousglass thermal insulation maintained by the NIST Building and Fire Research Laboratory.

2. Reference Material

The reference material of interest in this report is a low-density fibrous-glass blanket having a nominal bulk density of 9.6 kg-m"^ (0.6 Ib-ff ). The material lots were manufactured in July 1980

in the form of large sheets (1.2 m by 2.4 m) at nominal thicknesses of 28 mm and 81 mm. After

receipt and preparation of the material, the National Bureau of Standards' announced in December

1980 a program [8] to provide thick "cahbration transfer specimens" (CTS) on request for use in

conjunction with the "representative thickness" provision of the U.S. Federal Trade Commission

(FTC) rules published in 1979 [9] hereafter, called the "R-value Rule". The specimens were 610 mm

square and were originally issued at thicknesses of 25 mm, 75 mm, or 150 mm (two 75 mm

specimens stacked). Recently, however, in order to satisfy more stringent energy efficiency

requirements mandated in U.S. building codes, insulation manufacturers have begun requesting

CTS at thicknesses up to 225 mm (three 75 mm specimens stacked). In accordance with test guidelines in the "R-value Rule", measurements for customers are usually conducted at a mean

temperature of 297 K and a temperature difference of either 22.2 K or 27.8 K (40 °F or 50 °F,

respectively) across the specimen [9].

' In 1901, Congress established the National Bureau of Standards (NBS) to support industry, commerce, scientific

institutions, and all branches of government. In 1988, as part of the Omnibus Trade and Competiveness Act, the name

was changed to the National Institute of Standards and Technology (NIST) to reflect the agency's broader mission. For

historical accuracy, this report uses, where appropriate, NBS for events prior to 1988.
3. Steady-State Thermal Transmission Properties

ASTM Practice C 1045 [10] provides a uniform calculation procedure for thermal transmission

properties of materials based on measurements from steady-state one dimensional methods such as ASTM Test Method C 177. Table 1 summarizes the generahzed one-dimensional equations for

thermal resistance (R), conductance (C), resistivity (r), and conductivity (A,).

Table 1 - Steady-State One-Dimensional Thermal Transmission Property Equations

Thermal Resistance

R. m-KW'

Thermal Conductance

C, Wm"-K"'

Thermal Resistivity

r, m-KW'

Thermal Conductivity

X, Wm-^-K-^

Equation

Q AAT

AAT

r = QL AAT

Relationships

C X R L

_ 1 '^~ X x = i

r

Here, Q is the time-rate of one-dimensional heat flow (in units of watts, W) through the meter area of

the guarded-hot-plate apparatus, A is the meter area of the apparatus normal to the heat flow direction

(in units of square meters, m"), AT is the temperature difference across the specimen (in units of

kelvins, K), and L is the specimen thickness (in units of meters, m). As a rule, NIST provides value

assignments and uncertainty for only R and, to a lesser extent, X for thermal insulation reference

materials. Consequently, this report presents uncertainty assessments only for thermal resistance (R)

and thermal conductivity (X).

4. Measurement Principle

A guarded-hot-plate apparatus having appropriate plate temperature controllers can be operated in

either a double sided mode or in a single-sided mode (also known as two-sided or one-sided mode,

respectively). In principle, both modes of operation are covered in Test Method C 177; however,

additional information on the single-sided mode is available in ASTM Practice C 1044 [11]. For

completeness, this report presents both modes of operation but only the single-sided mode is examined

in the uncertainty analysis.

Double-Sided Mode

Figure 1 shows the essential features of a guarded-hot-plate apparatus designed for operation near ambient temperature conditions. The plates are shown in a horizontal configuration with heat flow (Q) in

the vertical (up/down) direction through the specimens. The apparatus is cylindrically symmetric about

the axis indicated in Figure 1. In the traditional double-sided mode of operation, specimens of the same

material having nearly the same density, size, and thickness are placed on each surface of the guarded hot

plate and clamped securely by the cold plates. Ideally, the guarded hot plate and the cold plates provide

constant-temperature boundary conditions to the specimen surfaces. Ideally, lateral heat flows (Qgap and

Qecif,e) are reduced to negligible proportions with proper guarding and, under steady-state conditions, the

apparatus provides one-dimensional heat flow (Q) normal to the meter area of the specimen pair.

Typically, a secondary guard is provided by an enclosed chamber that conditions the ambient gas (usually

air) surrounding the plates to a temperature near to the mean specimen temperature (i.e., average surface

temperatures of the hot and cold plates in contact with the specimens).
Cold plate 1

Cold plate 2- Ambient gas

Guarded hot plate

(Meter plate)

edge

I —Guard plate

edge

1

.

Principle: Tc < Th; Td = Tc2 = Tc

2. Practice: Tc < Th; Td ~ Tc2 * Tc

Figure 1 - Guarded-hot-plate schematic, double-sided mode of operation - vertical heat flow

Under steady-state conditions, the operational definition [10] for the mean (apparent) thermal

conductivity of the specimen pair (X^xp) is

"'P A[(AT/L\ + {AT/L)^]

(1)

where:

Q = the time rate of one-dimensional heat flow through the meter area of both

specimens and, under ideal conditions, is equal to Qm, the electrical power input to the meter plate (W);

A = the meter area normal to the specimen heat flow (m") (see Appendix A for

derivation); and,

(AT/L)\ = the ratio of the surface-to-surface temperature difference (Th - Tc) to the thickness

(L) for Specimen 1. A similar expression is used for Specimen 2.

For experimental situations where the temperature differences and the specimen thicknesses are nearly the same, respectively, Equation ( 1

) reduces to

'\ ^ average ""' ~ 2AATaverage

(2)

The thermal transmission properties of heat insulators determined from standard test methods typically include

several mechanisms of heat transfer, including conduction, radiation, and possibly convection. For that reason. some experimentalists will include the adjective "apparent" when describing thermal conductivity of thermal

insulation. However, for brevity, the term thermal conductivity is used in this report.
Using the relationship from Table 1, Equation (2) can be rewritten to determine the thermal

resistance of the specimen pair.

2A^T D average Q

(3)

In the double-sided mode of operation, the thermal transmission properties correspond to an

average temperature T given by T -{T,^+tJ/2.

Single-Sided Mode

Figure 2 shows the essential features of a guarded-hot-plate apparatus designed for operation near ambient temperature conditions in the single-sided mode of operation. In the single-sided mode of

operation, auxiliary thermal insulation is placed between the hot plate and the auxiliary cold plate,

replacing one of the specimens shown in Figure 1.

Cold plate

Qedge

Q

Th

T'h

edge

TV

Guarded hot plate

(Meter plate)

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<edge

Guard plate

^edge

Auxiliary cold plate

Ambient gas

1Principle: Tc < Th; Th = T'h = T'c; Q' =

2. Practice: Tc < Th; Th * T'h * T'c; Q' «

Figure 2 - Guarded-hot-plate schematic, single-sided mode of operation - heat flow up

The auxiliary cold plate and the hot plate are maintained at essentially the same temperature. The

heat flow {Q') through the auxihary insulation is calculated as follows [11]:

(2' = c'A(r;-r)-o (4)
where the prime (') notation denotes a quantity associated with the auxihary thermal insulation and

C is the thermal conductance of the auxiliary insulation. The specimen heat flow {Q) is computed

in the following equation:

*^ i^m i^ (5)

where Qm is the power input to the meter plate. Values of Q' are typically less than 1 % of 2,„. For

similar materials, Q from Equation (5) is approximately one-half the value obtained for Q in Equation (3) for the double-sided mode.

5. Apparatus

Figure 3 shows an illustration of the NIST 1016 mm Guarded-Hot-Plate apparatus. The apparatus

plates are typically configured in a horizontal orientation and are enclosed by an insulated

environmental chamber that can be rotated ±180°. The plates are made from aluminum alloy 6061-

T6. The plate surfaces in contact with the specimens are flat to within 0.05 mm and are anodized

black to have a total emittance of 0.89. The hot plate is rigidly mounted on four bearing rods.

Each cold plate can translate in the vertical direction for specimen installation and is supported at

its geometric center by means of a swivel ball joint that allows the plate to tilt and conform to a nonparallel rigid sample. The clamping force is transmitted axially by extension rods that are driven by a stepper motor and a worm-drive gear. A load cell measures the axial force that the

plate exerts on the specimen. The cold plates are constrained in the radial direction by steel cables

attached to four spring-loaded bearings that sHde on the bearing rods.

ij 1 p

Figure 3 - NIST 1016 mm Guarded-Hot-Plate Apparatus
Guarded Hot Plate

The 1016 mm guarded hot plate is nominally 16.1 mm thick and consists of a meter plate^ 405.6 mm

in diameter and a co-planar, concentric guard plate with an inner diameter of 407.2 mm. The

circular gap (also known as "guard gap") that separates the meter plate and guard plate is 0.89 mm

wide at the plate surface. The cross-sectional profile of the gap is diamond shaped in order to minimize lateral heat flow across the gap. The meter plate is supported within the guard plate by

three stainless steel pins, equally spaced around the circumference of the meter plate, that are used

to adjust the gap to a uniform width and maintain the meter plate in plane with the guard plate.

Across its diameter, the meter plate is flat to within 0.025 mm.

The hot-plate heater design, described previously in detail by Hahn et al. [12], utilizes circular hneheat sources located at prescribed radii. The circular line-heat-source for the meter plate is located

at a radius of v2/2 times the meter-plate radius which yields a diameter of 287 mm. This

location for the heater results in a temperature profile such that the temperature at the gap is equal

to the average meter-plate temperature [12]. The heating element is a thin nichrome ribbon

filament network, 0. 1 mm thick and 4 mm wide, electrically insulated with polyimide, having an

electrical resistance at room temperature of approximately 56 Q.

There are two circular line-heat-sources in the guard plate located at diameters of 524.7 mm and

802.2 mm. The heating elements are in metal-sheathed units, 1.59 mm in diameter, and were

pressed in circular grooves cut in the surfaces of the guard plate. The grooves were subsequently

filled with a high-temperature epoxy. The electrical resistances at room temperature for the inner

and outer guard heaters are approximately 72 Q. and 108 Q., respectively.

Meter-Plate Electrical Power

Figure 4 shows the electrical circuit schematic for the meter-plate power measurement which

consists of a four-terminal standard resistor, nominally 0. 1 Q, in series with the meter-plate heater.

A direct-current power supply (40 V) provides current (0 to the circuit which is determined by

measuring the voltage drop (V^) across the standard resistor (Figure 4) placed in an oil bath at 25.00 °C. The voltage across the meter-plate heater (V^) is measured with voltage taps welded to the heater leads in the center of the gap (described above). The meter plate power {Qm) is the

product of V,n and /.

Cold Plates

The cold plates are fabricated from 6061-T6 aluminum and contain channels that circulate an

ethylene glycol/distilled water solution. Each plate is 25.4 mm thick and consists of a 6.4 mm thick cover plate bonded with epoxy to a 19.1 mm thick base plate. The base plate has milled grooves 9.5

mm deep and 19.1 mm wide arranged in a double-spiral configuration. This arrangement forms a

counter-flow heat exchanger, that is, the supply coolant flows next to the return coolant providing a uniform temperature distribution over the cold-plate surface. The temperature of each cold plate is maintained by circulating liquid coolant from a dedicated refrigerated bath regulated to within

^ Terminology for the 1016 mm guarded hot plate reflects current usage in ASTM Practice C 1043.
Oil Bath

25.00 °C

DC Power

Supply

Vn

0.1 Q

Standard

Resistor, Rg

«- -^ A/WV

56 Q

Meter-plate

Heater, Rm

Figure 4 - Electrical schematic for meter-plate power measurement

±0.05 K over a temperature range of -20 °C to 60 °C. The outer surfaces and edges of the cold

plates are insulated with 102 mm of extruded polystyrene foam.

Environmental Chamber

The environmental chamber is a large rectangular compartment having inside dimensions of 1 .40 m

square by 1 .60 m high supported by a horizontal axle on rotational rollers that allow the apparatus

to pivot by ±180° (Figure 3). Access to the plates and specimens is permitted by front-and-back

double-doors. Air is circulated by a small fan in the chamber and is conditioned by a small cooling

coil/reheat system located within the chamber. The air temperature ranges from about 5 °C to 60 °C

and is maintained to within ±0.5 K by using the average of five Type T thermocouples located in the chamber.

Primary Temperature Sensors

The primary temperature sensors are small capsule platinum resistance thermometers (PRTs). The

sensor construction is a strain-free platinum element supported in a gold-plated copper cyhnder

3.18 mm in diameter by 9.7 mm long backfilled with helium gas and hermetically sealed. The

sensors are designed for temperatures from 13 K to 533 K (-260 °C to 260 °C) and the nominal

resistance is 100 H at °C

multimeter (DMM) that is part of an automated data acquisition system

The electrical resistance of each 4-wire PRT is measured with a digital

Figure 5 shows the locations of the PRTs in the cold plates and hot plate. The cold plate PRT is inserted in a 3.26 mm diameter hole, 457 mm long, bored into the side of the cold plate (Figure 5a).

The hot plate PRT is located in the guard gap at an angle of 69° from the location where meter

plate heater wires cross the guard gap (Figure 5b) based on the theoretical temperature distribution

T(r, 0) determined by Hahn et al. [12] for a similar apparatus. The sensor is fastened with a small

bracket on the meter side of the gap at the mid-plane of the plate {z = 0) as illustrated in Figure 5c. The radius to the center of the PRT was computed to be 199.3 mm.
N N

W w

Figure 5a - Location of cold plate

PRT (top view)

Figure 5b - Location of hot plate PRT in guard gap on meter side of guard gap (top view)

Z A

PRT

Bracket

Figure 5c - Cross-section view of PRT in guard gap (guard plate not shown)

Temperature Sensors in the Guard Gap

The temperature difference across the guard gap (ATgap) is estimated using an eight junction (4 pairs)

Type E thermopile. The thermopile was constructed from No. 30 AWG (American wire gauge)

insulated thermocouple wire 0.25 mm in diameter welded in an argon atmosphere to form small bead

junctions. The wire lengths were taken from spools of wire that were scanned using a large

temperature gradient (i.e., a bath of liquid nitrogen) to isolate inhomogeneities in the wire. The wire

passed from ambient to liquid nitrogen temperature and back to ambient; sections that gave

thermoelectric voltages larger than 3 |iiV for EP wire and 1.7 |xV for EN wire were discarded.

Figure 6 shows the angular locations for the individual junctions in the guard gap. The reference

angle of 0° is the location where the meter-plate heater leads cross the gap (the same as Figure 5b).

Type E is a letter designation for an ANSI standard base-metal thermocouple. Thermoelectric elements are designated

by two letters where the second letter, P or N, denotes the positive or negative thermoelement, respectively.
The thermocouple beads are installed in brackets with a thermally conductive epoxy and fastened,

in alternating sequence, to either the meter plate or the guard plate similarly to the method used for

the meter-plate PRT (Figure 5c). Like the PRT, the junctions are located at the mid-plane of the

hot plate (that is, z = in the axial direction). The EN leads of the thermopile depart the guard gap

at an angle of 185° (as shown in Figure 6) and are connected to copper leads on an isothermal

block mounted inside a small aluminum enclosure. The aluminum enclosure is located inside the

environmental chamber surrounding the apparatus plates.

G315°

270° —

Guard Gap

(not to scale)

Wire Type Diameter

EP 0.25 mm

EN 0.25 mm

Cu

G212°

from / G 148° Thermopile j

1 —

1

/ Isothermal

/ /

1Diock

'/////^

i •//^// : /^//^

to DMM

Figure 6 - Angular locations of Type E thermopile junctions in the guard gap (not to scale)

Temperature Control

The three heaters in the guarded hot plate are controlled by a digital proportional, integral (PI)

control algorithm that operates by actively controlling the plate temperatures. In other words, the

power level is not fixed at a specific level which could lead to temperature drift. Under steadystate conditions, the meter plate temperature is controlled to within ±0.003 K.

6. Measurement Uncertainty Estimation

This section summarizes relevant uncertainty terminology consistent with current international

guidelines [1-2] and presents a procedure for the estimation of measurement uncertainty based on

practical experiences by analytical chemical laboratories [13]. Using this procedure, an example is given for computation of the measurement uncertainty of the low-density fibrous-glass thermal

insulation issued by NIST as a CTS.
Terminology

The combined standard uncertainty of a measurement result, Uc (y) is expressed as the positive

square root of the combined variance Uc^iy):

:(>') = JZC-"'U,). (6)

i = l

Equation (6) is commonly referred to as the "/aw ofpropagation of uncertainty'' or the "root-sumof-squares." The sensitivity coefficients (c,) are equal to the partial derivative of an input quantity

(df/dXi) evaluated for the input quantity equal to an input estimate (X, = x,). The corresponding

term, u (jc,), is the standard uncertainty associated with the input estimate jc,. The relative combined

standard uncertainty is defined as follows (where >' ^ 0):

Each u (xi) is evaluated as either a Type A or a Type B standard uncertainty. Type A standard

uncertainties are evaluated by statistical means. The evaluation of uncertainty by means other than

a statistical analysis of a series of observations is termed a Type B evaluation [1]. Type B

evaluations are usually based on scientific judgment and may include measurement data from

another experiment, experience, a calibration certificate, manufacturer specification, or other

means as described in References [1-2]. It should be emphasized that the designations "A" and

"B" apply to the two methods of evaluation, not the type of error. In other words, the designations

"A" and "B" have nothing to do with the traditional terms "random" or "systematic." Categorizing

the evaluation of uncertainties as Type A or Type B is a matter of convenience, since both are based on probability distributions^ and are combined equivalently. Thus, Equation (6) can be

expressed in simplified form as:

«c = aPa + ul. (7)

Examples of Type A and Type B evaluations are provided in references [1-2]. A typical example

of a Type A evaluation entails repeated observations. Consider an input quantity X, determined

from n independent observations obtained under the same conditions. In this case, the input

estimate jc, is the sample mean determined from

The standard uncertainty, u (xi) associated with Xi is the estimated standard deviation of the sample

mean (where s is the standard deviation ofn observations):

"'' Note that the probabiHty distribution for a Type B evaluation, in contrast to a Type A evaluation, is assumed based on

the judgment of the experimenter.

10
The expanded uncertainty, U, is obtained by multiplying the combined standard uncertainty, udy),

by a coverage factor, k when an additional level of uncertainty is required that provides an interval

(similar to a confidence interval, for example):

U = ku^(y) = ^^cfuHx,), + Y^cfu'ix^), . (9)

The value of k is chosen based on the desired level of confidence to be associated with the interval

defined by U and typically ranges from 2 to 3. Under a wide variety of circumstances, a coverage

factor of A; = 2 defines an interval having a level of confidence of about 95 % and k = 3 defines an

interval having a level of confidence greater than 99 %. At NIST, a coverage factor of /: = 2 is used, by convention [1]. The relative expanded uncertainty is defined as follows (where y^O):

' \y\

For Type A evaluations, the degrees offreedom, v, is equal to « - 1 for the simple case given in

Equation (8). For the case when Uc is the sum of two or more variance components, an effective

degrees of freedom is obtained from the Welch-Satterthwaite formula as described in References

[1-2]. For Type B evaluations, v is assumed to be infinity. As will be shown later in this report,

the Type B evaluation is the dominant component of uncertainty. Therefore, values for v are not

necessary and are not ultimately used in determination of the coverage factor, k.

Procedure

The EURACHEM/CITAC Guide [13] provides a practical guide for the estimation of

measurement uncertainty based on the approach presented in the GUM [2]. Although developed

primarily for analytical chemical measurements, the concepts of the EURACHEM/CITAC Guide

are apphcable to other fields. The primary steps are summarized below.

• Specification of the mathematical process (measurement) model - clear and unambiguous

statement of the measurand, i.e., Y =f(Xi, X2, ... X^).

Identification of uncertainty sources - a comprehensive (although perhaps not exhaustive)

list of relevant uncertainty sources. A cause-and-effect diagram is a useful means for

assembling this list.

• Quantification of the components of the uncertainty sources - a detailed evaluation of the

component uncertainties using Type A and/or Type B evaluations described above (for

example. Equation (8)) or in the GUM.

• Calculation of the combined standard uncertainty - propagate the component uncertainties

using the "law of propagation uncertainty" given in Equation (6).

Calculation of the expanded uncertainty - using a coverage factor of A; = 2, compute an

interval for the expanded uncertainty given in Equation (9).

11
7. Mathematical Process Model

Mathematical process models are specified for thermal conductivity (X) and thermal resistance (R)

as determined using the single-sided mode of operation (Figure 2). For X, the mathematical

process model is given by

, _ QL _ (Q,„-AQ)L _ (Q„r Q,ar Q' - Q,)L

A(AT) A(T,-TJ A{T,-T^)

where:

Qm = power input (W) to the meter plate heater;

AQ = parasitic heat transfer (W) from the meter area (defined more specifically as Qgap, Q',

and Qe);

Qgap - lateral heat flow (W) across the guard gap (i.e. the airspace separation between the

meter plate and guard plate shown in Figure 2);

Q = heat flow (W) through the meter section of the auxiliary insulation (Figure 2);

Qe = error due to edge heat transfer (W) (i.e., from Qedge shown in Figure 2);

L = in-situ thickness of the specimen during testing (m);

A = meter area normal to Q (m^);

AT = specimen temperature difference (K);

Th = temperature of hot plate (K); and,

Tc = temperature of cold plate (K).

For R, the mathematical process model is given by

^^ A{AT) ^

A{T,-T^) ^ A(T,-TJ

^^^^

One of the major differences between Equations (10) and (11) is the absence of the term for

specimen thickness (L) in Equation (11). With regards to sign convention for heat flow {Q), heat

gain to the meter area is assumed to be positive (+) and heat loss is assumed to be negative (-).

8. Sources of Uncertainty

Figure 7 shows a cause-and-effect diagram that has been developed for A^xp from Equation (10).

The cause-and-effect diagram is a hierarchical structure that identifies the main sources (shown as arrows directly affecting Aexp) and secondary factors (shown as arrows affecting Q, L, A, and AT) of

contributory uncertainty. Tertiary (and additional hierarchical) factors of contributory uncertainty

are not shown in Figure 7. In general, the uncertainty sources in Figure 7 can be grouped in one of

three major metrology categories - dimensional metrology for meter area (A) and thickness (L);

thermal metrology for temperature (7); and, electrical metrology for voltage (V) and resistance {Q)

measurements. The analysis of parasitic heat losses and/or gains {AQ) requires either additional

heat-transfer analyses or experiments (or both).

12
Heat flow (G

)

Thickness (L)

Input power

measurement

Parasitic losses or gains

Meter-plate

dimension

Spacers

Plate

flatness

Thermal

expansion

Resistance

measurement

Meter area (A)

Temperature

difference (A 7^

n-situ measurement

Short-term

repeatability

Plate

deflection ^ x,exp

Sensor

calibration

Figure 7 - Cause-and-effect chart for Xgxp (2 levels of contributory effects)

From Figure 7, a comprehensive, but not exhaustive, hst of uncertainty sources is developed as shown in Table 2. This particular list could be applied to other apparatus but is most applicable to the NIST 1016 mm Guarded-Hot-Plate apparatus for single-sided measurements of low-density

fibrous-glass blanket thermal insulation. Other materials, mode of operation, apparatus, etc. may

require a (slightly) different Usting of sources (see, for example, the uncertainty analysis for NIST

SRM 1450c [3]).

Table 2 - List of Uncertainty Sources for X for the NIST 1016 mm Guarded-Hot-Plate Apparatus

1) Meter area (A)

a) Plate dimensions

b) Thermal expansion effects 2) Thickness (L)

a) In-situ linear position measurement system

i) Multiple observations

ii) System uncertainty

Dimensions of fused-quartz spacers

i) Repeated observations

ii) Caliper uncertainty

Short-term repeatability

Plate flatness

i) Repeated observations

ii) Coordinate measuring machine (CMM) uncertainty

Plate deflection under axial loading of cold plate

b)

c)

d)

e)

3) Temperature difference (AT)

a) Measurement {Tf, , Tc)

i) Digital multimeter (DMM) uncertainty

ii) PRT regression uncertainty in fit for calibration data

b) Calibration of PRTs

13
c) Miscellaneous sources (not shown in Figure 7)

i) Contact resistance

ii) Sampling of planar plate temperature

iii) Axial temperature variations

4) Heat flow (0

a) DC power measurement (Q^)

i) Standard resistor calibration

ii) Standard resistor drift

iii) PRT power input

iv) Voltage measurement

b) Parasitic heat flows (A0

i) Guard-gap (Qgap)

ii) Auxiliary insulation (Q')

iii) Edge effects (0)

The list of contributory sources of uncertainty for R is the same as the list given in Table 2 except

the contributory source for L would be omitted, as shown in Equation (11).

9. Quantiflcation of Uncertainty Components

Analysis of the standard uncertainties for meter area (A), thickness (L), temperature difference

(AT), and power (Q) are presented in this section. A useful approach that is followed in this report

is to treat each uncertainty component separately and evaluate the uncertainty component as either a Type A or Type B standard uncertainty [1-2]. The example presented here is for specimens of

low-density fibrous-glass thermal insulation taken from the CTS lot of reference material in

thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, 228.6 mm. The guarded-hot-plate measurements

were conducted at a mean temperature of 297 K and a temperature difference of 22.2 K. The

apparatus was operated in the single-sided mode of operation utilizing a specimen of expanded

polystyrene foam having a nominal thickness of 100 mm as the auxiUary insulation (Figure 2).

Meter Area (A)

The meter area is the mathematical area through which the heat input to the meter plate (Q) flows

normal to the heat-flow direction under ideal guarding conditions (i.e., Qgap = Qe = 0) into the

specimen. It is important to emphasize that the meter area is not the same as the area of the meter

plate (shown in Figures 1 and 2). The circular meter area was calculated from Equation (12) below

(see Appendix A for derivation):

^ = ^{ro+r,')(l + aATj' (12)

where:

To = outer radius of meter plate (m);

r, = inner radius of guard plate (m);

a = coefficient of thermal expansion of aluminum (alloy 6061-T6) (K' ); and,

ATmp = temperature difference of the meter plate from ambient (K) = 7/j - 20 °C.

14
The application of Equation (6) to Equation (12) yields

uM) = ^j^y(0 +cy-(r^) + clu\a) + cl,y(AT^^)

with

c,^ =dA/dro= 7rr„(l + aA7;J'

c^^ = dA/dn = nr-(l + a AT^^ f

c« =dA/da= iiATjr^' + r^){l + aATj

c^r = dA/d(ATmp) = na{r^' + r,'){l + aAT)

Plate Dimensions: The design gap dimensions [5] for the meter plate and the guard plate diameters

are 405.64 mm (15.970 in.) and 407.42 mm (16.040 in.), respectively. In 1994, as part of an

extensive sensor calibration check, the meter plate was separated and removed from the guard

plate. Using a coordinate measuring machine, the roundness of the meter plate was checked at six

locations at the periphery and the diameter was determined to be 405.67 mm (15.971 in.). During

re-assembly, a uniform gap width of 0.89 mm (0.035 in.) was re-estabUshed using three pin gages

spaced at equiangular intervals between the meter plate and guard plate. The uncertainty of the pin

gages was +0.005 mm / -0.000 mm. Based on these check measurements, the input values for r^ and r, were determined to be 0.20282 m and 0.2037 1 m, respectively, and the standard uncertainty

for both input values was taken to be 0.0254 mm (0.001 in.).

Thermal Expansion: For a, an input value of 23.6x10"^ K"' was taken from handbook data for

aluminum alloy 6061-T6. The standard uncertainty for the value of a was assumed to be 10 %

(that is, 2.36x10"^ K'^). For tests conducted at a mean temperature of 297 K and a specimen

temperature difference of 22.2 K, the meter plate temperature (Th) was maintained at 308 K (35 °C,

95 °F); thus, AT^p was equal to +15 K. The standard uncertainty for ATmp was determined to be

0.086 K (and will be discussed later in the section on Ar uncertainty).

Uc (A): Substituting the above input estimates into Equation (12), yields a meter area (A) of 0.12989 m".

For Uc (A), the input estimates (xi), sensitivity coefficients (q), standard uncertainties (u (jc,)), and

evaluation method (Type A or B) are summarized in Table 3. The last column in Table 3 provides

values for Cru(Xi) to assess the uncertainty contribution for each input X,. The combined standard

uncertainty udA) and relative standard uncertainty Uc.riA) were determined to be 2.4732x10"'' m"

and 0.019 %, respectively. This estimate for udA) is quite small near ambient temperature but

increases as Th departs from ambient conditions.

Table 3 - Summary of Standard Uncertainty Components for Meter Area (A)

Xi Xi Ci u (a-,) Type crii (a-,)

ro 0.20282 m 0.63763 m 0.0000254 m B 16.20x10' m'

fi 0.20371 m 0.64042 m 0.0000254 m B 16.27x10' m' a 23.6x10-^ K' 3.8953 m-K 2.36x10"' K' B 9.19x10' mAT^p 15 K 6.13x10-' m--K"' 0.086 K B 0.53x10' m'

15
Thickness (L)

In the single-sided mode of operation, the in-situ thickness of the specimen (Figure 2) is monitored

during a test by averaging four Hnear position transducers attached to the periphery of the cold

plate at approximate 90° intervals^. Each device consists of a digital readout and a slider that

translates in close proximity to (but not in contact with) a 580 mm precision tape scale bonded to a

precision ground plate of a low thermal expansion iron-nickel (FeNi36) alloy. In operation, the

slider is excited with a pair of oscillating voltages which are out-of-phase by 90°. The electrical

windings on the scale are inductively coupled with the sUder and the resulting output signal from

the scale is resolved and processed by the digital readout. As the slider follows the axial

movement of the cold plate, the corresponding output signal represents the linear distance between

the translating cold plate and the stationary hot plate.

The digital readouts are reset by placing a set of four fused-quartz spacers of known thickness

between the cold plate and hot plate. Fused-quartz tubing was selected because of its low

coefficient of thermal expansion (5.5x10"^ K'^) and high elastic modulus (72 GPa). The tubes have

nominal inner and outer diameters of 22 mm and 25 mm, respectively. Loose-fill thermal

insulation was placed in the tubes to suppress any convective heat transfer. Because the fibrousglass blanket CTS is compressible, the plate separation is maintained during a test by four fusedquartz spacers placed at the periphery of the specimen at the same angular intervals as the four

linear position transducers described above. Four sets of spacers having lengths of 25.4 mm, 76.2 mm,

152.4 mm, and 228.6 mm cover the thickness range of interest for fibrous-glass blanket CTS.

The combined standard uncertainty for L is given by

M, (L) = ylu\L^) + u\L,) + u\L,) + m'(LJ + u\L,) (13)

where the sensitivity coefficients are equal to unity (c^= 1) and the contributory uncertainties,

identified in Figure 7, are

u{L\) = standard uncertainty of the in-situ linear position measurement (m); u (L?) = standard uncertainty of the fused-quartz spacers (m); u (L3) = standard uncertainty of the repeatability of the linear position measurement (m); u (L4) = standard uncertainty of the plate flatness (m); and, u (L5) = standard uncertainty of the cold plate deflection under axial loading (m).

The contributory uncertainties u (Li) are discussed in detail below.

m(Li) - In-situ Measurement: During a test, the digital readouts are recorded manually and the

estimate for x (Li) is determined from the sample mean of the four observations. Two contributory

effects comprise u(L]): 1) multiple observations (Type A evaluation); and, 2) the measurement

system uncertainty (Type B evaluation). Thermal expansion effects of the linear tape scales were

neglected because the iron-nickel (36 %) alloy has a low coefficient of thermal expansion and the

^ For a two-sided test (Figure 1), eight linear positioning devices (four for each specimen) determine the in-situ thickness of the specimen pair.

16
tests are conducted near ambient conditions of 297 K. Equation (8) is applied to evaluate the Type A

standard uncertainty where s is the standard deviation of the four transducers (n = 4). The Type B

evaluation is the uncertainty specification stated by manufacturer (k= 1) of 0.005 mm. Application

of Equation (7) yields

M^(Z,) = ^M^ + (5.0xlO-^r

where ua varies for a particular test. Estimates for u(Li)a for a test thickness of 25.4 mm are summarized at the end of this section (see Table 5).

u{L2) - Spacers: Two contributory effects comprise u(L2): 1) multiple length observations (Type A

evaluation); and, 2) caliper uncertainty (Type B evaluation). Thermal expansion effects were

neglected because fused quartz has a low coefficient of thermal expansion (5.5x10"^ K'^) and the

tests were conducted near ambient conditions of 297 K. Deformation of the spacers under load

was also neglected because of the cross-sectional area of tubing and the relatively high value for

elastic modulus. The length of each spacer was measured under ambient conditions with digital

calipers and x (L2) was determined from the sample mean of four observations. Equation (8) is applied to evaluate the Type A standard uncertainty where s is the standard deviation of the four

observations (n = 4). The Type B evaluation assumes a uniform distribution with an interval of 2a

[2]; thus, ub = «/V3 where a is the smallest length interval of the caliper. The estimates for ua and

ub vary for each set of spacers and for the type of measurement calipers, respectively. Estimates

for u(L2)a.b for a test thickness of 25.4 mm are summarized at the end of this section (see Table 5).

u(L3) - Repeatability: The short-term repeatability of the linear position transducers was

determined from a series of replicate measurements. For these measurements, the digital readouts were initially set to the length values of each set of fused-quartz spacers placed between the cold

plate and hot plate. The cold plate was lifted from the spacers and subsequently lowered in contact

with the spacers five times to check within-day variation. The procedure was repeated for four

consecutive days to check the day-to-day variation (20 observations total).

The standard uncertainty for uiL^) was determined using the Type A evaluation given in Equation

(14) [14]

(14)

where Sa is the standard deviation of the daily averages (between-day variation), s^ is the (pooled)

within-day standard deviation, and r is number of replicates per day (r = 5). Table 4 summarizes

replication statistics for nominal specimen thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, and 228.6 mm.

Values for within-day average and within-day standard deviation for the 5 replicates are given in

columns 4 and 5, respectively, and values for Sa, Sd, and w(L3) for each nominal level of thickness

are summarized in the last three columns of Table 4. Note that values of uiLi,) in Table 4 do not

appear to be correlated with L. The degrees of freedom (v) for Equation (14) were detennined

from the Welch-Satterthwaite formula [ 1 ] and the value is summarized at the end of this section

(see Table 5).

17
Table 4 - Summary of Replication Statistics for Uncertainty Component u(L^)

Nominal

L

(mm)

Day Replicates

Within-day

Average

(m)

Within-day

Standard Deviation

(m)

Sa (m) (m) (m)

25.4 1 5 0.0254051 3.96x10'^

2 5 0.0254144 4.28x10"^

3 5 0.0254156 3.29x10"^

4 5 0.0254159 5.20x10''

5.12x10"^ 4.24x10"^ 6.37x10'

76.2 1 5 0.0762217 0.70x10"^

2 5 0.0762325 1.93x10-^

3 5 0.0762376 1.38x10-^

4 5 0.0762325 3.77x10"^

6.69x10"^ 2.25x10"^ 6.98x10"'

152.4 1 5 0.152405 3.68x10''

2 5 0.152410 0.70x10'^

3 5 0.152411 3.45x10"*^ 4 5 0.152409 2.98x10"^

2.48x10'^ 2.95x10'^ 3.62x10"'

228.6 1 5 0.228578 10.79x10"^

2 5 0.228569 7.64x10"^

3 5 0.228582 2.75x10"^

4 5 0.228571 2.63x10-^

6.28x10"^ 6.88x10-^ 8.79x10"'

u(L4) - Plate Flatness: Two contributory effects comprise u(L4): 1) multiple thickness observations

(Type A evaluation); and, 2) coordinate measuring machine (CMM) uncertainty (Type B evaluation).

As discussed above, the meter plate dimensions were checked with a CMM in 1994. The thickness

of the plate was measured at 32 different locations using a CMM and the estimate for x (L4) was

determined from the sample mean of 32 observations. The standard deviation (s) was 0.0131 mm

and, thus, the relative flatness over the meter plate is (0.013 mm)/(406.4 mm) = 0.003 %. It is interesting to note that the flatness specification given in CI 77-04 is 0.025 % [7]. Apphcation of

Equation (8) to evaluate the Type A standard uncertainty yields:

^(4)^ = 1.31x10"' m/V32 = 2.32x10"' m.

The Type B evaluation is the uncertainty specification (^ = 1) for the CMM of 0.0051 mm. Because

the cold plate was fabricated with the same machine finish as the meter plate, the cold plate flatness

is assumed to be nearly the same as the meter plate. In this case. Equation (7) becomes:

i^m)=^2{ui+ui)

Substituting the values for the Type A and Type B evaluations given above yields a standard

uncertainty for L4 of 0.0079 mm. The value of u(L4) (0.0079 mm) is apparatus dependent and,

thus, is fixed for all values of specimen thickness.

18
uiLs) - Cold Plate Deflection: The potential deflection of the (large) cold plate under a mechanical

load is evaluated as a Type B uncertainty using classical stress and strain formulae developed for

flat plates. As will be discussed below, this approach is an approximation. Recall that the

clamping force on the specimen and auxiliary insulation is transmitted axially by extension rods

(Figure 3). The axial force is applied over a circular area at the center of each plate and is assumed

to be uniformly distributed through a ball-joint connection between the plate and extension rod. In

the single-sided mode of operation, the auxiliary insulation is a rigid specimen of expanded

polystyrene foam which supports the hot plate (Figure 2). For a uniform load over a concentric

circular area of radius r, the maximum deflection y^ax at the center of the cold plate is given by the

following formula from Reference [15]. In this case, simple edge support is assumed because the

test specimen is compressible and the plate separation is maintained by edge spacers.

um) = y^ =-

3W(m'-l)

2^3 \6'KEm t

(I2m + 4)a^ . 2, - m + l r m + l (15)

where:

W = total applied load (N);

m = reciprocal of Poisson's ratio (dimensionless);

E = modulus of elasticity (N-m"");

t = thickness of the plate (m); and, a = radius of the plate (m).

Based on load cell measurements, a conservative estimate for the net applied force (W) for the cold

plate was assumed to be 356 N (80 Ibf). The plate is 1.016 m in diameter and 0.0254 m thick and is fabricated from aluminum alloy 6061-T6. The values for m, E, and r were taken to be (0.33)"^ = 3.0,

6.9x10 kPa (10x10 Ibfin" ), and 0.305 m, respectively. Substituting into Equation (15) yields a value

of 0.031 mm for y^ax, which is the dominant component of the thickness uncertainty and is essentially fixed for each level of specimen thickness (for constant loading).

In general, the uncertainty due to plate deflection depends on the apparatus plate design (i.e., dimensions and material), the rigidity of the test specimen, and the magnitude and appUcation of

the load applied. The major limitations for this assessment approach are: The cold plate is not simply supported as assumed in Equation (15). The plate is actually

constrained by the fused-quartz spacers at four locations around the periphery of the plate.

The cold plate is not a solid plate. As discussed above, the cold plate is actually a composite construction to allow the flow of coolant internally within the plate.

Uc (L): Table 5 summarizes the sources, sensitivity coefficients (c,), uncertainty components w(L,),

and the evaluation method (Type A or B) for a thickness of 25.4 mm (L25.4). As described above,

the component uncertainties are either test dependent (u(L\)), spacer dependent (uiLj)), process

dependent (uiL^,)), or apparatus dependent (m(L4) and u(Ls)). The final two components are

essentially fixed for all thicknesses. Consequently, only the first three rows of Table 5 are applicable for 25.4 mm thick specimens. Application of Equation (13) yields a combined standard

uncertainty for L25.4 of 0.038 mm (uc,r (L) = 0. 15 %). It is interesting to note that C 177-04 requires

that the specimen thickness be determined to within 0.5 % [7].

19
Table 5 - Summan' of Standard Uncertainty Components for 25.4 mm Thickness (L25a)

i((.x,) Source Ci Value of z^L,) Type

u{U) In-situ measurement

multiple obser\'ations

system uncertainty

1 20xl0''m

19xlO"^m

5.0 X 10"^ m

B

A (degrees of freedom = 3)

B (equipment specification. ^ = 1) U(L2) Spacers (nominal 25.4)

repeated observations

caliper uncertainty

1 1.9xl0^m

l.lxlO"^m

1.5xlO"^m

B

A (degrees of freedom = 12)

B (a/V3 where a = 2.54x10^ m)

u(L,) Short-term repeatabilit}' 1 6.4 X 10^ m A (degrees of freedom = 6.8)

uiU) Plate flatness repeated observations

CMM uncertainty

1 7.9 X 10-^ m

2.3 X IQ-^ m

5.1xlO"^m

B

A ( degrees of freedom = 31)

B (equipment specification. ^ = 1) u{L,) Plate deflection under load 1 BlxlQ-'m B (calculation [15])

Table 6 summarizes u{L,), Uc (L), and Uc.r {Q for specimen thicknesses of 25.4 mm, 76.2 mm,

152.4 mm, and 228.6 mm. As discussed above, the dominant component for all levels of thickness

is uiLs), the uncertainty due to potential deflection of the cold plate. As a result, the variation of

Uc(L) is small over the range of thicknesses. One should note that the values given in Table 6 are valid only for the apparatus described herein. Other guarded-hot-plate apparatus would have

different sources and values for the thickness uncertainty components. For example, the

uncertainty due to plate flatness could be much larger if proper attention is not given to the plate

design and fabrication.

Table 6 - Combined Standard Uncertainty Mr (L)

L

(mm) (m)

U{L2)

(m) (m)

UiU)

(m)

u{L,)

(m)

Uc(L)

(mm)

Uc.r (L)

(%)

25.4 20 x 10-^ 1.9x10"* 6.4 X 10"* 7.9x10"* 31 xlO"* 0.038 0.15

76.2 12x10"^ 2.4x10-* 7.0x10"* 7.9x10"* 31 xlO"* 0.035 0.05

152.4 12x10"^ 7.7 x 10"* 3.6 X 10"* 7.9x10^ 31 xlO"* 0.035 0.02

228.6 9.6 xlO"" 9.5 X 10"* 8.8 X 10^ 7.9 X 10^ 31 xlO^ 0.035 0.02

Temperature Difference (AT)

As discussed above, the priTtiary plate temperatures (Figures 1-2) are monitored during a test by

computing temporal averages of three small capsule platinum resistance thermometers (PRTs) in = 240

observations taken over a steady-state interval of 4 h). The uncertainty sources u(Ti) for the

primar}' temperature sensors are discussed in detail below. Secondary temperature sensors such as thermocouples and thermistors located in the plates, and their corresponding uncertainties, are not

discussed because these sensors are not input quantities in the mathematical process models given

in Equations (10) and (11).

u{T\) - Measurement. During a typical CTS test (4 h in duration), the electrical resistances of the PRTs

are recorded e\'er}' minute by an automated data acquisition system (77 = 240). Two major contributory

effects comprise u{Ti): 1) regression equation coefficients (Type A evaluation); and, 2) the

measurement system uncertainty (Type B evaluation). (The standard uncertainty for repeated

observations of AT (Type A evaluation) was less than 0.0002 K and was neglected in further analyses.)

20
1) For each PRT, individual observations in ohims (Q.) were converted to temperature using a curve fit to the cahbration data (discussed below). The curve-fits were obtained using a

statistical plotting package from NIST. The residual standard deviation for the fit of each

set of calibration data was "pooled" and the resulting standard uncertainty is 0.0052 K.

The degrees of freedom from the regression analyses were aggregated for a value of 15.

2) The Type B standard uncertainty for the resistance measurement assumes a uniform

distribution with an interval 2a [2] where a was determined from the specification of the

manufacturer for the digital multimeter (DMM). For a = 0.039 Q. at the 300 Q. DMM

range, ub = a/V3 = 0.022 Q. This standard uncertainty in ohms was propagated using the

above curve fit to yield a standard uncertainty for temperature of 0.058 K.

u(T2) - Calibration: The PRTs were calibrated by the NIST Thermometry Group by comparison

with a standard platinum resistance thermometer in stirred liquid baths. The thermometer was

inserted into a test tube partially filled with mineral oil which, in turn, was placed in the calibration

bath. In 1981, the thermometers were calibrated at the water triple point, 10 °C, 20 °C, 30 °C, 40 °C,

and 50 °C [4]. In 1993, the thermometers were removed from the apparatus and re-calibrated over an extended temperature range at -40 °C, °C, 40 °C, 80 °C, and 120 °C. All temperatures in the

1993 calibration were based on the International Temperature Scale of 1990 (ITS-90). Based on

the expanded uncertainty (k = 2) for the calibration bath temperatures of 0.01 K (Type B evaluation),

the standard uncertainty was 0.005 K (k= 1). Recently, the cold plate PRTs have been removed

from their respective plates and again submitted for calibration by the NIST Thermometry Group.

These results will be updated when the most recent calibration and analysis are completed.

u(T3) - Other Small or Negligible Contributors: Several small or negligible contributory effects

include the following: 1) PRT self heating/contact resistance; 2) sampling of temperatures in the

meter area (r, 9); and, 3) temperature variations in the axial (z) direction (Figure 5c). It is difficult to quantify the uncertainties of these contributors by separate experiments and, in some cases, the

uncertainties are based on theoretical calculations or experimenter judgment. Hence, in all cases, the uncertainties are Type B evaluations.

1) PRT self-heating/contact resistance - The PRT excitation current is 1 mA which, for a nominal 100 iQ PRT, dissipates about 0.0001 W. For the meter plate PRT, a thin layer of

thermally conductive silicone paste has been applied around the sensor to improve thermal

contact (Figure 5c). For the cold plate PRTs, the thermal conductance of the metal-to-airto-metal interface between sensor and plate is estimated to be 0.058 W-K'\ Thus, the

temperature rise (0.0001 W/0.058 W-K'^) is 0.0017 K.

2) Sampling (planar) - Rennex [4] and Siu [16] empirically determined the temperature

profiles of different NIST meter plates utilizing independent thermopile constructions. In each experiment, the thermopiles were placed on the plate surfaces and a test conducted

with semi-rigid specimens. Based on the thermopile measurements, Rennex [4] ascribed an estimate for the sampling uncertainty to be 0.015 K.

3) Axial temperature variations - A rigorous analytical analysis by B. A. Peavy published in Hahn et al. [12] shows that, for typical insulations, the differences between the temperature

at the guard gap and the average surface temperature of the meter plate is less than 0.05 %

of the temperature differences between the hot and cold plates. For a specimen

temperature difference of 22.2 K, the standard uncertainty is 0.01 1 K.

21
Uc (T): Table 7 summarizes the sources, sensitivity coefficients (c,), uncertainty components u(Ti),

and the evaluation method (Type A or B) for the plate temperature. Application of Equation (6) to the uncertainty components in Table 7 with c, = 1 yields a value for Uc (7) of 0.061 K (Table 7, last row). For a AT of 22.2 K, Uc,r (T) is 0.27 %. By comparison, C 177-04 specifies an uncertainty for

temperature sensors of less than 1 %. The dominant component for Uc (T) in Table 7 is the

uncertainty specification for the DMM measurement of the PRT electrical resistance.

Table 7 - Summary of Standard Uncertainty Components for T

uix,) Source Ci Value of u (Ti) Type

u{T,) Measurement (7), , Tc)

DMM uncertainty

regression uncertainty

0.058 K

0.058 K

0.0052 K

B

B (a/V3 where a = 0.039 Q)

A (degrees of freedom = 15)

iiiTi) Calibration of PRTs 0.005 K B(NIST Certificate, ^=1)

u{T,) Miscellaneous

Contact resistance Sampling (planar)

Axial variation in plate

0.019K

0.0017 K

0.015 K

0.011 K

B

B

B (Reference [4]) B (Reference [12])

Uc(T) 0.061 K

Uc (AT): Recall from Equation (10) that the specimen temperature difference (AT) was determined

from the following equation:

AT = (T,-TJ (16)

The application of Equation (6) to Equation (16) and setting Uj = Uj. = Uj yields

u^(AT) = Juj + Uj = ^Jlu^ (17)

Substitution of Uc (T) = 0.061 K (Table 7) into Equation (17) yields a value for Uc (AT) of 0.086 K.

For AT of 22.2 K, Uc,r (AT) is 0.39 % (and for single-sided tests conducted (for customers) at a AT

of 27.8 K, Uc,r (AT) decreases to 0.31 %). Note that the value for Uc (AT) of 0.086 K was used in

the uncertainty assessment for the meter area (A).

Heat Flow (Q)

Equation (10) defines the specimen heat flow (Q) as the difference between the power input to the

meter plate (Qm) and parasitic heat losses (Qgap, Qe, and Q'). Ideally, in the single-sided mode of

operation (Figure 2), the temperatures of the guard plate, ambient gas temperature, and auxiliary

cold plate are maintained such that the parasitic heat losses are reduced to negligible proportions in

comparison to Qm- Thus, Q is primarily determined by measuring the DC voltage and current

provided to the meter-plate heater (Qm)- The equation for Qm is:

(18)

22
where / is the current (Vs/Rs) measured at the standard resistor, and Vm is the voltage drop to the

meter-plate heater measured across the voltage taps located at the midpoint of the guard gap.

The application of Equation (6) to Equation (18) yields

«.(2J-V^v.w'(^j' + <«'(^.)'+<w'(Vj' (19)

V i s m

with

cy^ =d(Qr„)/dVs =VJR

V.R

Cy =d(Q„)/dVm = VjR^

Cn^ =d{Qm)ldRs =—±V^

u(Qm) - Power Input: The contributory sources u (Qm) for the meter-plate power input are discussed in detail below. Three contributory effects comprise u{Q,n): 1) cahbration of the standard

resistor (Type B evaluation); 2) PRT self-heating (Type B evaluation); and 3) voltage measurements

for Vs and Vm (Type A and Type B evaluations).

1) Standard resistor calibration: The 0.1 ohm standard resistor is a commercial, doublewalled manganin resistor [17] manufactured in 1913. The resistor has been calibrated by the

NIST Quantum Electrical Metrology Division in an oil bath at 25.00 °C for several years, most recently in 2(X)8. Figure 8 shows the historical control chart for the resistor from 1977

to 2008. Since January 1, 1990, the NIST calibrations have been based on the quantum Hall

effect used as the U.S. representation of the ohm [18]. The most recent calibration assigned

the resistor a value of 0.10006957 H and an expanded uncertainty {k = 2) of 0.0000005 Q.

Therefore, the standard uncertainty was 0.00000025 Q.{k=\).

Careful inspection of Figure 8 reveals a possible drift in the data and its presence could be

indicative of other detrimental factors affecting the resistor itself. Annual calibrations are now planned to investigate the extent of the possible drift. During operation with guardedhot-plate apparatus, the standard resistor is immersed in an oil bath maintained at 25.00 °C

as shown in Figure 4. Because the resistor is operated at the same temperature as the

calibration temperature of 25.00 °C, temperature effects during operation were neglected.

2) PRT power input - As shown in Figure 5c, the meter-plate PRT is fastened to the side of

the meter plate. Under normal operating conditions, the PRT will generate a small power

input to the meter plate due to self-heating effects. The excitation current for the meterplate PRT is 1 mA which, for the nominal 100 Q. PRT, dissipates a power of about 0.0001 W.

In the worst case for a 228.6 mm (9.0 in.) thick specimen, the power input to the meterplate heater is about 0.6 W and the PRT self-heating effect is 0.0001 W/0.6 W = 0.02 %.

Thus, the effect of PRT self-heating was neglected for all specimen thicknesses.

23
0.100076

0.100074

0.100072

sz 0.100070

o

0.100068

0.100066

0.100064

1975 1980 1985 1990 1995

Year

2000 2005 2010

Figure 8 - Control chart for 0.1 Q standard resistor, S/N 21736

3) Voltage measurement. Two contributory effects comprise the voltage measurement: 1)

multiple observations (Type A evaluation); and, 2) the DMM voltage measurement uncertainty

(Type B evaluation). During a typical CTS test (4 h in duration), Vs and Vm are recorded

every minute by an automated data acquisition system (n = 240). The Type A uncertainty

evaluations for V^ and Vn, are included later (in Table 16) as repeated observations for the

input power UA(Qm)- The Type B standard uncertainty for the voltage measurements V, and

Vm (Figure 4) assumes a uniform distribution with an interval 2a [2]; thus, ub =a/V3 ; where a was determined from the 1-year DMM specification. The DMM ranges for Vs and V^

are 30 mV and 30 V, respectively, and the corresponding values for «30mv and asov are 15.0 jiV

and 3.05 mV at L = 25.4 mm. Therefore, ub(Vs) and UB(Vm) are 8.7 )a,V and 1.76 mV,

respectively. Note that calibration checks for the DMM are conducted every other year; the

last check was in 2008.

Table 8 summarizes the input estimates (jc/j, sensitivity coefficients (c,), and standard uncertainties

(u (Xi) for a CTS specimen thickness of 25.4 mm. Only Type B evaluation methods are included in

Table 8. As stated above, the Type A uncertainty evaluations for V, and V,„ are included (in Table 16)

as repeated observations for input power UAiQm)- Substituting the values in Table 8 into Equation (19)

yields a combined standard uncertainty for Q,„ of 0.0016 W or about 0.03 % for an input to the

meter-plate heater of 5.1 1 W. The combined standard uncertainties at other specimen thicknesses

are summarized later in this section.

24
Table 8 - Summary of Standard Uncertainty Components for Power Input (Q,„) at L = 25.4 mm

Xi X; c, U (Xi) Type

K 0.03 V 169 A 8.7 X XQf^V B (a/V3 where a = 15.0 ixV)

R, 0.10006957^2 50.93 v-a- 2.5 X 10"^ a B(NIST Certificate, A: =1)

Vm 17 V 0.3 A 1.76 X 10"V B («/V3 where a = 3.05 mV)

u(Qgap), u{Q'), and u{Qe) - Parasitic Heat Flows: Although the parasitic heat flows are reduced

during steady-state conditions to very small values (on the order of 1 mW, or less), the uncertainty

associated with each term can be large as shown below. The sources (jc,) for the parasitic heat

flows are discussed individually below and, later, their respective uncertainties are determined

collectively as part of an imbalance study.

1) Qgap - Guard gap heat flow. The model for heat flow across the gap developed by

Woodside and Wilson [19] is given in Equation (20).

Qsa,={qo+c'kW^,^ (20)

where; qo represents the heat flow directly across the gap; ck is the heat flow distortion in

the insulation specimen adjoining the gap (Figure 2); and isTgap is the temperature difference across the guard gap. Here, the term X is the specimen thermal conductivity. The terms qo

and c are a function of the apparatus design, specimen thickness, and thermal conductivity

[19].

Empirically, Qgap is determined from the thermopile voltage {Vgap) of an eight junction (4

pairs) Type E thermopile across the guard gap.

Q,a,=S^,^{Sn^T^^^) = S^^^V^^^ , (21)

where Sgap is the heat flow sensitivity in W-p,V"\ S is the Seebeck coefficient for Type E

thermocouples in |xV-K"\ and n is the number ofjunction pairs. At a meter plate temperature

(7/0 near 308 K (35 °C), the value of 5 is equal to 61.87 ^iV-K"^ [20]; thus, the sensitivity of

the 8-junction (4 pair) thermopile is 4x61.87 ^iV-K'^ = 247.5 ^iV-K^ For a DMM

resolution of 0.1 |iiV, the temperature resolution of the thermopile is 0.0004 °C. Under

balanced control, the variabihty of the gap thermopile voltage {Vgap\ determined from

actual test data, is typically 1.5 |xV or about 0.01 K (at a 3 x standard deviation level).

2) Q' - Auxiliary insulation heat flow. Equation (4) predicts the heat flow {Q') through the

meter section of the auxiliary insulation, which under normal one-sided operation is approximately zero. With the exception of C ', the quantities in the right-side of Equation (4),

namely A, Th, and T'/,, are determined from the measurement test data. One method for

determining the thermal conductance (C ') of the auxihary insulation in (W-m""-K" ) is an

iterative technique described in Annex Al of Practice CI 044 [11]. After the value of C is obtained, the standard uncertainty of Q ' can be determined by propagation of uncertainty

in Equation (4). An alternate method is to determine the value of the product of C 'A (that

is, the thermal conductance per unit temperature (W-K"^)) from the imbalance study

described below. In this case, the standard uncertainty is propagated through the

mathematical model developed for the imbalance study.

25
3) Qe - Effect of edge heat transfer. In general terms, the edge heat flow error is the distortion

of one-dimensional heat flow through the specimen meter area due to heat flows at the

periphery of the specimen. Edge effects are controlled by appropriate guarding in the

design of the hot plate, limiting the specimen thickness, controlling the ambient

temperature at the specimen edge, and, if necessary, the use of edge insulation. The

empirical study by Orr [21] investigated the effects of edge insulation and changes in

ambient temperature on edge heat flow error. A similar approach to determine the

sensitivity of this error by varying the ambient air temperature for different specimen

thicknesses was implemented as part of the imbalance study discussed later. ASTM Practice C 1043 [5] provides a theoretical analysis of edge heat loss or gain based on analytical solutions derived by Peavy and Rennex [22] for both circular and square plate

geometries. The purpose of the analysis is to provide the user of Practice C 1043 with

design guidance in determining the proper diameter of the guard plate (Figure 2) for control

of edge heat loss or gain. An abbreviated version of the analysis is given below. The error

(e) due to edge heat transfer for either geometry is given by

£ = A + BX where ^^ 2{T^-Tj

T -T

(22)

where Th and Tc are the hot and cold plate temperatures, respectively (Figure 2); Ta is the

ambient temperature at the edge of the specimen (Figure 2); and, T^ is the mean specimen

temperature given by {Tf, + Tc)/2. For a circular plate geometry, coefficients A and B are given by:

n = l

where W =

71' (hL\

V A y n

A = Sw/,„ and B = YW,^_,

n = \

I^{n7id/yL)+ —-I^inndlyL) nnk

The terms /q and I\ are modified Bessel functions of the first kind of order and 1, respectively. The term b is the radius of gap center; d is the radius of the hot plate; L is the

specimen thickness; and, h is the convective film coefficient at the edge of the specimen.

The term QiLIX) is the Biot number; j^ = Xr / X^ where kr and Xr are the radial and axial

thermal conductivities of the specimen, respectively. The geometrical mean of the thermal

conductivities is X, = (Xr\y^.

The following results, for which the author is indebted to D. R. Flynn, retired from NIST,

are presented in Table 9 for the following parameters:

X = 40 and X = 2(T^-TJ_2(5)

T -T 22.2 = 0.450

26
Table 9 summarizes coefficients A and B, and the error (e) due to edge heat transfer for

specimen thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, and 228.6 mm.

Table 9

-- Predicted Values for s due to Edge Heat Transfer (Peavy and Rennex [22])

L

(mm)

A B £ X = -0.450 X = X= +0.450

25.4

76.2 0.0000021

152.4 0.0000021 0.0022826 -0.0010 +0.0013

228.6 0.0002111 0.0266257 -0.0118 0.0002 +0.0122

The error (s) for X = is essentially zero up to thicknesses of 228.6 mm. For an ambient

temperature difference of ±5 K from the mean specimen temperature (X = ±0.450),

predicted values of £ become significant at thickness of 152.4 mm.

AQ - Imbalance study: A series of imbalance tests were conducted to investigate empirically the

effects of moderate temperature differences on Qgap, Q \ and Qc. An imbalance test, as the name

impUes, is an operating condition in which a parameter (in this case, temperature difference) is purposely imbalanced (from zero) such that a parasitic heat flow is enhanced (i.e., the magnitude of

the heat flow is either increased or decreased). In this study, imbalance tests are conducted for

temperature differences across: 1) the guard gap (ATgap or, in this case, Vgap as shown in Equation (21));

2) auxiliary insulation (T'h - T'c); and, 3) the mean specimen temperature at the specimen edge

and the ambient air temperature (7^ - To).

These parameters were varied following an orthogonal experimental design illustrated in Figure 9,

where jci = Vgap', xj = T'h - T'c', and, X3 = Tm- Ta. The test plan for the imbalance tests is based on a

2^ full factorial design meaning that the three factors are each varied at one of two levels shown in

Yates order in the adjoining design matrix (with levels "coded" +1 for a high setting and -1 for a low setting). One advantage of an orthogonal design is that any interactions between factors (jciJC2, x\X3, X2X3, and JC1JC2JC3) can be detected; that is not possible for an experimental design in which

"one-factor-is-varied-at-a-time." Note that the experimental design given in Figure 9 also contains

a center point (#9), that is, a balanced test point where all imbalance settings have been set equal to zero.

7

1 VX2

8

3 / A

Q / ^s^

5 g

1 /

/ / /2

2^ Full Factorial Design Matrix - Yates Order

# Xi X2 X3

1 -

2 + - -

3 - + -

4 + + -

5 - +

6 + - +

7 - + +

8 + + +

9

Figure 9 - Full-factorial experimental design for 3 factors, 2 levels

27
For a 2 full factorial design, the useful underiying mathematical model would be

where:

jc, = "coded" level (+1 or -1) for factor /; Pi = main effect for coded jc,;

Pij = two-factor interaction effect for coded x,; and,

Pijk = three-factor interaction effect for coded xi.

Table 10 summarizes the input settings for the nine test conditions following the procedure given

in Figure 9. For tests #1 through #8, the "non-coded" settings for jci, xj, and x^ are ±50 |xV, ±0.5 K,

and ±5 K, respectively. The input values for the test settings were selected with the objective of

providing adequate responses in the parasitic heat flows. Test #9 is a balanced condition where

each parameter is set to zero and the corresponding parasitic heat flows consequently approach

zero. Note that Test #9 is an independent check test for the original test measurement.

Table 10 - Nominal Settings for Imbalance Study in Yates Order

#

y gap (^iV) (K)^ m 'a (K)

1 -50 -0.5 -5

2 +50 -0.5 -5

3 -50 +0.5 -5

4 +50 +0.5 -5

5 -50 -0.5 +5

6 +50 -Q.5 +5

7 -50 +0.5 +5

8 +50 +0.5 +5

9

The actual test sequence (not shown in Table 10) was randomized in order to minimize the

introduction of bias in the results. The guarded-hot-plate imbalance tests were conducted with lowdensity fibrous-glass CTS specimens having thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, and

228.6 mm (n = 36 data points) at a mean temperature of 297 K and a AT of 22.2 K across the

specimen. The apparatus was operated in the single-sided mode of operation utilizing a specimen

of expanded polystyrene foam having a nominal thickness of 100 mm as the auxihary insulation

(Figure 2).

Table 1 1 summarizes the 36 values for Q,n, AT, Vgap, T), - T'c, and, r,„ - Ta at specimen thickness of

25.4 mm, 76.2 mm, 152.4 mm, and 228.6 mm. For a particular specimen thickness, values of Qm

varied considerably, depending on the imbalance settings for x\ = Vgap, xj = T'h - T'c, and, x^, = TmTa. The balanced condition provided in Test #9 was used to establish a basehne value for analyses

of the data. At the balanced condition (Test #9), values of Q„, decreased from about 5.1 W to 0.6 W

as the sp)ecimen thickness increased from 25.4 mm to 228.6 mm. This 8 fold decrease in Qm is important and will be treated later.

28
Table 1 la - Imbalance Test Data in Yates Order

25.4 mm 76.2 mm

# (W) (K)

^ gap (K)

T -T

(K) (W) (K)

1/

* nap (K)

T -T

(K)

I 4.9756 22.22 ^9.96 -0.503 -5.01 1.6541 20 o"* -49.98 -^).499 -5.00

2 5.2325 22.22 +50.05 -0.498 -5.01 1.9186 22.22 +50.03 -0.500 -5.00

3 4.9281 22.22 ^9.94 +0.506 -5.01 1 .6065 22.22 -50.02 +0.497 -5.00

4 5.1829 22.22 +50.01 +0.501 -5.01 1.8712 22.22 +49.96 +0.497 -5.00

5 4.9847 22.22 -^9.97 -0.496 +4.99 1.6518 22.22 -50.00 -0.502 +5.00

6 5.2440 22.22 +50.03 -0.500 +4.99 1.9165 22.22 +50.00 -(J.500 +5.00

7 4.9392 22.22 -49.96 +0.504 +4.99 1.6037 22.22 ^9.96 +0.496 +5.00

8 5.1932 22.22 +50.02 +0.508 +4.99 1.8681 22.22 +49.99 +0.491 +5.00

9 5.0871 22.23 0.06 1.7609 22.22 -€.004

Table 1 lb - Imbalance Test Data in Yates Order

152.4 mm 228.6 mm

# (W)

Ar

(K)

ygap (hV) (K)

T -T

(K)

Qm

(W)

Ar

(K)

* f;ap (|iV) (K)

' m 'a (K)

1 0.7709 22.22 -50.06 -0.500 -5.01 0.4867 22.22 -49.97 -0.499 -5.00

2 1.0392 22.22 +50.02 -0.503 -5.01 0.7612 22.22 +49.96 -0.493 -5.00

3 0.7224 22.22 -50.00 +0.497 -5.00 0.4370 22.22 -49.99 +0.505 -5.00

4 0.9913 22.22 +50.04 +0.502 -5.00 0.7155 22.21 +50.07 +0.493 -5.00

5 0.7704 22.23 ^9.98 -0.498 +5.00 0.4884 22.22 ^9.91 -0498 +5.00

6 1.0396 22.22 +49.98 -0.497 +5.00 0.7666 22.22 +50.03 -0.502 +5.00

7 0.7231 22.21 ^9.98 +0.497 +5.00 0.4398 22.22 ^9.99 +0.502 +5.00

8 0.9916 22.22 +49.99 +0.498 +5.00 0.7180 22.22 +49.97 +0.500 +5.00

9 0.8797 22.22 +0.01 0.6004 22.21 +0.03

Values of Qm from Table 1 1 and coded values (±1) of jci, x?, and x^ were input into Equation (23) and

the least-squares estimated effects (fit, Pip and Pijk) were calculated using a NIST statistical graphical

analysis program [23] that employed Yates algorithm [24]. Table 12 summarizes whether the

estimated effects {fii, pj, and Pijk) are statistically significant at the 5 % level or the 1 % level. In all cases, factors JCi and xi were significant. Somewhat surprisingly, however, factor x^ was determined

to be insignificant for L = 152.4 mm and 228.6 mm. The effects estimates for all interactions (jciJC2,

JC1JC3, X2X-i, and xixjx^) across all thicknesses were insignificant. Based on these results, the (noncoded) data in Table 1 1 were subsequently fitted to a simphfied model (discussed below).

Table 12 - Statistical Significance for Estimated Effects for Imbalance Study

Factor 25.4 mm 76.2 mm 152.4 mm 228.6 mm

Xi **   ** 

Xi * ** ** *

^3

* * — —

X\X2 — — — —

X\X^ — — — —

XlX-i — — — —

X\X-2_Xs — — — —

**Statistically significant at the 1 % level

*Statistically significant at the 5 % level

29
The parasitic heat flows from the meter area, expressed mathematically in Equation (10), represent

lateral heat flows due to temperature differences across the guard gap and the auxiliary insulation,

and due to specimen edge effects. Based on the results from Table 12, the data from Table 1 1 were fit to an empirical model for Ag given in Equation (24) below,

AG - Q^ - e^ = «i V,., +«,(r;-r;)+«3(7; -rj (24)

where:

Q^ = the power input to the meter-plate heater for test / = 1 to 9; Q^ = the power input to the meter-plate heater for test / = #9 (balanced condition);

Uj = regression coefficients (j = 1,2,3);

Vgap = guard gap voltage corresponding to ATgap (fiV);

T'h = hot plate temperature (K);

T'c = auxiliary plate temperature (K);

Tm = mean specimen temperature (K) = (7), + Tc)/2; and,

Ta = ambient gas temperature near the edge of the specimen (K).

The presence of an offset coefficient uq was considered for the model given in Equation (24) but,

because the term is predicted to be nearly zero from theory (Equation 22), the term is not included here.

Table 13 summarizes estimates and approximate standard deviations determined by multiple

variable Unear regression for coefficients, au a2, and 013 as a function of specimen thickness. The

"goodness of fif denoted by the residual standard deviation (RSD) in the last column is less than 3

mW (Type A evaluation). If an offset coefficient ao is incorporated in Equation (24), the RSDs for

the fits are about 1 mW, or less. Regression coefficients ai, ai, and qt, are discussed below.

Table 13 - Estimates and Standard Deviations for a1, ai, and a^ in. Equation (24)

L

(mm) (W-^V') (W-^iV') (W-K"^)

s{a2)

(W-K-^)

^3

(W-K"')

5(^3)

(W-K"^)

RSD*

(W)

25.4 +0.002563 2.03 X 10"' -0.04817 2.02 X 10-^ +0.001064 2.03 x 10-^ 0.0029

76.2 +0.002646 4.36 X 10"^ -0.04801 4.39 X 10"^ -0.000266 4.36 X 10"' 0.0006

152.4 +0.002688 1.22 xlO'^ -^.04806 1.22x10"^ +0.000025 1.22x10^ 0.0017

228.6 +0.002773 1.26 X 10-' -0.04823 1.27x10"^ +0.000305 1.26x10^ 0.0018

* Residual standard deviation for fit

1) a\: Estimates for a\ in Table 13 represent the heat flow sensitivities {Sgap) in W-p,V'^ for the

4-pair thermopile as defined in Equation (21). At 25.4 mm, the a\ estimate is consistent

with previous results^ obtained for SRM 1450c, Fibrous Glass Board [3]. Note, however,

that estimates for a\ increase 8.2 % from 25.4 mm to 228.6 mm indicating that, for a fixed

^ Rennex [4] obtained a value of approximately 0.00057 W-^V"' for an 18-pair Type E thermopile used in the guard

gap. In 1994, the 1 8-pair thermopile was replaced, as part of upgrade and regular maintenance, with the 4-pair Type E

thermopile presented in this report. Note that the difference in sensitivities between the 4-pair and 18-pair thermopiles

is 4.5, roughly the same ratio observed between the value reported by Rennex [4] and the values for a\ reported in Table 13 (0.0026/0.00057 = 4.56).

30
temperature difference, the heat-flow across the guard gap increases with the specimen

thermal resistance. From Equation (20) and Table 1 , the ratio of gap heat flow and specimen

heat flow is

^^gap ^To 'gap

Q (AAT)/R

or ^= -^^(q^R + cL) (25)

Q A AT "

Therefore, for a particular apparatus (i.e., fixed A) and a given temperature imbalance {ATgap),

the ratio of lateral and specimen heat flows increases as the specimen resistance (R) and

thickness (L) increase if the other factors (namely AT) remain unchanged. Consequently, if one

assumes a constant coefficient for a\ based on L equal to 25.4 mm, an error in uncertainty is introduced for tests conducted at greater thicknesses.

2) aj. Estimates for ai in Table 13 represent the thermal conductance per unit temperature

(CflO of the auxiliary specimen (100 mm thick polystyrene foam) at 308 K (35 °C). The

average value for Table 13 is -0.04812 W-K"' and range is 0.00022 W-K'' (or 0.5 %). As a

check, the values for a? were compared with computed values obtained for the thermal

conductance (Q') computed from Equation (26) below:

C:=CA = ^. (26)

Substitution oik' = 0.0373 W-m'^-K'^ obtained from independent guarded-hot-plate tests at 311 K, A = 0.12989 m^ at 308 K, and V = 0.1013 m in Equation (26) yields a value of

0.0478 W-K* which is within 0.7 % of \a,\.

3) a^: Although the estimates for a^ in Table 13 represent a thermal conductance per unit

temperature, it is more useful to express these estimates as an edge heat flow error (e). By setting

Vgap = 0, (T'h - T'c) = 0, and dividing by Q^ , Equation (24) becomes

AQ Qm,-Q" ^{T^-Tj (27)

Table 14 summarizes values of 8 computed from Equation (27) for {T,„ - Ta) equal to K

and ± 5 K and values of 8 reproduced from Table 9 for L ranging from 25.4 mm to 228.6 mm.

Theoretical values of 8 are based on calculations from Equation (22) for the same values of

temperature imbalances. There are two general conclusions from the results presented in Table 14:At specimen thicknesses less than or equal to 76.2 mm, the measured effect is small

but greater than predicted by theoretical analysis. The fact that the empirical results are

31
not monotonic with L suggests that the observed variations for 152.4 mm and below

are due to experimental variations of factors other than L.

• At specimen thicknesses greater than or equal to 152.4 mm, the measured effect is much smaller than predicted by theoretical analysis.

Further work is recommended to investigate the differences between edge heat flow errors computed from empirically-derived models and from theoretical based models.

Table 14 - Comparison of Empirical and Theoretical Values of Edge Heat Flow Error (s) L

(mm)

Empirical i T„-Ta= -5 K 7^-? (Equation (27))

-Ta=OK r„-r,= +5K

Theoretical e (Table 9)

X = -0.450 X = X = +0.450

25.4 -0.0010 +0.0010

76.2 +0.0008 -0.0008

152.4 -0.0001 +0.0001 -0.0010 +0.0013

228.6 -0.0025 +0.0025 -0.0118 0.0002 +0.0122

Uc(AQ) - Standard uncertainty for parasitic heatflows: Substitution of xi = Vgap', xj = T'h - T'c, and,

X3 = Tm - Ta for the quantities in Equation (24) and application of Equation (6) yields:

u^ (AQ) = ^ciy{a^) + c\u^{x^) + cl^y{a^) + ciy^{x^) + c]y-{a^) + cl^u^(x^) (28)

with

c^=d{AQ)ldaj =Xj

c^^=d{AQ)/dxj =aj

Table 15 summarizes input values for Equation (28) and the corresponding Uc (AQ) for specimen

thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, and 228.6 mm. The input values for u(ai) and a, were

obtained from Table 13. Under steady-state test conditions, the input estimates for jc, are nearly zero (Table 15). The standard uncertainties for xi (ATgap), X2 (T'l-T'c), and x^ {T,n-Ta) were estimated to be

±0.01 K, which corresponds to ±2.48 |iV for Vgap, ±0.086 K and ±0.5 K, respectively. Note that the

resulting values of udAQ) are nearly constant across all levels of specimen thickness.

Table 15 - Estimates for w, (A^)

L (mm).V, u(ax) ax mU,)

(^V) rw-nv') (w-nv') (K) (K)

u{a^) a^ u(x2) (W-K') (W-k"') (K)

•^3 (K)

M(a3) 03 u(x^) (W-K') (W-K-^) (K)

Uc (AG)

(W)

25.4 0.01 2.03x10- +0.002563 2.48 0.005 2.02x1 0'^ -0.04817 0.086 0.004 2.03x10-^ +0.001064 0.5 0.0087

76.2 0.02 4.36x10"* +0.002646 2.48 0.003 4.39x10"^ -O.04801 0.086 0.003 4.36x10-' -0.000266 0.5 0.0083

152.4 0.01 1.22x10-' +0.002688 2.48 0.002 1.22x10-^ -0.04806 0.086 0.004 1.22x10"* +0.000025 0.5 0.0087

228.6 0.01 1.26x10-' +0.002773 2.48 0.002 1.27x10'^ -O.04823 0.086 0.002 1.26x10-^ +0.000305 0.5 0.0083

Uc (Q) - Standard uncertainty for specimen heatflow: Recalling the heat balance for the meter area given in Equations (10) and (11), the specimen heat flow (Q) was determined from the following

equarion:

Q - Q,n - ^Q (29)

32
Application of Equations (6) and (7) to Equation (29) yields

«. (Q) = ylul(Qj+ul{QJ+u'-iAQ) (30)

where the sensitivity coefficients are equal to unity and the contributory uncertainties are

Ma (Qm) = standard uncertainty of repeated power measurements {n = 240) during a test (W);

ub (Qm) = standard uncertainty of the meter-plate power input - see Table 8 (W); and,

Uc (A<2) = combined standard uncertainty for parasitic heat flows computed in Equation (28) -

see Table 15 (W).

Table 16 summarizes the estimates for the combined standard uncertainty Uc (Q) across all thicknesses. The dominant source by, in many cases, two orders of magnitude is the uncertainty in

the parasitic heat flows which, in turn, is due primarily to the uncertainty in the gap thermopile

voltage and temperature difference across the auxiliary insulation. The relative combined standard

uncertainty Uc,r (Q) increases considerably with specimen thickness from 0. 17 % at 25.4 mm to 1.4 %

at 228.6 mm.

Table 16 - Combined Standard Uncertainty udQ)

L

(mm)

Q

(W)

UAiQm)

(W)

UBiQm)

(W)

Uci^Q)

(W)

liciQ)

(W)

UcAQ)

(%)

25.4 5.1133 0.0006 0.0016 0.0087 0.0089 0.17

76.2 1.7920 0.0004 0.0007 0.0083 0.0083 0.46

152.4 0.8707 0.0003 0.0004 0.0087 0.0087 1.0 228.6 0.6075 0.0003 0.0003 0.0083 0.0083 1.4

10. Calculation of Combined Uncertainty

The standard uncertainties for meter area (A), thickness (L), temperature difference {1ST), and power

{Q) calculated in the previous section are combined using the law of propagation uncertainty given in

the GUM (Equation (6)). The combined standard uncertainties {udy) and Uciy)) for A^xp and R are presented for specimens of low-density fibrous-glass thermal insulation at thicknesses of 25.4 mm,

76.2 mm, 152.4 mm, and 228.6 mm. The expanded uncertainties {U, Ur) for X^xp and R are also

presented for the same thicknesses using a coverage factor of k = l.

Combined Standard Uncertainty and Expanded Uncertaintyfor X^xp

Application of Equation (6) to Equation (10) yields the combined standard uncertainty for X^xp:

"c (^exp ) = V4 "c (S) + Cl "c (^) + c\ u] (A) + cIj u~ (AT) (31)

with

33
L

eg — lyAexp/^v A{AT)

cl = 5Aexp/5L Q

A{AT)

ca = dKxpfdA

-QL

A- (AD

Cat = 5Aexp/5Ar

-QL

A / \ '-r\2

Table 17 summarizes the input estimates (xi), sensitivity coefficients (c,), standard uncertainties

(uc(xi)) for 2, L, A, and Ar at thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, 228.6 mm. The last two columns of each table provide the absolute and relative contributions of each component

uncertainty. The last two rows of each table provide Mc(^xp), "cX^xp), U, and Ur using a coverage

factor of /: = 2. The estimate for X^xp and corresponding bulk density (p) are also given on the last row of each table.

Table 17a - Combined Standard Uncertainty for Thermal Conductivity (kexp) for L = 25.4 mm

Xi X, d U (Xi)

\Ci u\

(W-m"'-K"')

(q u)/y

(%)

Q 5.1133 W 0.00881 m'-K-' 0.0089 W 0.00008 0.17

L 0.0254 m 1.77183 W-m"-K' 3.8x10'^ m 0.00007 0.15

A 0.12989 m' -0.34666 W-m"-K"' 2.47x10"' m' 0.00001 0.02

AT nil K -0.00203 W-m'-K" 0.086 K 0.00017 0.39

y = Xe^p = 0.0450 W-m"'-K"' (p = 9.4 kg-m"') Uc(K

U(k =

\p)

2)"

0.00020

0.00041

0.45

0.9

Table 17b - Combined Standard Uncertainty for Thermal Conductivity (k^xT^) for L -= 76.2 mm

Xi Xi Ci u (x,)

\Ci u\

rW-m"'-K"')

(c; u)/y

(%)

Q 1.7920 W 0.02641 m'-K' 0.0083 W 0.00022 0.46

L 0.0762 m 0.62078 W-m'-K"' 3.5x10"' m 0.00002 0.05

A 0.12989 m' -0.36429 W-m'-K"' 2.47x10"' m' 0.00001 0.02

AT 22.22 K -0.00213 W-m"'-K"' 0.086 K 0.00018 0.39

y = K.p = 0.0473 W-m'-K'(p = 9.0 kg-m"^) UcCK

U(k =

\p) 2)~

0.00029

0.00057

0.61

1.2

Table 1 7c - Combined Standard Uncertainty for Thermal Conductivity (kexp) for L = 152.4 mm

X, A- C, U (.V/)

\Ci u\

(W-m"'-K"')

(c, u)/y

(%)

Q 0.8707 W 0.05280 m"'-K' 0.0087 W 0.00046 1.0 L 0.1524 m 0.30168 W-m"-K"' 3.5x10"'

m

0.00001 0.02

A 0.12989 m' -0.35393 W-m"'-K"' 2.47x10"'

m' 0.00001 0.02

AT 22.22 K -0.00207 W-m"'-K"' 0.086 K 0.00018 0.39

y = K.p = 0.0460 W-m '-kVp = 9.6 kg-m"') Uc(K

U(k =

xp) 2)

0.00049

0.00099

1.1 2.1

34
Table 17d - Combined Standard Uncertainty for Thermal Conductivity (X^xp) for ^ =:228.6 mm

Xi X; c, u ix) (W-m'-K"')

(c, u)ly

(%)

Q 0.6075 W 0.07920•m"'•K"• 0.0083 W 0.00066 1.4 L 0.2286 m 0.21046 W-m"-K"' 3.5x10"''

0.00001 0.02

A 0.12989 m' -0.37039 W-m'-K' 2.47x10' m' 0.00001 0.02

A7 22.22 K -0.00217 W-m'-K' 0.086 K 0.00019 0.39

y = A^,p = 0.0481 W•m'•K (p = 8.7 kg-m"') "c/^xp)

U(k^2)

" 0.00068

0.00137

1.4 2.8

Using the values summarized in Table 17, Figure 10 plots the relative combined standard

uncertainty for X and individual uncertainty components for Q, L, A, and AT as a function of L. The graphical analysis clearly shows that the uncertainty contribution from Q is the major

component of uncertainty, especially at thicknesses of 75 mm and above. At 25 mm, the major

component is the standard uncertainty of AT which is fixed at 0.39 % across all levels of L. The

uncertainty contribution due to L is largest at 25 mm and declines dramatically, as expected, as L

increases. The uncertainty contribution due to A is nearly zero for all thicknesses.

2.0

1.8

1.6

1.4

„ 1-2

=" 0.8

0.6

0.4 -

0.2-

0.0

-X— Combined Uncertainty (l< = 1

)

A contribution

-0- - L contribution

• AT contribution ^- Q contribution

^'--.

:&--

50 100 150

L, mm

200 250

Figure 10 - Combined standard uncertainty and individual components for X

Figure 1 1 plots the estimates for thermal conductivity (A,) of the fibrous-glass blanket CTS given in Table 17 as a function of specimen thickness (L). The error bars shown in Figure 1 1 are equal to the

expanded uncertainties U (k = 2) given in Table 17. The data in Figure 1 1 show a "thickness effect"

which was the basis for the R-value Rule [9], and the subsequent development of the NIST CTS lot of material. The data in Figure 1 1 also reveal a material variability factor for the NIST CTS lot of

material.

35
0.050

0.049

0.048

E 0.047

0.046

0.045 H

0.044

Error bars (± U)

P (kgm"')

9.4

9.0

1

' A 9.6

ri 8.7

A

o

50 100 150

L, mm

200 250

Figure 1 1 - Thermal conductivity measurements of Fibrous-Glass Blanket NIST CTS as a function of Thickness

Combined Standard Uncertainty and Expanded Uncertaintyfor R

Application of Equation (6) to Equation (11) yields the combined standard uncertainty fori?:

u,m = -^clu-{Q) + cW^iA) + cl,u'^(AT) (32)

with

cq = dRIdQ = -A(AT)

A T' ca = dRJdA = Q

CAT = dR/dAT = —

Table 18 summarizes the input estimates (jc,), sensitivity coefficients (c,), standard uncertainties (udx,))

for Q, A, and AT at thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, and 228.6 mm. The last two

columns of each table provide the absolute and relative contributions of each component uncertainty.

The last two rows of each table provide R, udR), UcA^)^ ^^ and Ur using a coverage factor of A: = 2.

36
Table 18a - Combined Standard Uncertainty for Thermal Resistance (R) for L = 25.4 mm

X,- Xi c, U (Xi)

|c, u\

(m'-K-W"')

(c, ii)ly

Q 5.1133W -0.1 1038 m'-K-W' 0.0089 W 0.00098 0.17

A 0.12989 m' 4.34510 K-W' 2.47x10"'

m' 0.0001

0.02

AT 22.22 K 0.02540 m'-W

0.086 K 0.00219 0.39

y = R = 0.564 m-K-W ' u

U(k-.

(R)

= 2)

0.0024

0.0048

0.43

0.85

Table 1 8b - Combined Standard Uncertainty for Thermal Resistance (R) for L = 76.2 mm

x> Xi c, 11 ixd

\c, u\

(m'-K-W"')

(c, u)/y

Q 1.7920 W -0.89896 m'-K-W' 0.0083 W 0.00749 0.46

A 0.12989 m' 12.402 K-W' 2.47x10"-' m' 0.00031 0.02

AT 22.22 K 0.07249 m'-W"' 0.086 K 0.00625 0.39

y

= /?=1.61m-KW^ u

U(k--

~(R) = 2)0.010

0.020

0.61

1.2

Table 18c - Combined Standard Uncertainty for Thermal Resistance (R) for L = 152.4 mm

Xi X,- Ci u (x,)

\c.u\

(m'-K-W"')

(c, u)/y

Q 0.8707 W -3.8070 m'-K-W"' 0.0087 W 0.03319 1.0 A 0.12989 m' 25.5199 K-W"' 2.47x10"' m' 0.00063 0.02

AT 22.22 K 0.14918 m'-W"' 0.086 K 0.01286 0.39

y

= /? = 3.31m'K-W"' u

U(k--

(R)

= 2)

0.036

0.071

1.1 2 2

Table 18d - Combined Standard Uncertainty for Thermal Resistance (R) for L = 228.6 mm

Xi Xi Ci U {Xi)

k, "1 (m'-K-W"')

(f,i/)/v

(%)

Q 0.6075 W -7.82196-m'-K-W' 0.0083 W 0.06496 1.4 A 0.12989 m' 36.5804 K-W"' 2.47x10"' m' 0.00090 0.02

AT 22.22 K 0.21370 m'-W"' 0.086 K 0.01843 0.39

y

=:^ = 4.75m'-K-W' u

U(k-.

:(R) = 2)

0.068

0.135

1.4 2.8

Using the values summarized in Table 18, Figure 12 plots the relative combined standard

uncertainty for X and individual uncertainty components for Q, A, and AT as a function of L. The

graphical analysis is very similar to Figure 10 except for the absence of any uncertainty

contribution from L in Figure 12. The analysis again clearly shows that the contribution from Q is the major component of uncertainty, especially at thicknesses of 75 mm and above. The major

component is the standard uncertainty of AT which is fixed at 0.39 % across all levels of L. The

uncertainty contribution due to A is nearly zero for all thicknesses.

37
6^

if

2.0

1.8

1.6 -

1.4

1.2

1.0 -I

0.8

0.6

0.4 ^

0.2

0.0

—X— Combined Uncertainty (l< = 1)

" A contribution

- -»- - AT contribution

- -X- - Q contribution ^--^^^^^

- JX.^^' ^^^' X

'

__,-—

'^""^^^

-

'

_—--—

" -X

'

X '

50 100 150

L, mm

200 250

Figure 12 - Combined standard uncertainty and individual components for R

11. Reporting Uncertainty

The measurement result issued to a customer for a CTS specimen is for thermal resistance (/?), not

for thermal conductivity (Xexp)- Therefore, the expanded uncertainty U and relative expanded

uncertainty Ur for thermal resistance {R) are reported as follows:

Thermal Resistance (R): R, m^-K-W^ ± U, m--K-W"' (Ur, %)

where the reported uncertainty is an expanded uncertainty as defined in the International

Vocabulary of Basic and General terms in metrology, 2 ed., ISO 1993 calculated using a

coverage factor (k) of 2. The results given in this report apply only to the specimen tested, and

not to any other specimen fi-om the same orfrom a different lot ofmaterial.

The relative expanded uncertainty Ur for R is provided to a customer with two significant digits that are rounded up to the nearest 0.5 %. For example, the values of Ur for fibrous-glass blanket NIST CTS

given in Table 18 are rounded as shown in Table 19. Values of U for the customer are rounded to be

consistent with Ur.

Table 19 - Typical Values of ^ and Ur for Low-Density Fibrous-Glass Blanket NIST CTS

L

(mm)

R

(m--K-W"')

Vr

(%)

25.4 0.564 1.0 76.2 1.61 1.5 152.4 3.31 2.5 228.6 4.75 3.0

38
It is important to emphasize that other low-density fibrous-glass insulation materials may (and probably

will) have different values of R and, consequendy, different values of Ur than those given in Table 19. Furthermore, it usually inappropriate to include in the uncertainty for a NIST result any component that arises from a NIST assessment of how the result might be employed [1]. These uncertainties may

include, for example, effects arising from transportation of the measurement artifact to the customer

laboratory including mechanical damage; passage of time; and differences between the environmental

conditions at the customer laboratory and at NIST [ 1 ].

12. Discussion

NIST issues low-density fibrous-glass blanket CTS taken from an internal lot of insulation at nominal

thicknesses of 25 mm, 75 mm, 150 mm, or 225 mm. In general, measurements for customers are usually conducted at a mean temperature of 297 K and a temperature difference of either 22.2 K or 27.8 K. Figure 13 plots the nominal thermal resistance (R) of the reference material as a function of

specimen thickness (L) at Tm of 297 K and AT of 22.2 K. The error bars shown in Figure 1 3 are equal

to ±U (k = 2) given in Table 18.

6.0

5.0

4.0

3.0

2.0

1.0

0.0

Error bars (± U)

<^ P (kgm" ')

9.4

9.0 A 9.6

8.7

-

'

e

50 100 150

L, mm

200 250

Figure 13 - Thermal resistance measurements of Fibrous-Glass Blanket NIST CTS (T„ of 297 K, AT of 22.2 K)

Figure 14 examines the expanded uncertainties shown in Figure 13 in greater detail by plotting Ur

from Table 18 as a function of R. It is interesting to note that the trend for the expanded

uncertainty data is non-linear. There are two possible related explanations for this non-linear

behavior. At high levels of R, the specimen heat flow is reduced considerably thereby increasing

the sensitivity coefficients (c,) and udQ) as shown in Equation (32). Also, high levels of R are generally due to thick insulation specimens which will increase the edge heat fiow effects.

39
3.5 T

3.0

2.5

2.0

1.5

1.0

0.5

0.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0

Figure 14 - Relative expanded uncertainties (/c=2) versus thermal resistance of Fibrous-Glass Blanket NIST CTS

Dominant Uncertainty Components

Table 20 summarizes the individual contributions in percent (%) at the ^ = 1 level for A, L, hJ, and Q

(presented in Tables 17 and 18) for specimen thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, and

228.6 mm. At 25.4 mm, the dominant uncertainty component is for AJ and, at thicknesses greater

than 25.4 mm, the dominant component is for Q which increases considerably with L. The

contribution of u{lST) is fixed for the measurements presented herein at 22.2 K; however, the

contribution of m(A7) would change for different values of AT. It is interesting to note that the

contribution of L on Xexp is the largest at 25.4 mm, decreasing at higher specimen thicknesses.

Conversely, one would expect that, based on the results in Table 20, Uc.r(L) would increase for

Table 20 - Percent Contribution of Individual Components for Xexp and R

X, 25.4 mm

A^xp (Table 17)

76.2 mm 152.4 mm 228.6 mm 25.4 mm

R (Table 18)

76.2 mm 152.4 mm 228.6 mm

A 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02

L 0.15 0.05 0.02 0.02 — — — —

AT 0.39 0.39 0.39 0.39 0.39 0.39 0.39 0.39

Q 0.17 0.46 1.0 1.4 0.17 0.46 1.0 1.4

specimen thicknesses less than 25.4 mm. In that case, UcCKxp) could be larger than u^R) because,

strictly speaking, Uc (L) does not direcdy affect the uncertainty computation for R as shown in Equation

(11). Finally, the small contribution for A, which is fixed for the all thicknesses, could change

considerably at temperatures further from ambient due to thermal expansion effects (as noted above).

40
Dominant Contributory Sources

To investigate the results presented in Table 20 further, it is useful to re-examine the contributory sources for each component uncertainty of A, L, Ar, and Q. Figure 13 reproduces Figure 7 with the

dominant contributory source for each component identified (by circle). It is important to note that

these dominant contributory sources may be different for other measurements (i.e., different

specimen materials, operation temperatures, etc.). Furthermore, operators having other guardedhot-plates will note that the dominant contributory sources given here do not necessarily apply to

their own measurement process.

Heat flow (Q) Thickness (L)

Input power

measurement

Meter area (A)

Temperature

Difference (AT)

In-situ measurement

Short-term

repeatability

>^xp

calibration

Figure 1 3 - Cause-and-effect chart for X^xp with dominant contributory sources identified

The contributory sources identified in Figure 13 are discussed further below.

. Meter area (A): At Tm of 297 K and AJ of 22.2 K, UcM) is fixed at an estimate of 0.02 %

for all measurements presented in this report. The dominant contributory sources are the

uncertainties in plate dimensions of the guard gap (Table 3). However, at temperatures

departing from ambient, the contributory uncertainties for thermal expansion are expected

to increase noticeably.

• Specimen thickness (L): The dominant contributory source for L is due to the uncertainty

from the cold plate deflection under mechanical load (Table 6). This source is significant

for compressible materials, such as thermal insulation blankets, and is typically smaller for

more rigid and semi-rigid materials. In the case of rigid and semi-rigid materials, other

contributory sources in Table 6 could become more important.

• Temperature difference (AT): The dominant contributory source for AT is the digital

multimeter (DMM) measurement of the PRT temperature sensor which has a standard

uncertainty of 0.058 K (Table 7). The standard uncertainty is based on the manuraciurcr

specification and is probably a conservative estimate.

41
• Heat flow {Q): The dominant contributory source of Q is the uncertainty in the parasitic

heat flows (A^) across the guard gap and auxiliary insulation. The contributory sources for

these parasitic heat flows are the control variabihty of the gap thermopile voltage of 1.5 |iV or about 0.01 K (at the 3 standard deviation level) and the measurement of the PRT

temperature sensors.

Comparison with Previous Error Analysis

Officially, the uncertainty assessment presented herein supersedes the previous error analysis

prepared in 1983 [4]. This means that only the uncertainty values presented in this report should

be used for the NIST Calibrated Transfer Specimens. Nevertheless, there is technical merit in

discussing both analyses. Some of the obvious differences are the changes in measurement

processes from 1983 to 2008 that involve different operators as well as modifications to some of

the equipment (including both the apparatus and the data acquisition system). There is also a fundamental difference in the two approaches taken for the determination of combined standard

uncertainty as discussed below. An examination of both approaches also reveals similarities in

how individual and contributory uncertainties were determined. A brief discussion of the

differences and similarities of the two analyses is presented below.

Differences in approach: First and foremost, the error analysis prepared in 1983 [4] (hereafter,

1983 error analysis) preceded the official NIST policy [1] that adopted current international

guidehnes for the expression of measurement uncertainty [2]. Consequently, the terminology

given in the 1983 error analysis, as well as the approach for combining the uncertainties, has been

rendered out of date. Other specific differences in the two approaches are given below. The 1983

error analysis:

1) did not categorize the uncertainty components as either random or systematic as was done

in an uncertainty analysis of an earlier NBS guarded-hot-plate apparatus [16]; and,

2) estimated an "upper bound" of the "total uncertainty" by direct summation of the

uncertainty components.

In contrast, the GUM approach calculates a combined standard uncertainty by the root-sum-ofsquares approach shown in Equation (6). Further, the GUM and NIST pohcy [1] require that the

expanded uncertainty (U) be reported with a coverage factor (k) equal to 2 for international

comparisons. The 1983 error analysis does not report a coverage factor (k) and the "direct

summation approach" makes it difficult to assess the coverage factor. Without a coverage factor

(or a method to determine one) it is difficult, if not impossible, to compare the combined standard

uncertainties from the two analyses.

Similarities in individual uncertainty components: A brief review of the individual uncertainties in

the 1983 error analysis revealed that several of the estimates agree reasonably well with the values

presented in Table 1 7 for the combined standard uncertainty of Xgxp. Table 2 1 summarizes a sideby-side comparison of the individual uncertainties determined for this assessment and for the 1983

error analysis.

42
Table 21 -- Comparison of Individua Components in (%) for X^-xp

X, 25.4 mm

X,,p (Table 17)

76.2 mm 152.4 mm 228.6 mm

Atxp 25 mm

(from Table 1 75 mmin Reference [4])

150 mm 300 mm

A 0.02 0.02 0.02 0.02 0.01 0.01 0.01 0.01 L 0.15 0.05 0.02 0.02 0.1 0.03 0.02 0.01 AT 0.39 0.39 0.39 0.39 0.16 0.16 0.16 0.16

Q 0.17 0.46 1.0 1.4 0.04* 0.06* 0.08* 0.61*

*Value obtained by summing heat flow estimates from Table 1 in Reference [4].

For A and L, the standard uncertainties determined in this assessment and estimates of uncertainties for

the 1983 error analysis are in very good agreement. For AT, the standard uncertainties for this assessment are more conservative by a factor of approximately 2. As revealed in Table 21, the major

differences in the individual components are the estimates for the specimen heat flow Q. The

difference is attributed to the approach taken in this report for empirically determining the uncertainty

in parasitic heat flows of the meter area. The resulting Type B standard uncertainty ub(AQ), as observed in Table 16, dwarfs the other contributions for heat flow (Q) by, in some cases, two orders of

magnitude. The author believes that the approach presented here for determining estimates of udAQ)

using an experimental imbalance study is necessary to determine the standard uncertainty for the

specimen heat flow (Q).

13. Summary

An assessment of uncertainties for the National histitute of Standards and Technology (NIST)

1016 mm Guarded-Hot-Plate apparatus is described in this report. The uncertainties have been

reported in a format consistent with current NIST pohcy on the expression of measurement

uncertainty, which is based on recommendations by the Comite Intemational des Poids et Mesures

(CIPM) given in the Guide to the Expression of Uncertainty in Measurement. Strictly speaking,

these uncertainties have been developed for a particular lot of low-density fibrous-glass blanket

thermal insulation having a nominal bulk density of 9.6 kg-m"^. This reference material, known as a NIST Calibrated Transfer Specimen (CTS), is issued to customers as specimens 610 mm square

and at nominal thicknesses of 25.4 mm, 76.2 mm, 152.4 mm, and now 228.6 mm for use in

conjunction with the "representative thickness" provision of the U.S. Federal Trade Commission

(FTC) "R-value Rule." The relative expanded uncertainty at a coverage factor of k equal to 2 for

the thermal resistance of this material increases from 1 % for a thickness of 25.4 mm to 3 % for a thickness of 228.6 mm. The approach for the assessment of uncertainties that has been developed

herein is applicable to other insulation materials measured at 297 K.

Recommendationsfor Future Research

The uncertainty analyses given in this report have identified dominant components of uncertainty

and, thus, potential areas for future measurement improvement. For the NIST 1016 mm GuardedHot-Plate apparatus considerable improvement, especially at higher levels of R, may be realized by

developing better control strategies for the guard gap that include better measurement techniques

for the gap voltage and PRT temperature sensors. In some cases, determining the indi\idual

standard uncertainties has required establishing traceability to NIST metrology laboratories,

specifically for thermometry and primary electrical measurements. Recent calibrations from these

43
metrology laboratories have indicated that more frequent checks and/or calibrations are required in

the future. An extensive list of recommendations and future activities is given below.

• Annual check of platinum resistance thermometers (PRTs): The last calibration by the

NIST Thermometry Group revealed a stability problem with one of the cold-plate PRTs.

To track any drift or potential problem, an annual check of the cold-plate PRTs at the triple

point of water is now planned.

• Annual calibration ofstandard resistor. The last calibration by the NIST Quantum Electrical

Metrology Division revealed a possible drift in the standard resistor used for the meter-plate

power measurement. The presence of the drift could be indicative of detrimental factors

affecting the resistor itself. Annual cahbrations are now planned to investigate the extent of

the possible drift.

• Improvement in thickness measurement. The dominant component for the thickness uncertainty

is the analysis of the plate deflection under mechanical load. Because the approach presented

here has hmitations, an altemative technique to assess the plate deflection more accurately

could reduce the thickness uncertainty. This would be useful at specimen thicknesses of

25.4 mm or less.

• Improvement in temperature measurement. The dominant contributory source for the

specimen AT* is the digital multimeter (DMM) measurement of the PRT temperature sensor, which has a standard uncertainty of 0.058 K. The standard uncertainty for the DMM

measurement is based on the manufacturer specification and is probably conservative.

Further, the uncertainty probably includes systematic effects of unknown magnitude which

will largely cancel when the specimen difference is computed in Equation (16). A significant

improvement in the temperature measurement could be realized by development of new

instrumentation for measuring the PRT temperature sensors. This improvement would also

result in a lower uncertainty for the parasitic heat flows as discussed below.

• Reduction in the uncertainty of parasitic heat flows: The uncertainties of the parasitic heat

flows are due primarily to the uncertainty in temperature measurement (and control) of the

guard gap thermopile and the temperature difference of the auxiUary plate temperatures. As

discussed above, reduction in the temperature measurement uncertainty for PRTs would

significantly reduce the uncertainty in the parasitic heat flow, which is the dominant

component of the specimen heat-flow uncertainty.

• Analysis of error due to edge heat transfer. Further work is recommended to investigate the

differences between edge-heat flow errors computed from theory and those computed from

an empirically-derived model presented in this report.

44
Acknowledgements

The author is indebted to D. R. Flynn, retired from National Bureau of Standards, for his review,

discussions, and assistance, in particular, with the edge-heat flow analysis. The author appreciates

the many comments and discussions with D. L. McElroy, retired from the Oak Ridge National

Laboratory, Dr. Dean Ripple of the NIST Thermometry Group, and Will Guthrie of the NIST

Statistical Engineering Division.

45
14. References

I] Taylor, B. N. and C. E. Kuyatt, "Guidelines for Evaluating and Expressing the Uncertainty of NIST

Measurement Results," NIST Technical Note 1297, 1994 Edition.

2] ISO, "Guide to the Expression of Uncertainty in Measurement," International Organization for

Standardization, Geneva, Switzerland, 1993.

3]Zarr, R. R., ^'Standard Reference Materials: Glass Fiberboard, SRM 1450c, for Thermal Resistance from

280 K to 340 K," NIST Special Publication 260-130, AprU 1997.

4]Rennex, B. G., "Error Analysis for the National Bureau of Standards 1016 mm Guarded Hot Plate," NBSIR

83-2674, February 1983 (jepnnX&d'm Journal of Tliemial Insulation, Vol. 7, July 1983, pp. 18-51.)

5]ASTM C 1043-06, "Standard Practice for Guarded-Hot-Plate Design Using Circular Line-Heat

Sources," Annual Book ofASTM Standards, ASTM International, West Conshocken, PA, 2006.

6]ASTM Adjunct ADJC1043, "Line-Heat-Source Guarded-Hot-Plate Apparams," R.R. Zarr and M. H.

Hahn, Eds., (available from ASrM International, West Conshocken, PA,).

7] ASTM Standard C 177-04, "Standard Test Method for Steady-State Heat Flux Measurements and

Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparams," Annual Book of

ASTM Standards, ASTM hitemational. West Conshocken, PA, 2004.

S]Federal Register, "National Bureau of Standards: Availability of Calibration Transfer Specimens for

hisulation," December 9, 1980, p. 81089.

9]Federal Register, "Federal Trade Commission 16 CFR Part 460: Trade Regulations: Labeling and

Advertising of Home Insulation," August 27, 1979, pp. 50218-50245 (last updated in 2005).

10] ASTM C 1045-07, "Standard Practice for Calculating Thermal Transmission Properties Under SteadyState Conditions," Annual Book ofASTM Standards, ASTM Intemational, West Conshocken, PA, 2(X)7.

II] ASTM C 1044-07, "Standard Practice for Using a Guarded-Hot-Plate Apparatus or Thin-Heater

Apparatus in the Single-Sided Mode," Annual Book ofASTM Standards, ASTM Intemational, West

Conshocken, PA, 2007.

12] Hahn, M. H., Robinson, H. E., and D. R. Flynn, "Robinson Line-Heat-Source Guarded Hot Plate Apparatus," Heat Transmission Measurements in Thermal Insulations, ASTM STP 544, 1974, pp. 167-192.

13] EUison, S. L. R., Rossiein, M., and A. Williams, "EURACHEM/CITAC Guide: (^antifying Uncertainty

in Analytical Measurement," 2"^ Edition, EURACHEM/CITAC, 2000.

14] Burdick, R. K., Confidence Intervals on Variance Components, 1992, p. 66.

15] Roark, R. J., Formulas for Stress and Strain, 4th Ed., 1965, p. 216, Case 2. 16] Siu, M. C. I. and C. Bulik, "National Bureau of Standards line-heat-source guarded-hot-plate

apparatus," Review ofScientific Instruments, Vol. 52, No. 1 1, November 1981, pp. 1709-1716.

17] Thomas, J. L., "Stability of Double-walled Manganin Resistors," Journal ofResearch National Bureau

of Standards, 36, 1946, pp. 107-110.

18] Belecki, N. B., Dziuba, R. F., Field, B. F, and B. N. Taylor, "Guidelines for Implementing the New

Representations of the Volt and Ohm Effective January 1, 1990," NIST Technical Note 1263, June 1989.

19] Woodside, W. and A. G. Wilson, "Unbalance Errors in Guarded Hot Plate Measurements," Symposium

on Thermal Conductivity Measurements arui Applications of Thermal Insulation, ASTM STP 217,

1957, pp. 32-64.

20] Bums, G. W., Scroger, M. G., Strouse, G. F., Croarkin, M. C, and W. F. Guthrie, "TemperamreElectromotive Force Reference Functions and Tables for Letter-Designated Thermocouple Types

Based on the ITS-90," NIST Monograph 175, April 1993.

2 1 ] Orr, H. W., "A Study of the Effects of Edge Insulation and Ambient Temperamre on Errors in Guarded

Hot-Plate Measurements," Thermal Conductivity - /'' Conference, NBS Special Publication 302, 1968,

pp. 521-526.

22] Peavy, B. A, and B. G. Rennex, "Circular and Square Edge Effect Study for Guarded-Hot-Plate and

Heat-Flow-Meter Apparatuses," Journal of Thermal Insulation, Vol. 9, April 1986, pp. 254-300.

23] Filliben, J. J. "Dataplot - An Interactive High-Level Language for Graphics, Non-Linear Fitting, Data

Analysis, and Mathematics," Computer Graphics, Vol. 15, No., 3, April 1981, pp. 199-213.

46
[24] Box, G. E. P., Hunter, W. G., and J. S. Hunter , Statistics for Experimenters, (John Wiley & Sons, Inc.), 1978, p. 342.

47
Appendix A - Meter Area (A) Derivation

Test Method C 177 [7] defines the meter area as:

A

A = K, +

^

(A-l)

where:

A,„p = surface area of the meter plate (m"); and,

Agap = surface area of guard gap between meter plate and the guard plate (Figures 1 and 2),

For the circular meter area of the NIST guarded-hot-plate. Equation (A-l) is rewritten as

A = nr'^ + [nrl - nr^')/2 = |(rj + r^) (A-2)

where:

To = outer radius of meter plate (m); and,

r, = inner radius of guard plate (m).

Including thermal expansion effects. Equation (A-2) is rewritten in the following form:

A = ^[r^{l + aATJ- + r^{Ua^TJ] = |[(r>^.^)(l + aA7;J'] (A-3)

where:

a = coefficient of thermal expansion of aluminum (alloy 6061-T6) (K"'); and,

ATmp = temperature difference of the meter plate from ambient (K) = 7), - 20 °C.

Further simplification of Equation (A-3) can be obtained as shown in Equation (A-4):

A = 7ir/(l + aAr)' (A-4)

by substitution of:

J

2 , 2

48