NIST Special Publication 260-175 

Standard Reference Materials: 

SRM 1453, Expanded Polystyrene 

Board, for Thermal Conductivity from 

281 K to 313 K 

Robert R. Zarr 

Adam L. Pintar 

http://dx.doi.org/10.6028/NIST.SP.260-175
NIST Special Publication 260-175

Standard Reference Materials: 

SRM 1453, Expanded Polystyrene 

Board, for Thermal Conductivity from 

281 K to 313 K 

Robert R. Zarr 

Energy and Environment Division 

Engineering Laboratory 

Adam L. Pintar 

Statistical Engineering Division

Information Technology Laboratory

http://dx.doi.org/10.6028/NIST.SP.260-175 

December 2012

U.S. Department of Commerce 

Rebecca Blank, Acting Secretary 

National Institute of Standards and Technology 

Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology and 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 Special Publication 260-175

Natl. Inst. Stand. Technol. Spec. Publ. 260-175, 57 pages (December 2012) 

http://dx.doi.org/10.6028/NIST.SP. 260-175

CODEN: NSPUE2 
iii 

Authors’ Note: This document is based on NISTIR 5838, Room-Temperature Thermal 

Conductivity of Expanded Polystyrene Board for a Standard Reference Material (May 

1996). Most of the content remains the same; the thermal conductivity data and the corresponding uncertainty analysis have been updated to reflect recent modifications in the 

heat flow determination. The statistical analysis of the data has been updated following 

an errors-in-variables approach. In addition, the terminology has been updated in conformance with current documentary standards for the guarded-hot-plate test method. 

 
iv 

Abstract

Thermal conductivity measurements at and near room temperature are presented as the 

basis for certified values of thermal conductivity of SRM 1453, Expanded Polystyrene 

Board. The measurements have been conducted in accordance with a randomized full 

factorial experimental design with two variables, bulk density and temperature, using the 

NIST 1016 mm line-heat-source guarded hot plate apparatus. The thermal conductivity 

measurements were conducted over a range of bulk density from 37.4 kg/m3 to 

45.8 kg/m3 and mean temperature from 281 K to 313 K. Uncertainties of the measurements, consistent with current ISO guidelines, have been developed. Statistical analyses 

of the physical properties of the SRM are presented and include variations between 

boards, as well as within boards. Measurements of the foam surface roughness, microstructure, and compressive resistance are presented. 

Each unit of SRM 1453 is batch certified for thermal conductivity with the following 

equation: 

λ 0.00111 0.000 042 4ρ 0.000115   Tm

where λ is the predicted thermal conductivity (W/(m·K)), ρ is the bulk density (kg/m3) 

and, Tm is the mean temperature (K) valid over the bulk density and temperature ranges 

of 37 kg/m3 to 46 kg/m3 and 281 K to 313 K, respectively. The expanded uncertainty for 

λ is 1.5 % with a coverage factor of approximately k = 2. 

Keywords

calibration; bulk density; expanded polystyrene board; fenestration; guarded-hot-plate 

apparatus; heat-flow-meter apparatus; standard reference material; SRM 1453; thermal 

conductivity; thermal insulation; uncertainty 

 
v 

Table of Contents

1. Introduction ......................................................................................................................1 

1.1 Thermal Insulation SRMs ........................................................................................1 

1.2 Historical Background .............................................................................................2 

2. Certification Project Design .............................................................................................4 

2.1 Project Definition and Scope for Intended Use .......................................................4 

2.2 Material ....................................................................................................................4 

2.2.1 Fabrication ......................................................................................................4 

2.2.2 Surface Preparation .........................................................................................4 

2.2.3 Delivery...........................................................................................................4 

2.3 Project Design Plan ..................................................................................................5 

2.3.1 Inspection and Storage ....................................................................................6 

2.3.2 General Sampling Procedure ..........................................................................6 

2.4 Measurement Methods .............................................................................................6 

2.4.1 Bulk Density ...................................................................................................6 

2.4.2 Steady-State Thermal Transmission Properties ..............................................6 

2.4.3 Supplementary Measurements ........................................................................6 

3. Material Properties ...........................................................................................................7 

3.1 Bulk Density of Material Lot ...................................................................................7 

3.1.1 Graphical Analysis of Lb and ρb Data .............................................................7 

3.1.2 Summary Statistics for Material Lot .............................................................10 

3.2 Bulk Density within a Board..................................................................................10 

3.3 Microstructure ........................................................................................................13 

3.4 Surface Roughness .................................................................................................14 

4. Thermal Conductivity Measurement .............................................................................15 

4.1 Guarded-Hot-Plate Method ....................................................................................15 

4.2 Experimental Design and Initial Model .................................................................16 

4.3 Test Specimens ......................................................................................................17 

4.4 Test Sequence ........................................................................................................18 

4.4 Test Procedure .......................................................................................................19 

5. Data Evaluation ..............................................................................................................20 

5.1 Data Summary (Tabular Format) ...........................................................................20 

5.2 Uncertainty Budget, λexp ........................................................................................21 

5.3 Data Screening (Graphical Analysis) .....................................................................22 

5.4 Correction for Metered Section Bulk Density .......................................................22 

5.5 Standard Uncertainty for Metered Section Bulk Density Correction ....................25 

6. Regression Analysis .......................................................................................................26 

6.1 Model Selection .....................................................................................................26 

6.2 Estimation and Uncertainty....................................................................................27 

7. Certification ...................................................................................................................29 

7.1 Properties of Interest ..............................................................................................29 

7.2 Values and Uncertainties .......................................................................................29 

7.3 Instructions for Use ................................................................................................29 
vi 

7.3.1 Storage ..........................................................................................................29 

7.3.2 Preparation and Conditioning before Measurement .....................................29 

7.3.3 Thermal Conductivity Measurement ............................................................30 

7.3.4 Guidelines and Precautions ...........................................................................30 

Acknowledgments..............................................................................................................30

References ..........................................................................................................................31 

Annex 1 - Uncertainty Analysis for Thermal Conductivity (λ) .........................................33 

A1.1 Background .........................................................................................................33 

A1.2 Uncertainty for λexp .............................................................................................34 

A1.3 Specimen Heat Flow (Q).....................................................................................34 

A1.3.1 Uncertainty for Power Measurement (Qm) .................................................35 

A1.3.2 Guard Imbalance (Qg) ................................................................................36 

A1.3.3 Combined Standard Uncertainty u(Q) .......................................................39 

A1.4 Meter Area (A) ....................................................................................................39 

A1.4.1 Plate Dimensions........................................................................................40 

A1.4.2 Thermal Expansion Effects ........................................................................40 

A1.4.3 Combined Standard Uncertainty u(A) ........................................................40 

A1.5 In-situ Thickness (L) ...........................................................................................41 

A1.5.1 Fused-quartz Spacers, u1(L)A .....................................................................41 

A1.5.2 Micrometer, u2(L)B .....................................................................................41 

A1.5.3 Linear Positioning System, u3(L)B .............................................................42 

A1.5.4 Repeatability of Linear Positioning System, u4(L)A ..................................42 

A1.5.5 Plate Flatness, u5(L)A and u6(L)B ................................................................42 

A1.5.6 Plate Deflection, u7(L)B ..............................................................................43 

A1.5.7 Combined Standard Uncertainty u(L) ........................................................43 

A1.6 Temperature Difference (ΔT) ..............................................................................44 

A1.6.1 Sensor Calibration, u1(T)B ..........................................................................44 

A1.6.2 Regression Analysis, u2(T)B .......................................................................44 

A1.6.3 Electrical Resistance Measurement, u3(T)B ...............................................45 

A1.6.4 Temperature Rise due to PRT Self-heating, u4(T)B ...................................45 

A1.6.5 Radial Plate Temperature Variation, u5(T)B ...............................................45 

A1.6.6 Axial Plate Temperature Variation, u6(T)B ................................................45 

A1.6.7 Combined Standard Uncertainty u(T) ........................................................45 

A1.6.8 Combined Standard Uncertainty u(ΔT) .....................................................46 

A1.6.9 Combined Standard Uncertainty u(Tm) ......................................................46 

Annex 2 - Uncertainty Analysis for Bulk Density (ρ) .......................................................47 

A2.1 Digital Balance Uncertainty ................................................................................47 

A2.2 Length Uncertainties ...........................................................................................47 

A2.3 Uncertainty Budget .............................................................................................47 

A2.4 Combined Standard Uncertainty .........................................................................48 

A2.4.1 Square Specimen Geometries ....................................................................48 

A2.4.2 Circular Specimen Geometries ..................................................................49 

A2.4.3 Metered Section Correction for Specimen Pair .........................................49 

A2.5 Effect of Buoyant Force ......................................................................................50 

Annex 3 - Compressive Resistance of Expanded Polystyrene Foam ................................51 
vii 

Annex 4 – Bayesian Errors in Variables Analysis .............................................................53 

A4.1 Nomenclature (specific to Annex 4) ...................................................................53 

A4.2 Background .........................................................................................................53 

A4.3 Calculations .........................................................................................................54 

A4.4 Uncertainty ..........................................................................................................54 

Annex 5 - Sorption Isotherms for Expanded Polystyrene Foam .......................................56 

Annex 6 - General Precautions for Expanded Polystyrene Foam ......................................57 

A6.1 Upper Temperature Limit ...................................................................................57 

A6.2 Flammability .......................................................................................................57 

A6.3 Solvents ...............................................................................................................57

A6.4 Ultraviolet Degradation .......................................................................................57 
1 

1. Introduction

In general, a Standard Reference Material®1 (SRM) issued by the National Institute of 

Standards and Technology (NIST) is a homogeneous and stable material which is measured accurately and certified as a reference material for purposes of evaluating a measurement process [1]. Standard Reference Materials are provided by NIST as primary 

tools to assist user communities in achieving measurement quality assurance and metrological traceability. These materials are used by industry, academia, and government to 

verify or improve the accuracy of specific measurements and to advance the state-of-theart knowledge. New SRMs are developed, usually after a formal request, in order to satisfy the measurement needs of a user community. 

1.1 Thermal Insulation SRMs

Thermal insulation SRMs are issued by NIST for materials with certified value assignments for thermal resistance and thermal conductivity over a range of parameters, such as 

bulk density and temperature. These SRMs are utilized in standard test methods for the 

purposes of checking guarded-hot-plate apparatus [2], calibrating heat-flow-meter apparatus [3], and, when necessary, for checking or calibrating hot-box apparatus [4]. The 

systematic use of thermal insulation SRMs, including proper tracking with control charts, 

provides the means for accurate inter-laboratory comparison of thermal conductivity data. 

Value assignments for thermal insulation SRMs issued by NIST are developed using the 

guarded-hot-plate method [2]. The method is considered an absolute measurement procedure because the resulting thermal transmission properties are determined directly from 

basic measurements of length, area, temperature, and electrical power. 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. By accurately monitoring the plate separation and knowing the geometric shape factor for the heat flow, the steady-state heat transmission properties of the 

test specimen are determined using the Fourier heat conduction equation. Influence 

quantities such as plate clamping pressure, plate emittance, and ambient air temperature, 

among others, are controlled; while other quantities such as ambient air pressure are monitored during the measurement process. In principle, the method can be used over a wide 

range of insulating materials, mean temperatures, and temperature differences. 

The thermal resistance and thermal conductivity of a thermal insulation SRM are generally characterized as functions of bulk density and mean temperature. The characterization 

is typically accomplished by batch certification. A sampling scheme is used to select 

specific specimens from the material lot for testing in the guarded-hot-plate apparatus. 

The analysis of the thermal conductivity data of the sample sub-lot is used for certification of the SRM lot. Consequently, the uncertainty statement for a thermal insulation 

SRM contains a component of uncertainty (usually small) due to the material lot variability. It should be noted that a thermal insulation SRM unit issued to a customer has not 

been measured directly in a NIST guarded-hot-plate apparatus. The advantage of the 

 

1

 The term “Standard Reference Material” and the diamond-shaped logo which contains the term “SRM,” 

are registered with the United States Patent and Trademark Office. 
2 

batch approach is realized by characterizing a large quantity of units that are economical 

and available on demand. In practice, thermal insulation SRM lots are prepared with a 

sufficient number of units to meet anticipated demand for a period of ten years. 

Standard Reference Material 1453 is a rigid, high-density, molded expanded polystyrene 

board that was fabricated from a single production run by a commercial manufacturer of 

molded polystyrene foam products. Standard Reference Material 1453 is one of several 

certified thermal insulation reference materials issued by NIST. These related thermal 

insulation SRMs have been categorized by the NIST Standard Reference Materials Program (SRMP) in Table 203.17 – Thermal Resistance and Thermal Conductivity Properties of Glass, Silica, and Polystyrene (solid forms) reproduced in Table 1. 

Table 1. Thermal resistance and thermal conductivity of glass, silica, and polystyrene 

Designation Description Temperature range (K) 

1449 Fumed silica board 297.1 

1450d Fibrous glass board 280 to 340 

1452 Fibrous glass blanket 297.1 (100 to 330) 

1453 Expanded polystyrene board 285 to 310 

1459 Fumed silica board 297.1 

NIST Special Publication 260-175 (this publication), which is part of the “NIST Special 

Publication 260 Series,” provides supplemental documentation for the 1453 Certificate 

and covers the following subject matter: 

1) historical background of SRM 1453; 

2) project plan for certification including the fabrication and procurement of the material lot; 

3) measurement methods for the bulk density and thermal conductivity evaluations; 

4) uncertainty analysis; and, 

5) certification. 

1.2 Historical Background

The motivation for SRM 1453, Expanded Polystyrene Board, began in the late 1980s during the development of the ASTM Test Method C1199 [4] for the thermal evaluation of 

fenestration systems. The test method requires the use of a large calibration transfer 

standard having known thermal transmission properties in order to estimate the surface 

heat transfer coefficients of more complex fenestration systems. The motivation increased when the Energy Policy Act of 1992 [5] mandated that a voluntary window rating 

program be developed by the National Fenestration Rating Council (NFRC) according to 

accepted national testing procedures. 

On March 11, 1993, representatives from NFRC, industry, and the National Voluntary 

Laboratory Accreditation Program (NVLAP) met at the National Institute of Standards 

and Technology (NIST) to discuss, among other topics, a proposal for a new SRM. The 

primary purpose of the new SRM would be to assist the thermal testing community in the 

thermal evaluation of fenestration systems, especially windows. The meeting representatives discussed the criteria for the new SRM with regards to the design requirements for 
3 

the calibration transfer standard. The design for the transfer standard was based on work 

done at the National Research Council of Canada [6] and consisted of 13-mm thick expanded polystyrene foam (20 kg/m3

) bonded, on both sides, to 3 mm glass sheets [7]. 

Preliminary measurements were conducted by NIST on similar specimens of molded expanded polystyrene foam provided for this purpose by insulation manufacturers. The 

bulk density of the specimens ranged from 41 kg/m3

 to 49 kg/m3 and the average thermal 

conductivity (at 24 C) for the specimens was 0.033 W/(mK). The effect of density on 

thermal conductivity was found to be fairly insensitive for boards about 13 mm in thickness. Based on these preliminary studies, NIST procured a large number of boards of 

molded expanded polystyrene foam for production of SRM 1453. 
4 

2. Certification Project Design 

2.1 Project Definition and Scope for Intended Use

The certification project is defined as follows. 

“The preparation of thermal insulation SRM 1453 for thermal resistance and thermal 

conductivity measurements with expanded uncertainties (k = 2) associated with the certified values of less than or equal to 2 % over a mean temperature range of 281 K to 

313 K.” 

Standard Reference Material 1453 is intended for use as a proven check for the guardedhot-plate apparatus, hot box apparatus (or other absolute thermal conductivity apparatus), 

and for calibration of a heat-flow-meter apparatus over the temperatures 280 K to 340 K. 

This report cannot exclude the use of SRM 1453 for other purposes, but the user is cautioned that other purposes are not necessarily covered by the 1453 Certificate or by this 

report. Additional usage issues are covered in Sec. 7.3.4 and in the 1453 Certificate (under Instructions For Handling, Storage, And Use). 

2.2 Material

Standard Reference Material 1453 is a commercial rigid, high-density, molded expanded 

polystyrene board. The nominal physical properties of the material are as follows: 

 size: 660 mm by 930 mm 

 thickness: 13 mm 

 bulk density: 41 kg/m3

2.2.1 Fabrication

NIST purchased 300 boards of molded polystyrene foam from Polyfoam, Incorporated2

in April 1994. The foam beads were manufactured by expanding (under heating conditions) particles of a polystyrene polymer saturated with a volatile hydrocarbon, such as 

isopentane. The expanded beads were subsequently placed in a plank mold and heated 

under pressure until the beads fused together. 

2.2.2 Surface Preparation

To remove surface imprints from the molding process, the boards were shipped directly 

to a second vendor (Rollin Incorporated2

) where both sides of each board were sanded 

using a modified milling machine in order to obtain a uniform thickness. The final nominal dimensions of the sanded boards were 660 mm by 930 mm by 13 mm thick. 

2.2.3 Delivery

The boards were delivered to NIST on November 10, 1994, and placed in a storage room 

maintained at 21 C ± 2 C and a relative humidity that ranged from 30 % to 60 %. 

 

2

 Certain commercial equipment, instruments, or materials are identified in this report to specify adequately 

the experimental procedure. Such identification does not imply recommendation or endorsement by NIST, 

nor does it imply that the materials or equipment are necessarily the best for the purpose. 
5 

2.3 Project Design Plan

The schematic in Figure 1 outlines the project plan for the development and production of 

SRM 1453, Expanded Polystyrene Board. The experimental portion of the plan, including the selection of test specimens, was based on an experimental design provided by the 

NIST Statistical Engineering Division. 

Figure 1. Certification project plan for NIST SRM 1453, Expanded Polystyrene Board

Manufacturer 

300 boards 

660 mm  930 mm 

13 mm thick 

Finisher 

Surface sanding 

NIST 

300 boards 

NIST SRM Program 

 Documentation review 

 Pricing 

 Customer support 

 Storage and shipping 

Discard 

10 damaged boards 

Bulk Density 

300 boards

NIST 1016 mm 

Guarded Hot Plate 

30 boards (15 pairs) 

657 mm  657 mm 

Design of 

Experiment 

30 boards 

10 boards 

Material Properties 

 within-board bulk density 

 microstructure 

 surface roughness 

 compressive resistance 

 moisture sorption 

1 board (ID 049) 

Regression 

Analysis 

Thermal 

Conductivity 

Measurements 

Certification 

Documentation 

259 boards

Uncertainty 

Analysis 
6 

2.3.1 Inspection and Storage

The insulation boards were visually inspected for damage after delivery. Each board was 

identified with a temporary 3-digit number assigned from 001 to 300. The material lot 

was stored for several months in laboratory workspace at ambient conditions. 

2.3.2 General Sampling Procedure

The sampling plan was based on the certification project plan shown in Fig. 1 and, in particular, on the experiment design referred to as part of the project plan. For 100 % sampling of the material lot, the boards were individually removed from their shipping box 

containers for mass and dimensional measurements. The 300 boards were processed 

through a two-day measurement procedure described in Sec. 2.4.1. 

2.4 Measurement Methods

2.4.1 Bulk Density

The bulk density, as defined in ASTM Test Method C 177 [2], was determined for each 

individual finished board from established gravimetric and dimensional measurement 

procedures that are described in Sec. 3. The major objective of the bulk density study is 

to assess the material variability of the material lot (i.e., variability between insulation 

boards), thereby providing quantitative information for the following: 

 quantitative ranking of the material lot by bulk density; 

 the upper and lower bulk density limits of the material lot; and, 

 the detection of any anomalous thermal insulation boards for possible exclusion. 

2.4.2 Steady-State Thermal Transmission Properties

The steady-state thermal transmission measurements (i.e., thermal conductivity and thermal resistance) were determined in accordance with ASTM Test Method C 177 [2] using 

the NIST 1016 mm guarded-hot-plate apparatus. In contrast to the 100 % sampling process for the bulk density study, the thermal conductivity of 1453 was batch certified. The 

material lot was sub-sampled (Fig. 1) using 30 boards (15 pairs) at three levels of bulk 

density (low, mid, and high). The quantitative values for these density rankings were defined using the results of the bulk density study (Sec. 3). Detailed procedures of the 

guarded-hot-plate test method, apparatus, corresponding uncertainty, and thermal characterization are documented in Sec. 4-5. 

2.4.3 Supplementary Measurements

Figure 1 outlines supplementary measurements for material properties that were not part 

of the certification process for SRM 1453. These measurements were intended to provide 

further insight about the statistical characterization of the material lot and include the following: 

 within-board bulk density analysis; 

 microstructure using a scanning electron microscope; 

 surface roughness using a contact (stylus) profilometer; 

 compressive resistance; and, 

 moisture sorption using fixed-point humidities. 
7 

3. Material Properties

Section 3 describes the physical properties of the material lot, including density variations between boards and within one board, cellular microstructure, and surface profile. 

3.1 Bulk Density of Material Lot

The bulk density for a board (ρb, kg/m3

) was determined from gravimetric and dimensional measurements of the board using Eq. (1). 

1 2

ρ     

b b

b

bb bbb

m m

A Lll L (1) 

where the subscript b refers to the expanded polystyrene board, mb is the mass of the 

board (kg); Ab is the area of the board (m2) as determined by length and width dimensions 

(l1b and l2b, respectively); and, Lb is the board thickness (m). The mass (mb) of the insulation board was measured by a digital weighing balance (32.1 kg range, 0.0001 kg resolution). Dimensional measurements lib (i = 1,2) and Lb were determined using a steel rule 

(0.05 mm resolution) and hand calipers (0.01 mm resolution), respectively. 

3.1.1 Graphical Analysis of Lb and ρb Data 

Variations in Lb and ρb (board-to-board) were analyzed graphically using a four-step 

method. The method consisted of 1) a run-sequence plot that checked for systematic and 

random changes; 2) a lag plot that checked for randomness; 3) a histogram that checked 

the frequency distribution; and, 4) a normal probability (normality) plot that checked for 

the normality assumption. Examples of the method are illustrated in Figs. 2 and 3 for Lb

and ρb, respectively. 

The data in Fig. 2 reveal that, with the exception of five boards greater than 13.5 mm, the 

thickness of the 300 boards is, in general, very consistent from board to board. The presence of a few tail (extreme end) points is routinely observed with empirical data. The 

distribution of thicknesses is random as shown by the tight cluster (“bull’s-eye”) of data 

points in the center of the lag plot. The distribution is approximately normally distributed 

as shown in the histogram and normality plot about a sample mean value of 13.23 mm. 

In Fig. 3, there are suggestions of multimodality, that is, two or more underlying populations of bulk density each with a single peak. This is noticeable in the run sequence plot 

(Fig. 3a) and the lag plot (Fig. 3b). The run sequence plot reveals high/low excursions 

from the mean values and the lag plot reveals localized clusters of data points indicating 

that the distribution of data is skewed, particularly toward the lower bulk densities. The 

multimodality is evident in the histogram which confirms this observation and shows two 

peaks, a large peak at about 39 kg/m3

 and a smaller peak near 43 kg/m3. The normality 

plot (Fig. 3d) identified Board 049 for supplemental tests. 

 
8 

Figure 2. Graphical analysis of board thickness (n = 300): (a) run sequence plot, (b) lag plot, (c) histogram, 

(d) normal probability plot (normality index). Summary statistics: sample mean = 13.23 mm, sample 

standard deviation = 0.13 mm. 

 

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0 50 100 150 200 250 300

14.013.513.012.512.0

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NORMAL PROBABILITY PLOT

THEORETICAL VALUE

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9 

Figure 3. Graphical analysis of bulk density (n = 300): (a) run sequence plot, (b) lag plot, (c) histogram, (d) 

normal probability plot (normality index). Summary statistics: sample mean = 40.60 kg/m3

, sample standard deviation = 2.05 kg/m3

. 

 

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Board 049
10 

3.1.2 Summary Statistics for Material Lot 

Table 2 presents summary statistics for the mass, length, width, thickness, and bulk density of the 300 boards. Overall values for length and width are within acceptable limits 

(nominally 931 mm by 657 mm). The large range and small standard deviation for thickness indicate that some of the panels are somewhat oversized in thickness (Fig. 2). The 

bulk density mean and standard deviation values are acceptable. 

Table 2. Summary statistics for the SRM 1453 production run (300 boards) 

Mass 

(g) 

Length 

(mm) 

Width 

(mm) 

Thickness 

(mm) 

Bulk density 

(kg/m3

) 

Mean 328.3 930.7 656.9 13.23 40.6 

Std. dev. 16.9 0.4 1.3 0.13 2.1 

Maximum 371.1 931.5 658.0 13.88 45.7 

Minimum 285.1 927.5 636.5 12.86 35.2 

Board 049 was identified as having a particularly low density (Fig. 3), removed from the 

lot, and utilized for supplemental measurements that included within-board density analysis, scanning electron microscopy, and surface profile measurements (Sec. 3.2-3.4). 

3.2 Bulk Density within a Board

The density variation with respect to position within a board was examined by cutting 

Board 049 into 35 specimens, each 127 mm by 127 mm, as illustrated in Fig. 4. The bulk 

densities, ρ049-01 through ρ049-35, were determined using Eq. (1) and the resulting data 

were analyzed graphically in Fig. 5 using the four-step method described in Sec. 3.1.1. 

Figure 4. Diagram of Board 049 sampling plan (not to scale) 

01 02 03 04 05 

06 07 08 09 10 

11 12 13 14 15 

16 17 18 19 20 

21 22 23 24 25 

26 27 28 29 30 

31 32 33 34 35 

Excess Board 049 
11 

Figure 5. Graphical analysis of bulk density measurements (n = 35) from Board 049: (a) run sequence plot, 

(b) lag plot, (c) histogram, (d) normal probability plot (normality index). Summary statistics: sample mean 

= 35.09 kg/m3

, sample standard deviation = 2.26 kg/m3. 

 

X

X

X

X

X

XX

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

X

XX

XX

X

XX

X

X

X

403938373635343332313029

RUN SEQUENCE

Index

X

29 30 31 32 33 34 35 36 37 38 39 40

40393837363534333231

LAG PLOT

X(i-1)

X

20.0 25.0 30.0 35.0 40.0 45.0 50.0

6543210

HISTOGRAM

X

FREQUENCY

+

+ +

+

+++

+

+++

+

+

+

+

++ +++

+++++

++++++++

+ +

-3 -2 -1 0 1 2 3

403938373635343332313029

NORMAL PROBABILITY PLOT

THEORETICAL VALUE

X

0 10 20 30 40
12 

The histogram in Fig. 5c reveals a large peak near 36 kg/m3 and a small peak near 

33 kg/m3

. The mean bulk density for Board 049 was 35.09 kg/m3 (only slightly lower 

than the minimum value in Table 2) and the standard deviation was 2.26 kg/m3. 

The cyclic pattern in Fig. 5a is an artifact of the run order (01-35) shown in Fig. 4. The 

variation of bulk density within Board 049 is illustrated with the contour plot shown in 

Fig. 6. The higher values of bulk density were located near the center of the board; the 

lower values of bulk density near the edges and corners. 

Figure 6. Contour plot of bulk density (in kg/m3) for Board 049. 
13 

3.3 Microstructure

The microcellular structure of the foam was examined using scanning electron microscopy (SEM). Small samples of foam, approximately 5 mm by 13 mm by 28 mm, were cut 

from one board using a razor blade and were subsequently sputter-coated with a 20 nm 

film of gold to prevent surface charging. Secondary electron images of the surface topography were obtained under operating conditions of 4 keV and a current of about 

500 pA. Two images of the foam, at magnifications of 25 and 100, are shown in 

Figs. 7a and 7b, respectively. 

The SEM micrograph in Fig. 7a (magnification 25) shows the interfacial outlines of the 

fused beads of expanded polystyrene from the molding process. The size of the beads 

ranged from 1 mm to 3 mm and the voids at the intersection of three or more beads, if 

present, were relatively small as noted in the micrograph. Figure 7b (magnification 

100) reveals the distribution of microcells formed within the beads during the initial expansion process. The cells, which are somewhat elliptical in shape, have major diameter 

lengths ranging from 0.02 mm to 0.4 mm. As evident in the micrograph, a large percentage of the cell walls were intact indicating a relatively high content of closed cells. 

Figure 7a. SEM micrograph (25) shows outlines of molded beads and internal cellular structure 

 
14 

Figure 7b. SEM micrograph (100) shows close-up of cellular structure 

3.4 Surface Roughness

The surface roughness of the foam was examined by the NIST Precision Engineering Division with a contact profilometer (Form Talysurf1

 instrument) that generated a surface profile by means of a contact stylus. Sub-specimen 21 from Board 049 (Fig. 4) was securely 

fastened relative to the instrument’s stylus. The stylus was traversed over a sampling interval of the foam at three different locations. Displacements of the stylus in the z-direction 

(i.e., parallel to the specimen thickness) were referenced to a filtered mean line, thus generating a series of peaks and valleys comprising the surface texture of the foam. The filter 

was a standard 2RC (resistor-capacitor) type with a nominal cutoff of 2.5 mm [8]. The average roughness, Ra [8], of the surface was defined as the mean of the absolute values of 

the profile height deviations. Table 3 summarizes values of Ra at the three locations from 

the sanded specimen 049-21 and from an unsanded specimen of similar polystyrene foam. 

Table 3. Average roughness values for foam specimens surfaces 

 Average roughness, Ra (μm) 

Location Sanded Unsanded 

1 16 72 

2 20 80 

3 16 82 

 
15 

4. Thermal Conductivity Measurement

Thermal conductivity measurements of the expanded polystyrene foam specimens were 

determined in accordance with ASTM Test Method C 177 [2] using the NIST 1016 mm 

guarded hot plate apparatus. Section 4 describes the measurement method, experimental 

design, and specimen selection. 

4.1 Guarded-Hot-Plate Method

Figure 8 shows the essential features of the guarded-hot-plate apparatus designed for operation near ambient temperature conditions under operation in the double sided mode. 

The apparatus is cylindrically symmetric about the axis indicated in Fig. 8. The plates 

are horizontal and heat flow (Q/2) is vertical (up/down) through the pair of specimens. 

The NIST 1016 mm diameter guarded hot plate apparatus has been described previously 

[9-10] and its operation is summarized briefly, here. Two specimens having nearly the 

same density, size, and thickness are placed on either side of the guarded hot plate and 

clamped securely by the circular cold plates (F1 and F2). Ideally, the guarded hot plate 

and cold plates provide constant-temperature boundary conditions to the surfaces of the 

specimens. With proper guarding in the lateral direction, the apparatus is designed to 

provide one-dimensional heat flow (Q) through the meter area of the specimen pair. A 

secondary guard is provided by an enclosed chamber that conditions the ambient 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). 

Figure 8. Guarded-hot-plate schematic (not to scale) showing the double-sided mode of operation with vertical heat flow.

1. Principle: Tc < Th; Tc1 = Tc2 = Tc

2. Practice: Tc < Th; Tc1 ≈ Tc2 ≈ Tc

Guarded hot plate 

(Meter plate)

Guard plate 

Ambient gas 

Cold plate 2

Qg 

Qe 

Qe Qe 

Qe 

Cold plate 1

Qg 

Specimen 2 

Specimen 1 

Tc1 

Tc2 

Th 

Q 

2

Q 

2

F1 

F2 
16 

Equation (2) is the operational definition [11] for the experimental thermal conductivity3 of 

the specimen pair (λexp) 

exp

1 2

λ

[( / ) ( / ) ]

Q

A TL TL    (2) 

where; 

Q = heat flow rate through the meter area of guarded-hot-plate specimen, W; 

A = meter area normal to direction of heat flow, m2; 

 ΔT = Th - Tc, temperature difference across specimen, K; 

Th = hot plate temperature, K; 

Tc = (average) cold plate temperature, K; and, 

L = (in-situ) thickness of guarded-hot-plate specimen, m. 

The subscripts 1 and 2 refer to specimen 1 and specimen 2, respectively (Fig. 8). When the 

temperature differences and the specimen thicknesses are nearly the same, respectively, Eq. 

(2) reduces to 

exp λ

2

avg

avg

Q L

A T   (3) 

In the double-sided mode of operation, values of λexp were reported at the mean temperature (Tm) of the hot and cold plates, Tm = (Th + Tc)/2. 

4.2 Experimental Design and Initial Model

The experimental design for the batch certification of SRM 1453 is based on a model for 

bulk density and temperature. An initial model for thermal conductivity (λ), bulk density 

(ρ), and temperature (T) was assumed to be 

  2 3

01 2 3 4 λ ρ,T a a aT aT aT     ρ (4) 

Table 4 summarizes a full factorial design having three levels for ρ and five levels for T. 

This design checks the adequacy of Eq. (4) and also allows checking for the necessity of 

a quadratic term for ρ, a fourth-order term for T, and/or a cross-product term for ρ and T. 

Each cell in Table 4 represents one measurement of a different pair of specimens (15 tests 

in total). The benefit of testing a unique pair of specimens at each combined level of 

temperature and density is that independent information is obtained at each such level. 

 

3

 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 heat insulators. However, for brevity, the term thermal conductivity will be used in this report. 
17 

Table 4. Full factorial (35) experimental design 

Density level 

Temperature level (K) 

281 289 297 305 313 

Low 1 obs.a14 1 obs. 7 1 obs. 9 1 obs. 6 1 obs. 2

Mid 1 obs. 5 1 obs. 11 1 obs. 1 1 obs. 10 1 obs. 8

High 1 obs.3 1 obs. 4 1 obs. 15 1 obs. 12 1 obs. 13 a

 observation 

The experimental design presented in Table 4 is balanced in the sense that an equivalent 

amount of information is obtained at each setting of the independent variables. If either 

extra information had been obtained at some of the settings, or worse, critical information 

omitted at one setting, the design would be unbalanced and the resulting statistical analysis would suffer. The test sequence, shown as circled numbers () in Table 4, was randomized to mitigate systematic effects from other uncontrolled or partially controlled factors that might inadvertently affect the test. 

Three nominal levels of ρ were selected to cover the upper and lower limits of the material lot (36 kg/m3

 to 45 kg/m3). The temperature value of 281 K was effectively fixed by 

the low limit of the guarded hot plate apparatus. Unfortunately, the limiting value of 

281 K is somewhat higher than the low temperature values specified in ASTM Test 

Method C 1199 [4]. The temperature value of 313 K was selected based on the ASTM 

material specification for cellular polystyrene thermal insulation [12]. 

4.3 Test Specimens 

A major goal in the selection of the test specimens was to reduce the likelihood of the user 

extrapolating outside the range of values of bulk density given in the 1453 Certificate. Fifteen pairs of (30) boards, each pair having nearly the same bulk density, were selected from 

the material lot covering a range from 36 kg/m3

 to 45 kg/m3. The boards were subsequently cut using a table saw to nominally square dimensions of 657 mm by 657 mm. 

The bulk density (ρs) for a specimen was determined from Eq. (5): 

1 2

ρ s s

s

s s sss

m m

A L ll L     

 (5) 

where the subscript s refers to the specimen. The mass (ms) of the insulation board was 

measured by a digital weighing balance (32.1 kg range, 0.0001 kg resolution). Dimensional measurements lis (i = 1,2) were determined using a steel rule (0.05 mm resolution). 

The quantity Ls was averaged from five height gage measurements (0.1 mm resolution) 

on a granite surface plate. 

Table 5 lists the 15 pairs of test specimens and their corresponding bulk density measurements (board and specimen density). The subscripts 1 and 2 refer to two pieces of an 

individual pair (Fig. 8). The percentage difference (Δρrel) is defined by Eq. (6). 

 

2 1

1 2

ρ ρ

ρ 100

ρ ρ 2 rel

  



 (6) 
18 

Table 5. Bulk density measurements for 30 test specimens 

Pair 

Density 

level ID1 ID2

Board densitya Specimen densitya

ρ1b

(kg/m3

)

ρ2b

(kg/m3

)

Δρb, rel 

(%) 

ρ1s

(kg/m3

)

ρ2s

(kg/m3

) 

Δρs, rel 

(%) 

1 Low 023 182 36.62 36.97 0.95 35.93 36.73 2.20

2 Low 168 242 36.98 37.05 0.20 36.77 36.64 0.38

3 Low 073 223 37.32 37.08 0.66 37.69 36.91 2.09

4 Low 059 072 37.53 37.57 0.11 37.58 37.74 0.43

5 Low 019 297 37.67 37.64 0.10 37.95 37.86 0.23

6 Mid 104 156 39.95 39.94 0.03 40.12 39.73 0.96

7 Mid 038 258 39.96 39.97 0.01 40.00 39.58 1.05

8 Mid 126 251 40.01 40.05 0.10 40.14 39.44 1.75

9 Mid 113 210 40.08 40.10 0.04 39.91 39.93 0.04

10 Mid 202 207 40.13 40.11 0.05 39.66 39.90 0.62

11 High 080 120 44.45 44.49 0.09 44.15 43.54 1.38

12 High 042 164 44.59 44.62 0.07 44.26 44.37 0.26

13 High 176 260 44.71 44.78 0.14 44.24 43.76 1.08

14 High 165 238 45.14 44.97 0.37 44.38 44.30 0.18

15 High 239 240 45.16 45.67 1.12 44.44 44.97 1.18

a

 Extra digit included for rounding purposes 

4.4 Test Sequence 

The bulk density for a specimen pair (i.e., two pieces), ρ pair, was determined from Eq. (7). 

ρ ρρ  1 2  2 pair s s   (7) 

The subscripts 1 and 2 refer to specimen 1 and specimen 2, respectively (Fig. 8). Table 6 

summarizes the randomized test sequence for the 15 pairs of expanded polystyrene specimens following the experimental design given in Table 4. 

Table 6. Test sequence (fully randomized) for specimen pairs of expanded polystyrene foam 

Test 

number 

Density 

level 

T 

(K) ID1 ID2

ρ1s

(kg/m3

) 

ρ2s

(kg/m3

) 

ρpair 

(kg/m3) 

1 1995-018A Mid 297 104 156 40.1 39.7 39.9 

2 1995-019A Low 313 073 223 37.7 36.9 37.3 

3 1995-020A High 281 080 120 44.1 43.5 43.8 

4 1995-021A High 289 239 240 44.4 45.0 44.7 

5 1995-022A Mid 281 038 258 40.0 39.6 39.8 

6 1995-023A Low 305 059 072 37.6 37.7 37.7 

7 1995-024A Low 289 168 242 36.8 36.6 36.7 

8 1995-025A Mid 313 126 251 40.1 39.4 39.8 

9 1995-026A Low 297 023 182 35.9 36.7 36.3 

10 1995-027A Mid 305 113 210 39.9 39.9 39.9 

11 1995-028A Mid 289 202 207 39.7 39.9 39.8 

12 1995-029A High 305 176 260 44.2 43.8 44.0 

13 1995-030A High 313 042 164 44.3 44.4 44.3 

14 1995-031A Low 281 019 297 37.9 37.9 37.9 

15 1995-032A High 297 165 238 44.4 44.3 44.3 
19 

4.5 Test Procedure 

The insulation guard material for the thermal conductivity measurements was a fibrous 

polyester blanket insulation having a diameter of 1016 mm and a square section, 657 mm 

by 657 mm, removed from the center. This insulation material was selected because of 

its compressibility and similar thermal conductivity. In separate guarded hot plate tests, 

the thermal conductivities of 13.3 mm thick specimens of fibrous polyester blanket insulation were determined to be 0.034 W/(mK), 0.037 W/(mK), and 0.040 W/(mK) at Tm

of 281 K, 297 K, and 313 K, respectively. 

The foam specimens (657 mm by 657 mm) were installed in the guarded-hot-plate apparatus and encircled with the fibrous polyester blanket insulation. The thermal conductivity measurement of each pair of foam specimens was completed in one to two days. During the steady-state test period, data for Q, Th, and Tc were collected every 2 min. When 

the plate temperatures were within 0.05 K, or less, of their target temperatures and Q no 

longer changed monotonically, steady-state data were collected for 4 h and averaged over 

the time interval. 
20 

5. Data Evaluation 

Section 5 presents the thermal conductivity measurements (tabular and graphical formats) 

and uncertainty analyses (for λexp and ρ) for the foam specimens. 

5.1 Data Summary (Tabular Format) 

Table 7 summarizes the experimental results – specimen information, input estimates, 

influencing (secondary) factor estimates, and the output estimate for measured thermal 

conductivity (λexp) – for the 15 specimen pairs specified in the experimental design (Table 4). The rows of data in Table 7 are grouped by Tm from 281 K to 313 K and, within 

each level of Tm, the specimen pair bulk densities (ρpair) are arranged from lowest to 

highest value. The columns of data are grouped into the following major sections: specimen material property; input quantities for Eq. (3); secondary quantities; and resultant 

thermal conductivity (λexp). The subscripts 1 and 2 designate the top and bottom specimen, respectively, as illustrated in Fig. 8. 

Table 7. Thermal conductivity data, sorted by Tm and by ρpair

 Input quantities for Eq. (3) Secondary quantities 

Tm ρpair Th Tc1 Tc2 Q/2 A Lavg Ta pa RH f λexpa

(K) (kg/m3) (K) (K) (K) (W) (m2) (mm) (K) (kPa) (%) (kPa) (W/m·K)

281 37.9 291.15 271.15 271.16 6.177 0.129 79 13.36 281.2 100.8 24 0.75 0.031 81

281 39.8 291.15 271.15 271.16 6.221 0.129 79 13.30 281.2 101.4 23 0.84 0.031 88

281 43.8 291.15 271.15 271.15 6.226 0.129 79 13.25 281.2 101.3 24 1.05 0.031 77

289 36.7 299.15 279.15 279.14 6.346 0.129 84 13.46 289.2 101.4 17 0.69 0.032 89

289 39.8 299.15 279.14 279.15 6.258 0.129 84 13.61 289.2 100.5 18 0.55 0.032 80

289 44.7 299.15 279.15 279.15 6.250 0.129 84 13.52 289.2 101.3 18 0.33 0.032 54

297 36.3 307.15 287.16 287.15 6.505 0.129 89 13.53 297.2 100.5 13 0.98 0.033 90

297 39.9 307.15 287.15 287.15 6.564 0.129 89 13.32 297.2 100.5 14 0.94 0.033 66

297 44.3 307.15 287.15 287.16 6.421 0.129 89 13.53 297.2 100.4 12 1.00 0.033 46

305 37.7 315.15 295.15 295.15 6.741 0.129 93 13.38 305.2 100.3 10 1.38 0.034 70

305 39.9 315.15 295.16 295.14 6.675 0.129 93 13.52 305.2 100.3 10 1.59 0.034 71

305 44.0 315.15 295.15 295.15 6.641 0.129 93 13.48 305.2 100.4 10 1.46 0.034 46

313 37.3 323.15 303.15 303.15 6.955 0.129 98 13.29 313.2 101.2 8 1.65 0.035 56

313 39.8 323.15 303.15 303.15 6.898 0.129 98 13.44 313.2 101.1 8 1.70 0.035 68

313 44.3 323.15 303.15 303.15 6.835 0.129 98 13.41 313.2 100.1 8 1.66 0.035 26 a

 Extra digit included for rounding purposes 

For each test, the plate temperatures (Th, Tc1, and Tc2) were maintained within 0.01 K, or 

less, of their respective set-point temperatures such that the specimen ΔT was maintained 

at 20.0 K. The estimates of Q/2 ranged from 6.2 W to 7.0 W for Tm at 281 K and 313 K, 

respectively. For a fixed value of Tm, the variation of Q/2 due to changes in ρpair was 

much smaller. The estimates for A have been corrected for thermal expansion effects of 

the meter-plate (Annex A1.4). The estimates for the in-situ test thickness L were determined by averaging the digital outputs of the eight linear position transducers, four for 

each cold plate (Annex A1.5). The resultant estimates for λexp include an extra digit to 

reduce rounding errors. 
21 

During a test, the secondary influence quantities (Ta, pa, RH, and f ) were either controlled or only recorded. The chamber air temperature (Ta) was controlled to be the same 

temperature as Tm (within 0.1 K, or less). The chamber air pressure (pa) varied with the 

site barometric conditions from 100.1 kPa to 101.4 kPa. The chamber RH varied inversely with the chamber dry-bulb air temperature (Ta). The clamping pressure (f ) was determined by averaging the loading force (F1 and F2 as illustrated in Fig. 8) applied by each 

cold plate and dividing by the surface area of the specimen surface area in contact with 

the cold plate. Table 8 provides summary statistics for test quantities in Table 7 that were 

fixed across all tests. 

Table 8. Summary statistics for fixed input quantities and secondary quantities 

ΔT 

(K) 

Lavg 

(mm) 

pa

(kPa) 

f

(kPa) 

Mean 20.000 13.43 100.8 1.10 

Std. Dev. 0.003 0.11 0.5 0.44 

The maximum clamping pressure in Table 7 (fmax equal 1.7 kPa ) was much less than the 

mechanical yield point of the polystyrene foam established in Annex 3. 

5.2 Uncertainty Budget, λexp

The measurement uncertainty for λexp was derived in accordance with current international guidelines [13-14] and described, in detail, in Annex 1. For the multiplicative expression of Eq. (3), the relative combined standard uncertainty in λexp can be expressed as the 

relative uncertainties associated with each factor combined in quadrature. 

            2 222

exp

, exp

exp

λ

λ

λ

c

c rel

u uQ u T uL uA

u

Q TLA

     

              (8) 

The standard uncertainties and input quantities used in Eq. (8) are derived in Annex 1. 

The maximum combined standard uncertainty for λexp was determined at Tm of 281 K. 

 

2 22 2

, exp

0.0068 0.077 0.061 0.000023 λ

6.169 20.000 13.31 0.12979 c rel u

                  (9) 

       2222

, exp λ 0.00110 0.00385 0.00458 0.00018 0.0061 c rel u    (10) 

λexp , exp  2 λ  0.012 U u rel c rel   (11) 

Expressed as a percent (× 100), Urel is equal to 1.2 %. 
22 

The relative contribution for the first term in Eq. (10) is 3.3 %; for the second, 40.0 %; 

for the third, 56.6 %; and, for the fourth, 0.1 %. The major contributory uncertainties for 

SRM 1453 are due to the empirical determinations for specimen thickness (Lavg) and 

temperature difference (ΔT). 

5.3 Data Screening (Graphical Analysis) 

Values of λexp from Table 7 are plotted in Figs. 9a and 9b as a function of the model input 

variables ρ and T, respectively. Figure 9a plots the individual data points as filled circle 

symbols corresponding to T levels of 281 K, 289 K, 297 K, 305 K, and 313 K. The error 

bars represent expanded uncertainties of 1.2 % (Sec. 5.2). Figure 7b plots the individual 

data points as filled circle, square, and triangle symbols (without error bars for clarity) corresponding to the three levels selected for bulk density. 

The data in Fig. 9 strongly suggests that, in the range of ρ and T covered for the 300 

boards, comprising the current SRM: 

1) λexp is slightly sensitive to ρ; and, 

2) the dependency of λexp on T is linear. 

The first assertion is supported by the linear least squares fits to the 3-point negativelysloped horizontal profiles visible in Fig. 9a. For the bulk density range of 37.4 kg/m3

 to 

45.8 kg/m3

, the change in thermal conductivity with respect to bulk density was quite 

small and, for the most part, decreased as bulk density increased (Fig. 9a). As discussed 

in Sec. 5.4, the small dependency on ρ required a correction for the metered section bulk 

density. The second assertion is supported by the data in Fig. 9b that shows, over a range 

of 281 K to 313 K, λexp increased linearly by about 12 % (from 0.0316 W/(mK) to 

0.0354 W/(mK)). 

5.4 Correction for Metered Section Bulk Density 

The data in Fig. 9a indicate that λexp is slightly dependent on ρ. To determine the effect 

of the metered section density, the insulation cylinder above and below the meter plate 

(Fig. 8) was cut for mass determination (mm). The metered section bulk density [2] is defined in Eq. (12) as 

    2 2

4

ρ

π π

mm m

m

m m m m

mm m

A L rL dL     

 (12) 

where the subscript m refers to the metered section (in this case, a right-angle cylinder); 

and, r and d are the radius and diameter, respectively, of the round cylinder. 

In order to preserve particular specimens of interest for future measurements, a specific 

subset of only eight specimens (four pairs) was selected for cutting. Table 9, which follows the same format as the experimental design presented in Table 4, summarizes the 

specimen pairs that were selected for the determination of the metered section bulk density. 

 
23 

Figure 9. SRM 1453 a) Graphical analysis of thermal conductivity versus bulk density (uncorrected). Error 

bars represent expanded uncertainties of 1.2 %. b) Graphical analysis of thermal conductivity (without error 

bars for clarity) versus temperature. 

34 36 38 40 42 44 46 48 50

0.031

0.032

0.033

0.034

0.035

0.036

Bulk density, kg · m-3

Thermal conductivity, W· m-1 · K-1

281 K

289 K

297 K

305 K

313 K

270 280 290 300 310 320

0.031

0.032

0.033

0.034

0.035

0.036

Temperature, K

Thermal conductivity, W· m-1 · K-1

Low Bulk Density

Mid Bulk Density

High Bulk Density
24 

Table 9. Specimens selected for metered section bulk density measurements (see Table 4) 

Density level 

Temperature level (K) 

281 289 297 305 313 

Low --- 168, 242 --- 059, 072 ---

Mid --- --- --- --- ---

High --- 239, 240 --- 176, 260 ---

Using a jigsaw, 406 mm diameter cylinders were cut from the center of each of the eight 

specimens and weighed. The mass (mm) of the metered section was measured by a digital 

weighing balance (32.1 kg range, 0.0001 kg resolution). Dimensional measurements for 

the diameter d were determined using a steel rule (0.05 mm resolution). The quantity Lm

was averaged from five height gage measurements (0.1 mm resolution) of the metering 

section placed on a granite surface plate. 

For a specimen pair, Eq. (13) defines the correction for the metered section bulk density: 

, ρ ρρ pair corr pair corr   (13) 

Substituting Eq. (7) for the first term in Eq. (13) and expanding the second term yields: 

 

8

1 2

,

1

1

8

ρ ρ ρ ρρ 2

s s

pair corr m s i

i



  (14) 

Values for ρs, ρm, and ρm – ρs were determined for the eight specimens and are summarized 

in Table 10. The differences (ρm – ρs) range from 0.1 kg/m3

 to 2.1 kg/m3. As observed in 

Fig. 6, and confirmed in Table 10, the bulk-density variation within a board was generally 

higher near the center and lower near the edges. 

Table 10. Comparison of specimen and meter area bulk densities 

ID 

Density 

level 

ρs

a

(kg/m3) 

ρm

a

(kg/m3) 

(ρm – ρs)

a 

(kg/m3) 

242 Low 36.64 38.49 1.85 

072 Low 37.74 38.68 0.94 

059 Low 37.58 38.61 1.03 

168 Low 36.77 38.83 2.06 

176 High 44.24 44.92 0.68 

260 High 43.76 45.06 1.30 

239 High 44.44 44.84 0.40 

240 High 44.97 45.06 0.09 

 Mean 1.05 

 Std. dev. 0.67 a Extra digit included for rounding purposes
25 

From Table 10, the mean of the differences ( ρcorr ) and the standard deviation (s) were 

1.05 kg/m3 and 0.67 kg/m3, respectively. A 95 % confidence interval for the “true” density difference was determined from Eq. (15): 

ρcorr α 2,DoF

s

t

n

 (15) 

where (Student’s) t for 95 % and 7 degrees of freedom (DoF) is 2.36, and n is the number 

of measurements (8). The corresponding interval was 1.05 kg/m3

 ± 0.56 kg/m3 which 

does not contain zero. Therefore, the difference in densities for the meter area and the 

entire specimen was statistically significant and a value of 1.05 kg/m3 was added using 

Eq. (13) to the specimen pair bulk densities (ρpair) of Table 7 for the regression analyses 

(Sec. 6) to account for the metered section bulk density. 

5.5 Standard Uncertainty for the Metered Section Bulk Density Correction 

The measurement uncertainties for ρs, ρm, and ρpair,corr were derived in accordance with 

current international guidelines [13-14] and described, in detail, in Annex 2. The standard uncertainties for these quantities are as follows: 

 uc (ρs) is 0.18 kg/m3 (from Eq. (A2-5)); 

 uc (ρm) is 0.27 kg/m3 (from Eq. (A2-9)); and, 

 uc (ρpair,corr ) is 0.42 kg/m3 (from Eq. (A2-11)). 

 
26 

6. Regression Analysis

Section 6 describes the regression analysis of the data. There are two goals for this analysis: 

1) to estimate a predictive relationship between λ and the predictors ρ and T; and, 

2) to quantify the uncertainty in predictions from this estimated relationship. 

In the first stage of the analysis, a model is selected by considering all subsets of the regression terms {ρpair,corr, Tm, Tm

2

, Tm

3

} (with response λexp), where the Bayesian information criterion (BIC) [15] is used for comparison. In the second stage, the final estimate of the predictive relationship and the expanded uncertainty (U ) for predictions from 

that estimated relationship are calculated using a measurement error model [16] and the 

Bayesian inference paradigm [17]. 

6.1 Model Selection 

The experiment was designed so that each of the factors, ρpair,corr, Tm, Tm

2

, and Tm

3

 could be 

included as predictors in a multiple regression model (with response λexp). To choose a 

model, the BIC for all 16 subsets of regression terms are calculated. The BIC can be 

thought of as applying the parsimony principle to model selection. That is, it seeks the 

simplest model that provides a good description of the observed data. More directly, the 

BIC is the sum of two terms. The first term describes the fit of the model to the observed 

data, and the second term involves a positive penalty for the number of parameters (regression coefficients) in the model. Models providing a good fit to the data have small first 

terms, and models with more parameters receive a larger penalty. Since more parameters 

typically lead to a better fit, these two terms fight against one another, and we select the 

model with the smallest BIC. The BIC values for all 16 models under consideration are 

given in Table 11. From Table 11, it can be seen that the model with ρpair,corr and Tm is best 

(i.e., Rank 1). 

Table 11. BIC values for all models under consideration. 

Model BIC Rank 

ρpair,corr, Tm -82.13 1 

ρpair,corr, Tm

2

 -80.25 2 

ρpair,corr, Tm, Tm

2

 -79.52 3 

ρpair,corr, Tm, Tm

3

 -79.52 3 

ρpair,corr, Tm

2

, Tm

3

 -79.52 3 

ρpair,corr, Tm, Tm

2

, Tm

3

 -77.34 6 

ρpair,corr, Tm

3 -76.71 7 

Tm -63.85 8 

Tm

2

 -63.28. 9 

Tm

3

 -62.14 10 

Tm, Tm

3

 -61.20 11 

Tm

2

 Tm

3

 -61.20 11 

Tm, Tm

2

 -61.19 13 

Tm, Tm

2

, Tm

3

 -58.59 14 

Intercept only 0.00 15 

ρpair,corr 2.58 16 
27 

To verify that the model with ρpair,corr and Tm actually provides a good fit to the data, plots 

of the deviations (also known as residuals) from a least squares fit are examined. In 

Fig. 10, a normal quantile-quantile plot of the deviations as well as plots of the deviations 

against ρpair,corr, Tm, and the predicted values of λ are shown. 

Figure 10. Plots of deviations (also known as residuals) for assessing the model fit. 

The normal quantile-quantile plot indicates no problems as the points follow a reasonably 

straight line. The plots of the deviations versus ρpair,corr, Tm, and the predicted values of λ

also indicate no problems since the deviations form a symmetric horizontal band around 

zero. 

6.2 Estimation and Uncertainty 

The final estimation of the predictive relationship as well as the uncertainty quantification 

in predictions from this relationship is carried out using a measurement error model [16] 

and the Bayesian inference paradigm [17]. A detailed description of the analysis is found 

in Annex 4. The final equation and expanded uncertainty on the certificate is 
28 

λ    0.00111 0.000 042 4ρ 0.000115 1.5 % ( 2). T k (16) 

The uncertainty represents an interval in which a new hypothetical NIST measurement of 

thermal conductivity will lie with at least 95% posterior predictive probability. In 

Sec. 5.2, Eq. (11) the expanded uncertainty for λexp is 1.2 %. The additional uncertainty 

in predictions from the above equation comes from various other sources. For example, 

the predictive relationship, while quite good, does not perfectly predict the λexp. 

 
29 

7. Certification

Section 7 presents summary information on the properties of interest, values and uncertainty, and instructions for use for Standard Reference Material 1453. This information is 

intended to provide supplementary documentation for the 1453 Certificate. 

7.1 Properties of Interest 

Standard Reference Material 1453 is an expanded polystyrene foam board certified for 

thermal conductivity, λ. Each SRM unit consists of a rectangular panel of expanded polystyrene beads molded into a semi-rigid board. The nominal dimensions of a unit are 

931 mm by 657 mm by 13.2 mm (Table 1) and the bulk density ranges from 37 kg/m3 to 

46 kg/m3 (Table 6). 

7.2 Values and Uncertainties 

Each unit of SRM 1453 is batch certified for thermal conductivity with Eq. (17): 

λ 0.00111 0.000 042 4ρ 0.000115 1.5 % ( 2)    T k m (17) 

where λ is the predicted thermal conductivity (W/(m·K)) and Tm is the mean specimen 

temperature (K). The test temperature difference across the specimen (ΔT) was 20 K 

(Table 8). Equation (17) is only certified to be valid over the bulk density range from 

37 kg/m3 to 46 kg/m3 and temperature range from 281 K to 313 K. The expanded uncertainty for λ values from Eq. (17) is 1.5 % with a coverage factor of approximately k = 2. 

7.3 Instructions for Use 

Standard Reference Material 1453 is intended for use as part of a calibration transfer 

standard for the thermal evaluation of fenestration systems using a hot box test method. 

The SRM is also intended for use as a proven check for the guarded-hot-plate apparatus 

(or other absolute thermal conductivity apparatus) and for calibration of a heat-flowmeter apparatus over the temperature range of 281 K to 310 K. NIST cannot exclude the 

use of SRM 1453 for other purposes, but the user is cautioned that other purposes are not 

sanctioned by the 1453 Certificate. 

7.3.1 Storage 

For protection and identification, it is recommended that the reference material be stored 

in the original packaging in a clean, dry environment at temperatures between 15 °C and 

30 °C. 

7.3.2 Preparation and Conditioning before Measurement 

Prior to the thermal conductivity measurement, the reference material should be conditioned in laboratory conditions of 20 ° C to 25 °C and from 40 % RH to 65 % RH until 

the mass of the unit is stable (i.e., two successive measurements within 24 h vary by less 

than 1 %). It should be noted, however, that polystyrene foam is insensitive to changes in 

humidity as described in Annex 5. 
30 

7.3.3 Thermal Conductivity Measurement 

Thermal conductivity measurements should be conducted in accordance with the appropriate ASTM Test Method C 177 [2], C 518 [3], or other comparable standard. Measurement as part of a fenestration test should be conducted in accordance with ASTM Test 

Method C 1199 [4] or other similar standard. 

7.3.4 Guidelines and Precautions 

The following guidelines and precautions are provided for the user. 

3) Stacking: Certified values of thermal conductivity are valid for a single unit, and 

are invalid for stacked units. The thermal conductivity of polystyrene foam dramatically increases with specimen thickness due to the increased transmission of 

long-wave thermal radiation in thicker specimens [18-19]. 

4) Slicing: Certified values of thermal conductivity are invalid for a unit where the 

thickness of the material has been modified by slicing. 

5) Cutting: It is possible to cut the reference material unit into smaller pieces. It is 

imperative to verify that bulk density of each piece is within the certified range of 

bulk density (Sec. 7.1). 

6) Upper Temperature Limit: The upper temperature for SRM 1453 is limited to the 

softening point of the polystyrene polymer which is 347 K (74 C) [12]. It should 

be noted that oven drying, as opposed to desiccant drying, can remove other volatiles and potentially affect chemical or physical properties of the material. 

7) Lower Temperature Limit: A lower temperature limit for SRM 1453 has not been 

established but, in principle, there is no known lower limit. 

8) Atmospheric Pressure: The effect due to changes in ambient atmospheric pressure 

is negligible for this material. 

9) Clamping Pressure: For thermal testing, the specimens must be in firm contact 

with the apparatus plates. However, do not compress the material more than 

0.34 mm (2.5 %) of its original thickness. The compressive resistance of expanded polystyrene foam is described in Annex 3. 

10) General precautions for handling polystyrene foam are described in Annex 6. 

Acknowledgments

The original 1995 certification of SRM 1453 was accomplished with the assistance of the 

following individuals. Nancy Trahey and Robert Gettings provided support through the 

Standard Reference Materials Program. NIST Statistician, Dr. E. S. Lagergren provided 

the initial guidance in the experimental design and data analysis. The surfaces of the 

foam boards were prepared by R. L. Baumgardner of Rollin, Inc. Micrographs of the 

foam were filmed by P. E. Stutzman and compressive resistance measurements provided 

by J. Seiler and Dr. W. J. Rossiter. The surface roughness measurements were provided 

by C. K. Rymes and J. F. Song. Sorption isotherm measurements were supervised by 

D. Burch and C. Saunders assisted with the density measurements of the 300 foam 

boards. 
31 

References

[1] J. K. Taylor, “Standard Reference Materials, Handbook for SRM Users,” NIST Special Publication 260-100, Natl. Inst. Stand. Technol., February 1993. 

[2] ASTM, “C 177 Standard Test Method for Steady-State Heat Flux Measurements 

and Thermal Transmission Properties by Means of the Guarded Hot Plate Apparatus,” Annual Book of ASTM Standards, Vol. 04.06. 

[3] ASTM, “C 518 Standard Test Method for Steady-State Thermal Transmission 

Properties by Means of the Heat Flow Meter Apparatus,” Annual Book of ASTM 

Standards, Vol. 04.06. 

[4] ASTM, “C 1199 Standard Test Method for Measuring the Steady-State Thermal 

Transmittance of Fenestration Systems Using Hot Box Methods,” Annual Book of 

ASTM Standards, Vol. 04.06. 

[5] “Energy Policy Act of 1992,” Act of October 24, 1992, Public Law 102-486, 102nd 

Congress. 

[6] R. P. Bowen and K. R. Solvason, “A Calorimeter for Determining Heat Transmission Characteristics of Windows,” Thermal Insulation: Materials and Systems, 

ASTM STP 922, F. J. Powell and S. L. Matthews, Eds., American Society for Testing and Materials, Philadelphia (1987) 567-581. 

[7] W. P. Goss, H. A. Elmahdy and R. P. Bowen, “Calibration Transfer Standards for 

Fenestration Systems,” In-Situ Heat Flux Measurements in Buildings: Applications 

and Interpretations of Results, CRREL Special Report 91-3, S. N. Flanders, Ed., 

(1991) 251-260. 

[8] ASME, “Surface Texture: Surface Roughness, Waviness, and Lay,” B46.1-1995, 

ISBN: 0791823563. 

[9] F. J. Powell and B. G. Rennex, “The NBS Line-Heat-Source Guarded Hot Plate for 

Thick Materials,” Thermal Performance of the Exterior Envelopes of Buildings - II, 

ASHRAE SP 38 (1983) 657-672. 

[10] R. R. Zarr and M. H. Hahn. “Line Heat Source Guarded Hot Plate Apparatus,” Adjunct ASTM Practice C 1043, PCN: 12-310430-61. 

[11] ASTM, “C 1045 Standard Practice for Calculating Thermal Transmission Properties Under Steady-State Conditions,” Annual Book of ASTM Standards, Vol. 04.06. 

[12] ASTM, “C 578 Standard Specification for Rigid, Cellular Polystyrene Thermal Insulation,” Annual Book of ASTM Standards, Vol. 04.06. 

[13] B. N. Taylor and C. E. Kuyatt, “Guidelines for Evaluating and Expressing the Uncertainty of NIST Measurement Results,” NIST Technical Note 1297, Natl. Inst. 

Stand. Technol. 1994. 

[14] ISO, Guide to the Expression of Uncertainty in Measurement, International Organization for Standardization, Geneva, Switzerland, 1993. 

[15] G. Schwarz, “Estimating the Dimension of a Model,” The Annals of Statistics, 6, 

461-464 (1978). 

[16] W. Fuller, Measurement Error Models, John Wiley & Sons, Inc., New York (1985). 
32 

[17] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, 2nd 

Ed., Chapman & Hall/CRC, Boca Raton, Florida (2004). 

[18] J. R. Booth and J. T. Grimes, “The Determination of Radiation k-Factor for Foam 

Structures Using HFM Apparatus,” Thermal Conductivity 22, Technomic: Lancaster, Pennsylvania (1994) 783-793. 

[19] T. T. Jones, “The Effect of Thickness and Temperature on Heat Transfer Through 

Foamed Plastics,” Plastics and Polymers, February, 33-39 (1972). 

[20] Y. S. Touloukian, R. K. Kirby, R. E. Taylor, and P. D. Desai, Thermophysical 

Properties of Matter, The TPRC Data Series, Volume 12: Thermal Expansion: Metallic Elements and Alloys. IFI/Plenum, New York (1977) pp. 1028-1033. 

[21] R. K. Burdick, Confidence Intervals on Variance Components, 1992, p. 66. 

[22] R. J. Roark, Formulas for Stress and Strain, 4th Ed. McGraw-Hill Inc., New York 

(1965) p. 219. 

[23] B. G. Rennex, “Error Analysis for the National Bureau of Standards 1016 mm 

Guarded Hot Plate,” NBSIR 83-2674, Natl. Inst. Stand. Technol., February 1983 

(reprinted in Journal of Thermal Insulation, 7, 18-51 (July 1983)). 

[24] M. H. Hahn, H. E. Robinson, 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. 

[25] ASTM, “C 165 Standard Test Method for Measuring Compressive Properties of 

Thermal Insulation,” Annual Book of ASTM Standards, Vol. 04.06. 

[26] A. Gelman, “Prior distributions for variance parameters in hierarchical models,” 

Bayesian Analysis, 1, 515-533 (2006). 

[27] D. Lunn, D. Spiegelhalter, A. Thomas, and N. Best, “The BUGS project: Evolution, 

critique, and future directions,” Statistics in Medicine, 28, 3049-3067 (2009). 

[28] R Development Core Team, R: A language and environment for statistical computing, R Foundation for Statistical Computing, Vienna, Austria (2012); 

http://www.R-project.org [Accessed (Aug 2012)]. 

[29] S. Mehta, S. S. Biederman, and S. Shivkumar, “Thermal Degradation of Foamed 

Polystyrene,” Journal of Materials Science, 30, 2944-2949 (1995). 
Annex 1 

33 

Annex 1 - Uncertainty Analysis for Thermal Conductivity (λ)

A1.1 Background

In 1992, NIST officially adopted a new policy [13] for the expression of measurement 

uncertainty consistent with international practices set forth in the ISO Guide to the Expression of Uncertainty in Measurement [14], also known as the “GUM.” This policy 

provides a uniform approach at NIST to uncertainty analysis and is summarized briefly 

below. 

In many cases, a measurand, Y, is not determined directly from a measurement, but rather 

mathematically from a function of N other independent quantities Xi: 

Y fXX X   1 2 , ,..., N  (A1-1) 

The output estimate of Y, denoted as y, is obtained using input estimates xi for the values 

of the N independent quantities Xi: 

y fxx x   1 2 , ,..., N  (A1-2) 

The combined standard uncertainty of y, uc(y), is the positive square root of the combined 

variance, uc

2

(y); where 

2

2 2 22

1 1

() ( ) ( )

N N

c i ii

i i i

f u y u x cu x   x

           (A1-3) 

Equation (A1-3) is commonly referred to as the “law of propagation of uncertainty”. The 

partial derivatives are known as sensitivity coefficients (ci) and are equal to f /Xi evaluated at Xi = xi . The corresponding term, u(xi), is the standard uncertainty associated with 

the input estimate xi. 

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

Standard uncertainties are denoted as Type A if they are evaluated by statistical means. 

Standard uncertainties that cannot be determined directly from the experiment at hand 

and are evaluated by scientific judgement including information from (previous) measurement data from another experiment, experience, a calibration certificate, manufacturer 

specification, among others. [13-14]. Standard uncertainties evaluated using these types 

of analysis methods are denoted Type B. Categorizing the evaluation of uncertainties as 

Type A or Type B is simply a matter of convenience, since both are based on probability 

distributions4 and combined equivalently in Eq. (A-3). An example of a Type A evaluation is provided below. Examples of Type B evaluations are provided in Refs. [13-14]. 

It should be noted that the designations “A” and “B” apply to the two methods of evaluation, not the type of error. 

 

4

 The probability distribution for a Type B evaluation, as opposed to a Type A evaluation, is assumed based 

on the experimenter's judgment. 
Annex 1 

34 

As an example of a Type A evaluation, consider an input quantity Xi determined from n

independent observations obtained under the same conditions (i.e., repeated observations). In this case, the input estimate xi is the sample mean determined from: 

,

1

1 n

i i ik

k

x X X

n 

   (A1-4) 

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

the sample mean: 

() ( ) i i

s ux sX

n   (A1-5) 

When an uncertainty bound that provides an interval that will contain the true value of 

the measurand with high probability is required (similar to a confidence interval used for 

decision making, for example), an expanded uncertainty, U, is obtained by multiplying 

the combined standard uncertainty, uc(y), by a coverage factor, k: 

22 22 () ( ) ( ) U ku y k c u x c u x   c i iA i iB   (A1-6) 

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. Interpretation of the coverage factor 

requires a word of caution. The term “confidence interval” has a specific definition in statistics and is only applicable to intervals based on uc when certain conditions are satisfied, 

including that all components of uc be obtained from Type A evaluations. Under these circumstances, assuming the inputs with the most significant uncertainties are based on relatively large sample sizes, a coverage factor of k = 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, the value of the coverage factor is k = 2, by convention [13]. 

A1.2 Uncertainty for λexp

Referring to Eq. (3) in Sec. 4.1, the standard uncertainty for λexp was evaluated based on 

the standard uncertainties for the specimen heat flow (Q), meter area (A), in-situ thickness (L), and specimen temperature difference (ΔT). These individual standard uncertainties were evaluated by either statistical methods (Type A), other means (Type B), or both, 

and are discussed below. 

A1.3 Specimen Heat Flow (Q)

Under normal operation (Fig. 8), the guard plate and ambient air temperature were maintained such that lateral heat losses (Qg and Qe) were reduced to negligible proportions. Under these circumstances, the specimen heat flow (Q) was determined by simply measuring 

the direct current (DC) electrical power provided to the meter area (Qm) of the guarded hot 

plate. The electrical circuit for the measurement consisted of a standard resistor, nominally 

0.1 Ω, in series with the electrical heater of the meter area as illustrated in Fig. A1. 
Annex 1 

35 

The corresponding equation for the power input to the meter area is: 

s

mm m

s

V Q iV V R

       

 (A1-7) 

where i is the current (Vs /Rs) through the circuit and Vm is the voltage drop measured 

across the electrical leads to the meter plate. 

A1.3.1 Uncertainty for Power Measurement (Qm)

The Type A standard uncertainty for Qm was determined by pooling the standard deviations for each guarded hot plate test. The uncertainty estimates for u(Qm)A are tabulated 

at the end of this section. 

The Type B standard uncertainty for Qm was determined by application of Eq. (A1-3) to 

Eq. (A1-7) which yields 

22 2 2 2 2 2 2 2 ( ) () () ( ) c mB V s R s V m ssm uQ cuV cu R c uV   (A1-8) 

with 

s

m m

V

s s

Q V c

V R

   

s 2

m s

R m

s s

Q V c V

R R

   

m

m s

V

m s

Q V c

V R

   

Figure A1. Electrical schematic for meter-plate power measurement. 

DC Power 

Supply 

0.1 Ω

Standard 

Resistor, Rs 

56 Ω

Meter-plate 

Heater, Rm 

i 

Oil Bath Vs Vm 

25.00 °C 
Annex 1 

36 

The input values for Rs and u (Rs) in Eq. (A1-8) were equal to 0.100071 Ω and 0.000001 Ω

(k = 1), respectively. The standard uncertainties for Vs and Vm were determined from 

Eq. (A1-9). 

( ) 3

a u V  (A1-9) 

The Type B standard uncertainties for Vs and Vm (Fig. A1) were assumed to have a uniform distribution in the interval ±a; where ±a was determined from the manufacturer 

specification for the integrating digital voltmeter (DVM). The interval half-widths, a, for 

Vs and Vm (300 mV and 30 V ranges, respectively) were based on the 1 year manufacturer 

specification for the integrating voltmeter and were computed from Eqs. (A1-10a) and 

(A1-10b), respectively, where reading is in volts. 

: 0.00008 8 μV 0.0001 V a reading reading s     (A1-10a) 

: 0.00008 300 μV 0.0001 V a reading reading m     (A1-10b) 

Table A1.1 summarizes the standard uncertainty components for u (Qm) for Tm at 281 K, 

297 K, and 313 K. The Type B evaluations in Table A1.1 are about 4 times greater than 

the Type A evaluations. There is a slight increase in the uncertainty with Tm. 

Table A1.1. Summary of standard uncertainty components for u (Qm) 

Tm u (Qm)A Vs u (Vs) Vm u (Vm) u (Qm)B

(K) (W) (V) (V) (V) (V) (W) 

281 0.000 67 0.047 0.000 009 5 26.3 0.0029 0.0028 

297 0.000 72 0.048 0.000 009 6 27.0 0.0030 0.0030 

313 0.000 72 0.049 0.000 009 7 27.8 0.0031 0.0031 

A1.3.2 Guard Imbalance (Qg)

The terms Qg and Qe (Fig. 8) represent both the lateral heat loss at the guard gap between 

the meter plate and guard plate (Qg) and the heat loss at the edges of the specimens (Qe). 

These effects were quite small (approximately zero) for the 15 guarded hot plate tests 

(Sec. 4.1) because guarding at the gap and edges of the specimen reduced the lateral heat 

flows Qg and Qe to negligible proportions (Fig. 8). An imbalance condition is defined 

when a significant temperature difference develops either across the gap (Vg) or at the 

edge of the specimens (Ta – Tm). The first parameter, Vg, is the voltage output from an 8-

junction Type-E thermopile across the gap. The second parameter, Ta – Tm, is the temperature difference between the ambient air and mean temperature. 

Table A1.2 summarizes the imbalance settings for two treatments (Vg and Ta – Tm) for 

five test conditions. The last row in Table A1.2 is a center point where Vg and Ta – Tm

were both set to zero; that is, under normal operating conditions for negligible heat flows 
Annex 1 

37 

at the guard gap and specimen edges. This experimental design checks previous results 

and any interaction between the independent factors. 

Table A1.2. Nominal settings for imbalance study 

Vg Ta – Tm

Index (μV) (K) 

1 –50 –5 

2 +50 –5 

3 –50 +5 

4 +50 +5 

5 0 0 

The test sequence shown in Table A1.2 was randomized for each level of Tm at 281 K, 

297 K, and 313 K (5+5+5 = 15 tests). The tests were conducted with one pair of specimens of expanded polystyrene boards (202, 207) installed in the guarded-hot-plate apparatus and encircled with fibrous polyester. During each test, the steady-state power input 

to the meter plate (Qm) was recorded and averaged for the test. The imbalance effect, 

ΔQ, was defined as: 

QQQ   m m0 (A1-11) 

where Qm0 was the power input to the meter plate for the gap and edge temperatures 

thermally balanced, i.e., at the center point where Vg and Ta – Tm were both set to zero. 

Table A1.3 summarizes the test results from the imbalance study at Tm of 281 K, 297 K, 

and 313 K. 

Table A1.3. Test results for imbalance study (Yates order) 

Index Tm ΔT Qm

y

(Qm – Qm0) 

x1

(Vg – Vg0)

x2

[(Ta – Tm) 

– (Ta – Tm)0] 

 (K) (K) (W) (W) (μV) (K) 

1 281 20.00 11.9901 -0.1458 -50.01 -4.97 

2 281 20.00 12.2175 0.0816 49.96 -4.99 

3 281 20.00 12.0512 -0.0847 -50.00 5.00 

4 281 20.00 12.2727 0.1368 50.00 5.00 

5 281 20.00 12.1359 0. 0. 0. 

1 297 20.00 12.7124 -0.1240 -50.00 -5.00 

2 297 20.00 12.9375 0.1011 49.96 -5.00 

3 297 20.00 12.7330 -0.1034 -50.03 5.01 

4 297 20.00 12.9549 0.1185 49.97 5.00 

5 297 20.00 12.8364 0. 0. 0. 

1 313 20.00 13.3952 -0.1260 -49.99 -4.99 

2 313 20.00 13.6351 0.1139 50.00 -4.99 

3 313 20.00 13.4123 -0.1089 -49.97 5.01 

4 313 20.00 13.6449 0.1237 50. 5.01 

5 313 20.00 13.5212 0. 0. 0. 
Annex 1 

38 

The response variable (y) and input variables (x1, x2) were normalized with respect to the 

balance point. Negative values for x1 indicate that the guard plate is warmer than the meter 

plate (Fig. 8). Conversely, positive values for x1 indicate that the guard plate is cooler than 

the meter plate. 

The data in Table A1.3 were fit to the model given in Eq. (A1-10). The presence of an offset coefficient b0 was initially considered but, because the term is predicted to be nearly 

zero from theory, the term was not included. 

11 2 2 y Q bx bx    (A1-12a) 

1 02       g g am am 0 y bV V b T T T T          (A1-12b) 

Table A1.4 summarizes the parameter estimates and approximate standard deviations 

from multiple variable linear regression for coefficients b1 and b2 at Tm of 281 K, 297 K, 

and 313 K. 

Table A1.4. Parameter estimates and standard deviations for b1 and b2 in Eq. (A1-12) 

Tm b1 s(b1) b2 s(b2) Residual SD 

(K) (W/V) (W/V) (W/K) (W/K) (W) 

281 2.24510-3 3.9410-5 5.81910-3 3.9510-4 0.0039 

297 2.23610-3 2.3910-5 1.90210-3 2.3910-4 0.0024 

313 2.36310-3 2.2310-5 1.33910-3 2.2310-4 0.0022 

Application of Eq. (A1-3) to Eq. (A1-12a) yields 

           11 2 2

22 22 2 2 2 2

cc b x b x 112 2 u y u Q cu b cu x cu b cu x      (A1-13) 

with 

  i b i

i

Q

c x

b

    

  i x i

i

Q c b

x

    

Note that no term for the correlation between b1 and b2 is required in Eq. (A1-13) because 

the experiment design used in the assessment of the uncertainty form this source was an 

orthogonal design. Table A1.5 summarizes the input values for Eq. (A1-13) and the corresponding estimates for uc (ΔQ) at Tm levels of 281 K, 297 K, and 313 K. Under steadystate test conditions, the input estimates for x1 and x2 are nearly zero. The standard uncertainties u (x1) and u (x2) were estimated to be ±0.01 K (converted to microvolts in Table 
Annex 1 

39 

A1.4 for the guard gap thermopile voltage), and ±0.5 K, respectively. The input values for 

bi and u (bi) = s (bi) were obtained from Table A1.4. 

Table A1.5. Estimates for uc (ΔQ) 

Tm x1 u(b1) b1 u(x1) x2 u(b2) b2 u(x2) uc(ΔQ) 

(K) (μV) (W/μV) (W/μV) (μV) (K) (W/K) (W/K) (K) (W) 

281 -0.0230 3.94×10-5 0.002 245 2.41 0.0051 3.95×10-4 0.005 82 0.5 0.0061 

281 -0.0374 3.94×10-5 0.002 245 2.41 0.0058 3.95×10-4 0.005 82 0.5 0.0061 

281 0.0365 3.94×10-5 0.002 245 2.41 0.0015 3.95×10-4 0.005 82 0.5 0.0061 

297 0.0298 2.39×10-5 0.002 235 2.47 0.0008 2.39×10-4 0.001 90 0.5 0.0056 

297 0.1300 2.39×10-5 0.002 235 2.47 -0.0043 2.39×10-4 0.001 90 0.5 0.0056 

297 0.0071 2.39×10-5 0.002 235 2.47 0.0020 2.39×10-4 0.001 90 0.5 0.0056 

313 -0.0523 2.23×10-5 0.002 363 2.54 0.0032 2.23×10-4 0.001 34 0.5 0.0060 

313 -0.0470 2.23×10-5 0.002 363 2.54 -0.0038 2.23×10-4 0.001 34 0.5 0.0060 

313 -0.0375 2.23×10-5 0.002 363 2.54 -0.0017 2.23×10-4 0.001 34 0.5 0.0060 

A1.3.3 Combined Standard Uncertainty u	(Q)

The combined standard uncertainty u (Q ) was computed from Eq. (A1-14). 

222

A B () ( ) ( ) ( ) c mm uQ uQ uQ u Q    (A1-14) 

Table A1.6 summarizes the input values for Eq. (A1-14) and the corresponding estimates 

for uc (Q) at Tm levels of 281 K, 297 K, and 313 K. 

Table A1.6. Combined standard uncertainty (k = 1) for uc (Q) 

Tm u (Qm)A u (Qm)B uc(ΔQ) uc (Q) Q /2 uc, rel (Q)

(K) (W) (W) (W) (W) (W) (%) 

281 0.000 67 0.0028 0.0061 0.0068 6.169 0.11 

297 0.000 72 0.0030 0.0056 0.0064 6.456 0.10 

313 0.000 72 0.0031 0.0060 0.0068 6.853 0.10 

A1.4 Meter Area (A)

The meter area is the mathematical area through which the heat input to the meter plate Qm

flows normally under ideal guarding conditions (i.e., Qg  0) into the specimen. The circular meter area, corrected for thermal expansion effects, was calculated from Eq. (A1-15): 

    π 22 2 1 α

2

A rr T     o i mp (A1-15) 

where ro is the outer radius of the meter plate (m); ri is the inner radius of the guard plate 

(m); α is the coefficient of thermal expansion of 6061-T6 aluminum (K-1); and, ΔTmp is 

the temperature difference of the meter plate (Th) minus 20 °C (in kelvin). 
Annex 1 

40 

A1.4.1 Plate Dimensions

The design dimensions for the meter plate and the guard plate diameters are 405.64 mm and 

407.42 mm, respectively. A coordinate measuring machine (CMM) measured the roundness 

of the meter plate at six locations at the periphery and the diameter was determined to be 

405.67 mm (within 0.03 mm of the design dimension). A uniform gap width of 0.89 mm 

was established using three precision pin gages spaced between the meter plate and guard 

plate at equiangular intervals. The uncertainty of the pin gages was +0.005 mm/–0.000 mm. 

Based on these check measurements, the input values for ro and ri for Eq. (A1-15) were determined to be 0.20282 m and 0.20371 m, respectively, and the standard uncertainty (k = 1) 

for both input values was taken to be 0.0254 mm. 

A1.4.2 Thermal Expansion Effects

For α, an input value of 23.610-6 K-1 at 20 °C was obtained from aggregated handbook data 

[20] for 6061-T6 aluminum. The standard uncertainty (k = 1) for the value of α was estimated conservatively to be 10 % (2.3610-6 K-1). The standard uncertainty (k = 1) for ΔTmp was 

based the uncertainty for the plate temperature (Sec. A1.6.7). 

A1.4.3 Combined Standard Uncertainty u	(A)

Application of Eq. (A1-3) to Eq. (A1-15) yields 

0

2 2 22 22 2 2 () ( ) () () ( ) i mp c r o r i T mp u A cu r cu r cu c u T        (A1-16) 

with 

or c = 

o

A

r





 =  

2

π r T

o mp

1 α

ir c = 

i

A

r





 =  

2

π r T i mp 1 α

cα = 

α

A



 =     2 2 π    Trr T mp o i mp 1 α

Tmp c =   mp

A

T



 

 =     2 2 π α rr T o i mp    1 α

Table A1.6 summarizes the contributory uncertainties for the meter-area calculation. 

Table A1.6. Standard uncertainty components (k = 1) for A

Source Description Estimate Standard 

Uncertainty 

Type of 

Evaluation

u1(ro ) Meter-plate outer radius 202.82 mm 0.0254 mm B 

u2(ri ) Guard-plate inner radius 203.71 mm 0.0254 mm B 

u3(α ) Linear thermal expansion coefficient 23.610-6 K-1 2.3610-6 K-1 B 

u4(ΔTmp ) Temperature difference (Tm = 280 K) –0.65 K 0.063 K B 

u4(ΔTmp ) Temperature difference (Tm = 310 K) 29.4 K 0.063 K B 

u4(ΔTmp ) Temperature difference (Tm = 340 K) 59.4 K 0.063 K B 
Annex 1 

41 

Table A1.7 summarizes the combined standard uncertainties for A at three levels of Tm. Values 

of A were computed with Eq. (A1-15). The estimates for uc (A) are, as evident in Table A1.7, quite small. 

Table A1.7. Combined standard uncertainty (k = 1) for uc (A) 

Tm A uc(A) uc, rel(A) 

(K) (m2) (m2) (%) 

281 0.12979 0.000023 0.018 

297 0.12988 0.000025 0.019 

313 0.12998 0.000029 0.023 

A1.5 In-situ Thickness (L)

The thickness of each pair of specimens was measured during testing using the average of 

eight (four top and four bottom) linear positioning transducers at the periphery of the 

plates. The equation for the in-situ thickness measurement is: 

1

1 n

i

i

L L

n 

  (A1-17) 

where Li is the measurement value for an individual positioning system and n equals 8. 

Each positioning system consists of a digital readout and a slider translating on a tape 

scale bonded to a precision ground plate of a low thermal expansion iron-nickel (FeNi36) 

alloy. The electrical windings on the scale are inductively coupled with the slider and the 

resulting output signal from the scale is resolved by the digital readout having a resolution of 2.5410-6 m. The digital readouts are reset by placing one set of fused-quartz 

spacers of known thickness between the cold plate and hot plate. Fused-quartz was selected for the spacers because of its extremely low coefficient of thermal expansion, 

5.510-7 cm/cmK. 

The standard uncertainties associated with the mean of the eight linear positioning transducers were determined for each guarded-hot-plate test using Eq. (A1-5) where n was 

equal to 8. The standard uncertainty u (L )A was subsequently computed from the pooled 

experimental standard deviations for the 15 tests and found to be 0.0508 mm. Contributory 

uncertainty components for the thickness measurement are described below. The contributory uncertainty for the gage blocks was 35 nm and neglected in further analysis. 

A1.5.1 Fused-quartz Spacers, u1	(L)A

The standard uncertainty was computed from the pooled standard deviations of multiple 

measurement locations of the four fused-quartz spacers and was 0.00422 mm (k = 1). 

A1.5.2 Micrometer, u2	(L)B

The standard uncertainty for the micrometer used to determine the lengths of the fusedquartz spacers was based on a uniform rectangular distribution with a symmetrical half-
Annex 1 

42 

width a of 0.0025 mm. Substitution of the half-width in Eq. (A1-9) yields a standard uncertainty of 0.0015 mm (k = 1). 

A1.5.3 Linear Positioning System, u3	(L)B

The standard uncertainty for the linear position transducers was based on the accuracy 

specification from the manufacturer as 0.0051 mm (k = 1). 

A1.5.4 Repeatability of Linear Positioning System, u4	(L)A

The short-term repeatability of the linear positioning system was determined from a sequence of replicate thickness measurements taken over four days using the fused-quartz 

spacers as reference values. The linear positioning indicators for the cold plates were initially reset using the four fused-quartz spacers. The cold plates were subsequently 

opened and closed until the plates were in complete contact again with the spacers. The 

readings from the digital indicators were recorded and the procedure was repeated five 

times (per day). To check the variation from day to day, readings were taken over four 

days for a total of twenty thickness averages. Table A1.8 gives summary statistics for the 

short-term repeatability test. 

Table A1.8. Summary statistics for short-term repeatability of linear positioning system 

Day Replicates 

Day average 

(m) 

Within-day 

standard deviation 

(m) 

1 5 1.266  10-2 4.98  10-6

2 5 1.268  10-2 5.88  10-6

3 5 1.267  10-2 1.99  10-6

4 5 1.267  10-2 5.09  10-6

The standard deviation of the daily averages (sa) was 7.9110-6 m and the (pooled) within-day standard deviation (sd) was 4.7210-6 m. The standard uncertainty was determined 

to be 8.9710-6 m using Eq. (A1-18) from [21]: 

  2 2

4

1

a d

r uL s s

r

   (A1-18) 

where r is equal to the number of replicates per day. The DoF (degrees of freedom) was 

determined from the Welch-Satterthwaite formula in Ref. [14]. 

A1.5.5 Plate Flatness, u5	(L)A and u6	(L)B

The standard uncertainty associated with the mean of 32 thickness measurements of the 

meter plate was determined to be 0.0023 mm. The standard uncertainty for the coordinate measuring machine (CMM) used to determine the meter-plate thickness was based 

on the accuracy specification from the manufacturer as 0.0051 mm (k = 1). The standard 

uncertainty for the plate flatness was estimated from Eq. (A1-19) to be 0.0079 mm. 
Annex 1 

43 

      2 2

,56 5 6 2 c u L uL uL   (A1-19) 

A1.5.6 Plate Deflection, u7	(L)B

The mechanical deflection of the (large) cold plates under mechanical loading was evaluated as 

a Type B uncertainty using classical stress and strain formulae for flat plates. Clamping forces 

(F1 and F2) were transmitted axially (Fig. 8) and distributed over a circular area at the center of 

each cold plate. For a uniform load over a concentric circular area of radius rf, the maximum 

deflection ymax at the center of the cold plate is given by the following formula [22]. Uniform 

support loading was assumed because the test specimen was a semi-rigid material. 

  4 2 24 2

22 2

7 max 23 2 2 2

3 ( 1) 31 73 ( ) 4 ln 2 16π 1 1

p ff f f c c

ff f

c p pf pc c

r r r rr F m t t uL y r r r Em t r r r rt t

                          

 (A1-20) 

where: 

F = applied load (N); 

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

E = modulus of elasticity (N·m-2); 

tc = thickness of cold plate (m); and, 

rp = radius of (cold) plate (m). 

From Table 8, the average clamping pressure f for the guarded-hot-plate tests was 1.1 kPa resulting in an applied load of approximately 477 N. For aluminum alloy 6061-T6, the values for 

m and E were taken to be (0.33)-1 = 3.0 and 6.9107 kPa, respectively. The cold plate thickness 

was 25.4 mm; radius of uniform loading was 108 mm; and, the cold plate radius was 508 mm. 

Substituting these values into Eq. (A1-20), yields a value of 0.032 mm for ymax. A major limitation for this assessment approach is that the cold plate is not a solid plate, but is actually a composite construction to allow the flow of coolant internally within the plate. 

A1.5.7 Combined Standard Uncertainty u	(L)

Table A1.9 summarizes the contributory uncertainties ui (L ) for the thickness measurement. 

Table A1.9. Standard uncertainty components (k = 1) for L

Source Description Standard 

Uncertainty (mm) 

Type of 

Evaluation 

u(L)A Multiple measurement locations 0.0508 A 

--- Gage block calibration Negligible B 

u1(L ) Fused-quartz spacers – multiple measurements 0.0042 A 

u2(L ) Fused-quartz spacers – micrometer 0.0015 B 

u3(L ) Linear positioning system 0.0051 B 

u4(L ) Repeatability of linear positioning system 0.0090 A 

uc,5,6(L ) Plate flatness 0.0079 B 

u7(L ) Cold plate deflection 0.032 B 
Annex 1 

44 

Equation (A1-21) computes the combined standard uncertainty (k = 1) for the temperature measurement. 

  

7

2 2

A

1

c i

i

uL uL uL   (A1-21) 

Substituting the component standard uncertainties from Table A1.9 into Eq. (A1-21) yields 

a value of 0.061 mm. For L equal to 13.43 mm (Table 8), uc, rel (L) was equal to 0.46 %. 

A1.6 Temperature Difference (ΔT	)

The temperature difference of the specimens was determined from the following equation: 

 1 2  2      TTT T T T hc h c c (A1-22) 

where the subscripts h and c refer to hot and cold, respectively, and the subscripts 1 and 2 

refer to the two cold plates (Fig. 8). 

The temperature difference across the specimen is determined by small capsule industrial 

platinum resistance thermometers (PRTs) located in the hot and cold plates (Th and Tc, 

respectively). Each thermometer is constructed of a strain-free platinum element supported in a gold-plated copper cylinder 3.18 mm in diameter by 9.7 mm long backfilled 

with helium gas and hermetically sealed. The operational range is from –260 °C to 

260 °C and the nominal electrical resistance is 100 Ω at 0 °C. The electrical resistance of 

each 4-wire PRT is measured with integrating digital voltmeter (DVM). 

The standard uncertainties u (ΔTi )A associated with the time-averaged observations taken 

over the steady-state measurement period were determined using Eq. (A1-5) where n was 

equal to 120. The standard uncertainty u (ΔT )A was subsequently computed from the 

pooled experimental standard deviations for the 15 tests and found to be 0.0002 K which, 

in comparison to other temperature uncertainty estimates, was neglected. 

A1.6.1 Sensor Calibration u1	(T	)B

The expanded uncertainty (k = 2) for the bath temperature measurements for the meter-plate 

PRT calibration was reported to be 0.01 K. Therefore, the standard uncertainty estimate of 

0.005 °C (k = 1) was used for the uncertainty analysis. 

A1.6.2 Regression Analysis u2	(T	)A

For each PRT, the individual observations (in ohms) were converted to temperature with 

the curve fit for the NIST Thermometry calibration data. The standard uncertainty was 

computed from the pooled residual standard deviations for each curve fit of the calibration 

data and was 0.0052 K (k = 1). 
Annex 1 

45 

A1.6.3 Electrical Resistance Measurement u3	(T	)B

The standard uncertainty for electrical resistance measurement of the PRT was based on 

the 1 year manufacturer specification for the integrating voltmeter (DVM). A uniform 

rectangular distribution was assumed for the accuracy specification with a symmetrical 

half-width a computed from Eq. (A1-23) where reading is in ohms. 

300 Range : 0.00015 8 m 0.0001       a reading reading (A1-23) 

For a reading of 131.372 Ω, a300Ω is 0.041 Ω. Substitution of 131.372 Ω ± (0.041 Ω/2) in 

the PRT calibration curve fit, yields a corresponding temperature half-width of 0.105 K. 

Final substitution of the half-width of 0.105 K in Eq. (A1-9) yields a standard uncertainty 

of 0.0607 K. 

A1.6.4 Temperature Rise due to PRT Self-heating u4	(T	)B

The nominal 100 Ω PRT dissipates about 0.0001 W for a 1 mA excitation current. For 

the cold plate PRTs, the thermal conductance of the metal-to-air-to-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. For the meter-plate PRT, a thin layer of thermally 

conductive silicone paste was applied to the sensor exterior surface to improve thermal 

contact. 

A1.6.5 Radial Plate Temperature Variation u5	(T	)B

From previous measurement data [23], an estimate for the radial sampling uncertainty was 

taken to be 0.015 K. In these separate experiments, the temperature profiles of the meter 

plates were estimated utilizing independent thermopile constructions placed between the 

plate surfaces and semi-rigid specimens. 

A1.6.6 Axial Plate Temperature Variation u6	(T	)B

A rigorous analytical analysis by Peavy published in Hahn et al. [24] shows that, for typical 

specimen insulations, the differences between the temperature of the meter-plate PRT at the 

mid-plane of the guard gap and the average surface temperature of the meter plate is less 

than 0.01 % and, thus, neglected in further analyses. 

A1.6.7 Combined Standard Uncertainty u	(T	)

Table A1.10 summarizes the standard uncertainty sources ui (T ), their descriptions, and 

corresponding uncertainty estimates for the PRT temperature measurement. 

Equation (A1-24) computes the combined standard uncertainty (k = 1) for the temperature measurement. 

 

6

2

1

c i

i

uT uT   (A1-24) 
Annex 1 

46 

Table A1.10. Standard uncertainty components (k = 1) for T

Source Description Standard 

Uncertainty (K) 

Type of 

Evaluation 

u(T )A Repeated observations Negligible A 

u1(T ) PRT calibration by NIST Thermometry Group 0.005 B 

u2(T ) PRT calibration data curve fit 0.0051 A 

u3(T ) Electrical resistance measurement/conversion 0.0607 B 

u4(T ) Temperature rise due to PRT self-heating 0.0017 B 

u5(T ) Temperature variation - radial dimension 0.015 B 

u6(T ) Temperature variation - axial dimension Negligible B 

Substituting the standard uncertainty components from Table A4-7 into Eq. (A1-24) 

yields a value of 0.063 K. 

A1.6.8 Combined Standard Uncertainty u	(ΔT	)

For a double-sided guarded-hot-plate test, the temperature difference (ΔT) across the 

specimen pair was determined from Eq. (A1-25). 

 1 2  / 2    TT T T h cc (A1-25) 

Applying Eq. (A1-3) to Eq. (A1-25) and setting u(Th) = u(Tc1) = u(Tc2) = u(T ) yields 

    3 2 2 1.5 0.063 0.077K

2 c u T uT      (A1-26) 

For ΔT equal to 20.000 K (Table 8), uc, rel (ΔT) was equal to 0.39 %. 

A1.6.9 Combined Standard Uncertainty u	(Tm	)

The mean temperature was determined from the Eq. (A1-27) 

   1 2  1 2 2

2 2 24 4

hc h c c hc c

m

TT T T T TT T T

 

    (A1-27) 

Applying Eq. (A1-3) to Eq. (A1-27) and setting u(Th) = u(Tc1) = u(Tc2) = u(T ) yields 

    6 2 2 0.375 0.063 0.039K

16 c m uT uT     (A1-28) 
Annex 2 

47 

Annex 2 - Uncertainty Analysis for Bulk Density (ρ)

A2.1 Digital Balance Uncertainty

The basic measurement equation for the digital balance is: Indication = Applied load ± 

Uncertainty. Assuming a uniform (rectangular) distribution, the standard uncertainty and 

the process standard deviation (sp) for the balance were determined from Eq. (A2-1). 

,

3 c bal p

d

u s   (A2-1) 

where d is equal to one display unit (i.e., the resolution) of the balance. For d equal to 

0.1 g, uc,bal is determined to be 0.058 g. 

A2.2 Length Uncertainties

The Type A standard uncertainties for the length measurements were computed by pooling the standard deviations from a relatively small number of measurements for each 

(square) specimen or metered section cylinder. 

The Type B standard uncertainties for the length measurements were assumed to have a 

uniform distribution in the interval ±a; where ±a was the smallest length interval of the 

length scale. The values of the interval ±a for the steel rule, calipers, and height gage 

were 0.5 mm, 0.01 mm, and 0.1 mm, respectively. The standard uncertainties were computed using Eq. (A2-1) with a set equal to d. 

A2.3 Uncertainty Budget

Table A2.1 summarizes the Type A and Type B standard uncertainties for the determination of bulk density for the 30 specimens (Sec. 4.3). 

Table A2.1. Summary of standard uncertainty components for bulk density 

Source 

u(xi), k = 1 

(Type A or Type B) DoFa

 for Type A 

evaluation or 

Specimen Cylinder Type B source 

1) Digital balance (m) 0.058 g (B) 0.058 g (B) Eq. (A2-1) Eq. (A2-1) 

2) Length measurement (l) 

 Repeated measurements 

 Steel rule 

0.01 mm (A) 

0.29 mm (B) 

0.67 mm (A) 

0.29 mm (B) 

180 

Eq. (A2-1) 

16 

Eq. (A2-1) 

3) Thickness measurement (L) 

 Repeated measurements 

 Height gage 

 Datum surface (granite plate) 

0.005 mm (A) 

0.058 mm (B) 

0.015 mm (B) 

0.039 mm (A) 

0.058 mm (B) 

0.015 mm (B) 

150 

Eq. (A2-1) 

Mfg. 

40 

Eq. (A2-1) 

Mfg. 

 a Degrees of freedom 
Annex 2 

48 

A2.4 Combined Standard Uncertainty

A2.4.1 Square Specimen Geometries

For square geometries, Eqs. (1) and (5) are re-written as 

1 2

ρs

m

ll L   

 (A2-2) 

where l1 and l2 are the length and width, respectively, of the specimen. For equal length 

sides, l1 and l2 are equal to l. 

The combined standard uncertainty uc (ρ) was determined using the following equation 

  22 22 22 ρ ( ) 2[ ( )] ( ) cs m l L u cu m cu l cu L  (A2-3) 

with 

2

ρs 1 mc

m lL

   

3

ρs

l

m

c

l lL

    

2 2

ρs

L

m

c

L lL

    

The factor 2 in Eq. (A2-3) occurs to account for the two equal sides of the specimen. 

Average values for cm, cl, cL, were 172.3 m-3, -61.5 kg/m4, and -3007 kg/m4, respectively. 

Substituting the uncertainty values for the specimen from Table A2.1 into Eq. (A2-3), the 

combined standard uncertainty uc (ρs) for the specimen bulk density determination was 

determined from the following equation: 

 

   

 

2 2

2 2

222

2

0.058 0.29 172.3 2 61.5

1000 1000

ρ

0.005 0.058 0.015 3007

1000 1000 1000

c s u

                                        

 (A2-4) 

   52 6 3 uc s ρ 9.987 63.62 3268 10 kg m 0.18 kg / m     (A2-5) 

 
Annex 2 

49 

A2.4.2 Circular Specimen Geometry

For a circular geometry, Eq. (12) is re-written as 

    2 2

4

ρ = ρ

π π

m cyl

m m

rL dL    

 (A2-6) 

where r and d (d = 2  r) are the radius and diameter, respectively, of the cylinder. 

The combined standard uncertainty uc (ρcyl) was determined using the following equation 

        2 2 22 22 ρ ρ m c cyl m c L u u cu m cu d cu L    (A2-7) 

with 

2

ρ 4

π

   

cyl

mc

m dL

3

ρ 8

π

    

cyl

d

m

c

d dL

2 2

ρ 4

π

    

cyl

L

m

c

L dL

Average values for cm, cd, cL, were 589.3 m-3, -208.8 kg/m4

, and -3103 kg/m4, respectively. Substituting the uncertainty values for the cylinder from Table A2.1 into Eq. (A2-7), 

the combined standard uncertainty uc (ρcyl) for the meter area bulk density determination 

was determined from the following equation: 

   

   

 

2 22

2 2

222

2

0.058 0.67 0.29 589.3 208.8

1000 1000 1000

ρ ρ

0.039 0.058 0.015 3103

1000 1000 1000

m c cyl u u

                                                 

 (A2-8) 

      52 6 3

u u ρ ρ m c cyl 116.8 2324 4920 10 kg m 0.27 kg / m       (A2-9) 

A2.4.3 Metered Section Correction for Specimen Pair

Application of Eq. (A1-3) to Eq. (14) yields: 

           1 2

2 2 2 2 2 2 22 2 ρ ρ ρ ρ ρρ , ρ 1 ρ 2 ρ ρ s s ms c pair corr s s m s u c u c u cu cu s     (A2-10) 
Annex 2 

50 

with 

  ,1 ,2

,

ρ ρ

,1

1

2

ρ

ρ s s

pair corr

s

c c

   

  ,

ρ ρ 1 ρ

ρ m s

pair corr

m

c c

   

    2

2 ρ

ρ i s

s

n 

Substituting the above sensitivity values (ci), the standard uncertainties from Eqs. (A2-5) 

and (A2-9), and the experimental standard deviation (s) and n from Table 10 yields: 

          

 

22 2

2 2 2 6

,

3

,

0.18 0.18 0.67

0.27 0.18 kg m 44 8

0.42 kg/m

ρ

ρ

c pair corr

c pair corr

u

u

 

    

 



 (A2-11) 

A2.5 Effect of Buoyant Force

Corrections for the effect of the buoyant force on the polystyrene solid polymer were estimated to be 0.1 % and neglected. 
Annex 3 

51 

Annex 3 - Compressive Resistance of Expanded Polystyrene Foam

The compressive resistance of SRM 1453 was determined in accordance with ASTM 

Test Method C 165 [25], Procedure A, using a universal testing machine. The purpose 

was to provide users of SRM 1453 a maximum pressure limit before the material would 

mechanically yield, i.e., deform permanently. Seven cylindrical specimens, 100 mm in 

diameter by 13 mm thick, were cut from Board 049 and conditioned in an environment of 

(23 ± 0.8) C and (53 ± 0.2) % relative humidity (aqueous salt solution) for about 11 days 

to a steady mass. Since the compressive resistance is a function of bulk density, the specimens were specifically selected from the lowest densities available. That is, the measured compressive resistance would be expected to represent the low end of the range of 

the SRM. 

During testing, the machine was operated at a crosshead speed of 1 mm/min. The compressive load and corresponding deformation were sampled in real-time at a rate of 

9.1 points per second in order to construct a load-deformation curve for each specimen. 

Using a straightedge, the straight portion of the curve was extended to the x-axis, establishing a “zero deformation point”. All deformations were determined with respect to 

this (zero) point. The compressive resistance was determined from: 

W S

A  (A3-1) 

where; 

W = compressive load at a given deformation, N; and, 

A = original (undeformed) area, m2. 

The compressive resistance for the 7 specimens at an average deformation of 0.34 mm 

(2.5 %) is summarized in Table A3.1. The load-deformation curve departed from linearity above this deformation. 
Annex 3 

52 

Table A3.1. Compressive resistance of specimens of SRM 1453, Expanded Polystyrene Board 

ID 

Bulk 

density 

(kg/m3

) 

Undeformed 

area 

(m2) 

Deformation 

(mm) 

Strain 

(%) 

Load 

(N) 

Compressive 

resistance 

(kPa) 

1 31.3 0.00785 0.34 2.5 2474 315.0 

2 32.6 0.00778 0.27 2.0 2275 292.6 

3 30.8 0.00781 0.33 2.5 2121 271.4 

4 31.8 0.00781 0.38 2.8 2469 316.0 

5 32.2 0.00785 0.35 2.6 2667 339.6 

6 31.2 0.00785 0.35 2.6 2375 302.4 

7 29.5 0.00785 0.36 2.7 2307 293.7 

Mean 31.3 0.00783 0.34 2.5 2384 304.4 

Std. dev. 1.0 0.00003 0.03 0.3 175 21.7 
Annex 4 

53 

Annex 4 – Bayesian Errors in Variables Analysis

A4.1 Nomenclature (specific to Annex 4)

Symbol Description 

ܶ෡݉,݅ mean measured temperature at which thermal conductivity for specimen pair ݅

is measured 

Tm,i true unknown mean temperature at which thermal conductivity for specimen 

pair ݅ is measured 

βi regression parameters 

λexp,	i	 measured thermal conductivity of specimen pair i

λtrue, i true unknown thermal conductivity of specimen pair i

ρˆ pair,corr,i corrected mean measurement of bulk density for specimen pair i

ρi true unknown mean bulk density for specimen pair i 

A4.2 Background

The quantities λୣ୶୮,௜, ܶ෠

௠,௜, and ρො௣௔௜௥,௖௢௥௥,௜ can be thought of as approximations to their 

true unknown counterparts5

. The following model for the measurements, which explicitly relates the measurements to their true unknown counterparts, is employed. 

 

 

 

 

8

exp, true,

, ,

, ,

2

true, 0 1 2 ,

λ λ ,3.61 10

ˆ , 0.0015

ρ ρ ˆ , 0.1764

λ ρ ,

i i

mi mi

pair corr i i

i i mi

N

T NT

N

N T  

 

 









 (A4-1) 

The variances for the normal distributions governing ܶ෠

௠,௜ and ρො௣௔௜௥,௖௢௥௥,௜ are given in 

Eq. (A1-28) and Eq. (A2-11), respectively. The variance for the normal distribution governing λ௘௫௣,௜ can be derived by applying Eq. (A1-3) to Eq. (3) using the quantities from 

Eq. (9), which are derived in Annex 1. 

Inference for the parameters of (A4-1) (ܶ௜, ρ௜, ߚ଴, ߚଵ, ߚଶ, and ߪ (is performed under the 

Bayesian paradigm, so priors are assigned to them. The following priors are used: 

 

5

 In accordance with the International vocabulary of basic and general terms in metrology (known as the 

VIM), a true value “in the classical approach to describing measurement, is considered unique and, in practice, unknowable”. 
Annex 4 

54 

 

 

5

01 2

50 50 300, 3 3

10 10

ρ 40, 3 3

, , and 0,1.0 10

Half-Cauchy 0, 25

i

i

T N

N

  N



    

 

      











 (A4-2) 

The Half-Cauchy(0, 25) may be thought of in the following way. 

  2

2

1

X N 0, 25

Y

X

Y



 



 (A4-3) 

This specific Half-Cauchy distribution as a prior for a standard deviation is proposed in 

[26]. The means for the prior distributions on ܶ௜ and ρ௜ are chosen to be the approximate 

center of the measured values. The variances for the prior distributions on ܶ௜ and ρ௜ are 

chosen so that 50 and 10, respectively, represent 3 standard deviations. The selection of 

3 standard deviations is an arbitrary choice. The sensitivity of the results to this choice 

was examined, and under reasonable choices (i.e. reasonably diffuse priors) the results 

were not affected. The values 50 and 10 encompass the range of the temperature and 

bulk density measurements, respectively. 

A4.3 Calculations

All calculations of posterior summaries were done using the open-source software2

OpenBUGS [27] and R [28]. The posterior means of ߚ଴, ߚଵ, and ߚଶ are 

0.00111, -0.0000424, and 0.000115, respectively. Thus, given a new measurement of 

temperature and density, ܶ෠

௠,௡௘௪ and ρො௣௔௜௥,௖௢௥௥,௡௘௪, respectively, the prediction of measured thermal conductivity, λ෨௘௫௣,௡௘௪ is 

exp,new , , , ˆ λ 0.00111 0.0000424ρˆ 0.000115   pair corr new m new T  (A4-4) 

A4.4 Uncertainty

Since the uncertainty in predictions from Eq. (A4-4) can depend on the specific ܶ௠,௡௘௪

and ρ௣௔௜௥,௖௢௥௥,௡௘௪, a regular grid (10×10) of ሺܶ෠

௠,௡௘௪,௝, ρො௣௔௜௥,௖௢௥௥,௡௘௪,௝ሻ pairs over 

ൣmin൛ܶ෠

௠,௜ൟ, max൛ܶ෠௠,௜ൟ൧ ൈ ൣmin൛ρො௣௔௜௥,௖௢௥௥,௜ൟ , max	ሼρො௣௔௜௥,௖௢௥௥,௜ሽ൧ is constructed. Note that 

i = 1, 2, ..., 15 indexes the original 15 measurements, and j = 1, 2, ..., 100 indexes the 

rectangular grid. Then, the posterior predictive distribution of λ௘௫௣,௡௘௪,௝ a new NIST 

measurement of thermal conductivity at ሺܶ෠

௠,௡௘௪,௝, 	ρො௣௔௜௥,௖௢௥௥,௡௘௪,௝ሻ, treating 

൫ܶ෠

௠,௡௘௪,௝, ρො௣௔௜௥,௖௢௥௥,௡௘௪,௝൯ as measured values, is calculated for each point in the grid 
Annex 4 

55 

݆ ൌ 1, 2, ... , 100. We calculate the posterior predictive distribution of ߣ௘௫௣,௡௘௪,௝ instead 

of λ௧௥௨௘,௡௘௪,௝ to ensure that our final expanded uncertainty accounts for the known 

sources of uncertainty, including those described in Annexes 1 and 2. Thus, the uncertainty is interpreted in terms of a new NIST measurement instead of the unknown true 

value. The posterior predictive distributions are then used to construct 95% credible intervals for λ௘௫௣,௡௘௪,௝, and the upper and lower bounds of those intervals are denoted as 

λ௘௫௣,௡௘௪,௝

௨ and λ௘௫௣,௡௘௪,௝௟ , respectively. The relative expanded uncertainty for a prediction 

from Eq. (A4-4) is then taken to be 

  exp,new, exp,new, exp,new, exp,new,

, exp,new, exp,new,

λ λλ λ

max max , 1.5 % λ λ

l u

j jj j

j lu j j

U              

       

 

  (A4-5) 

where 

exp,new, , , , , , ˆ λ 0.00111 0.0000424ρˆ 0.000115 . j   pair corr new j m new j T  (A4-6) 

Of course for some j, U will represent a credible interval that covers more than 95 % of 

the appropriate posterior predictive distribution, but to give a single expanded uncertainty 

for all predictions from Eq. (A4-4) that must be the case. The effective value of k for this 

expanded uncertainty is 2, which is calculated by dividing U by the standard deviation of 

the posterior predictive distribution of ߣ௘௫௣,௡௘௪,௝ that provides U. 
Annex 5 

56 

Annex 5 - Sorption Isotherms for Expanded Polystyrene Foam

Sorption isotherms for expanded polystyrene foam were determined using fixed-point 

humidities provided by aqueous salt solutions. To increase the surface area and thus enhance the sorption process, small rectangular billets, 0.5 mm by 2 mm by 13 mm, of foam 

were cut. One set of billets was prepared for adsorption measurements; the other for desorption measurements. Both sets of foam billets were conditioned in an environment 

over calcium-chloride desiccant to obtain a “dry” or tare mass. The billets for desorption 

were subsequently removed from the desiccant containers and placed over distilled water. 

A fixed number of billets from each set were subsequently placed in containers at 11 %, 

33 %, 43 %, 58 %, 79 %, 84 %, 94 % and 97 % relative humidity at laboratory temperature of 24 C. 

The billets from each container were collectively removed and weighed to monitor 

changes in mass due to sorption of water vapor. The weighing process was repeated until 

the conditioned mass of the billets achieved a steady value (i.e., less than 1 % change 

over a 2-day interval). The equilibrium moisture content, γ(φ), was determined by taking 

the differences between the conditioned mass, m(φ), and initial mass, mi, and dividing by 

mi. 

  φ γ φ i

i

m m

m

  (A4-1) 

For the relative humidity range of 11 % to 94 %, γ(φ) was found to be less than 0.4 % 

and less than 1 % at 97 % relative humidity. 
Annex 6 

57 

Annex 6 - General Precautions for Expanded Polystyrene Foam

A6.1 Upper Temperature Limit

Polystyrene is a thermoplastic and, by definition, has a well-defined temperature at which 

the polymer softens. The upper temperature limit before cellular polystyrene foam softens, as specified in the ASTM Specification C 578 [12], is 74 C. Mehta et al. [29] reported that similar polymer polystyrene beads when exposed to elevated temperatures 

collapsed about 110 °C to 120 C. The collapsed beads melted at 160 C and began to 

vaporize above 275 C [29]. 

A6.2 Flammability

Standard Reference Material 1453 is a commercial grade of expanded polystyrene foam 

that does not contain fire-retarding additives. It is an organic thermoplastic material that 

is combustible and, at elevated temperatures melts (at a temperature greater than 160 C, 

[29]). Do not expose the material to sources of ignition. 

A6.3 Solvents

Polystyrene is soluble in many organic solvents such as chlorinated and aromatic hydrocarbons, esters, and ketones. The aromatic chemical structure of polystyrene is inherently water repellant. As noted in Annex 5, the material is very insensitive to low levels of 

humidity. 

A6.4 Ultraviolet Degradation

When exposed to sunlight (ultraviolet radiation), polystyrene degrades as evidenced by 

discoloration of the surface. To minimize degradation, protect SRM 1453 by covering 

when exposed to direct sunlight.