NAT'L INST. OF STAND A TCru R 83-2674

Amob 2t0£7 b Qr Ana|ySj s for the National

Bureau of Standards 1016 mm

Guarded Hot Plate

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

February 1983

Issued April 1983

This work was jointly sponsored by the Department of Energy and the National

Bureau of Standards.

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

&c

i o o

.US(,

fk.ss-W

NATIONAL BUR£A17 SUSDAJiDS ubrauy

apr zo 1993

NBSIR 83-2674

ERROR ANALYSIS FOR THE NATIONAL

BUREAU OF STANDARDS 1016 mm

GUARDED HOT PLATE

Brian Rennex

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

National Engineering Laboratory

Center for Building Technology

Washington, DC 20234

February 1983

Issued April 1983

This work was jointly sponsored by the Department of Energy and the National

Bureau of Standards.

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler. Director

PREFACE

This report is one of a series documenting NBS research and analysis efforts to

support the Department of Energy/National Bureau of Standards' Measurements

Program. The work reported in this document was performed cooperatively and

supported by DoE/NBS Task Order A008 under Interagency Agreement No.

DE-AIO 1-7 6PR060 10.
ABSTRACT

An error analysis is given for the 1-meter Guarded Hot Plate at the National

Bureau of Standards. This apparatus is used to measure the thermal resistance

of insulation materials. The individual contributions to uncertainty in

thermal resistance are discussed in detail. The total uncertainty is estimated

to be less than 0.5 percent at sample thicknesses up to 150 mm (6 inches) and

less than 1 percent at a thickness of 300 mm (12 inches).

Keywords: apparent thermal conductivity; error analysis; guarded hot plate;

thermal insulation; thermal resistance.

iv
TABLE OF CONTENTS

Page

ABSTRACT lv

LIST OF FIGURES vi

LIST OF TABLES ..... vi

ACKNOWLEDGMENTS vli

1. INTRODUCTION 1

2. PRINCIPLE OF MEASUREMENT 2

3.

ERROR ANALYSIS 3

3.1 SUMMING OF INDIVIDUAL UNCERTAINTIES 3

3.2 SUMMARY LIST OF INDIVIDUAL UNCERTAINTIES 3

3.3 METER-AREA 4

3.4 THICKNESS 5

3.5 HEAT FLOW 7 3.5.1 Qmeter 8

3.5.2 Qresistor 9

3.5.3 Qgap ••••••••• 9

3.5.4 Gap Theory 12

3.5.5 Qedge 14

3.6 TEMPERATURE MEASUREMENT 16

3.6.1 Overview 16

3.6.2 PRT Circuit Rationale 16

3.6.3 PRT Calibration 16

3.6.4 PRT Self-Heating Uncertainty 17

3.6.5 Temperature Distribution Over Meter Area 17

3.6.6 Summary of Temperature Uncertainties 19

3.7 PLATE EMITTANCE 19

4.

TOTAL UNCERTAINTY 22

5. VERIFICATION OF UNCERTAINTY ESTIMATE 24

6. REFERENCES 25

v
LIST OF TABLES

Page

Table 1. Percentage Estimate of Uncertainties in the Measured Apparent

Thermal Conductivity for the NBS Guarded Hot Plate 26

LIST OF FIGURES

Pa8>e

Figure 1. Heat flow (Q) in a two-sided guarded hot plate 27

Figure 2. Side view of gap 28

Figure 3. Plate support and thickness calibration 29

Figure 4. Thickness transducer for L-map 30

Figure 5. Circuit for meter-area power measurement 31

Figure 6. Hot-plate, meter-area power, C^, versus gap voltage 32

Figure 7. Calculated angular temperature distribution at gap 33

Figure 8. Diagram of gap thermopile junctions 34

Figure 9. Voltage circuit and wire configuration for the gap

thermopile • 35

Figure 10. The change in apparent thermal conductivity, X, as a function of ambient temperature 36

Figure 11. Circuit for the platinum-resistance thermometer (PRT)

resistance measurement 37

Figure 12. Theoretical plot of the hot-plate, meter-area radial

temperature distribution 38

Figure 13. Measured hot-plate, meter-area radial temperature

distribution 39

Figure 14. Figure showing temperature uncertainty due to hot plate PRT

location 40

vi
ACKNOWLEDGMENTS

The author expresses his appreciation to Dave Ober, who developed most of

the instrumentation and many of the concepts herein, to Rocky Somers and

John McAuley for their tireless efforts to improve and run the apparatus, and

to Robert Jones and Prank Powell for their leadership in this task.

vii
.

'
1 . INTRODUCTION

This report gives the error analysis of the National Bureau of Standards (NBS)

1016 mm Guarded Hot Plate (GHP) apparatus. It begins with a brief description

of the apparatus and of the measured quantities necessary to calculate the

thermal transmission properties of a sample material. A method to add the

individual uncertainty contributions is rationalized. There follows a detailed

discussion of upper-bound estimates of the random and/or systematic uncertainties associated with each measured or modeled parameter. The propagation of errors is accomplished by summing these estimates. There is a discussion of

the steady state condition and repeatability. Finally, there is a discussion

of how to compare results among different apparatuses, such as would pertain

in a round-robin test series. The uncertainties herein apply to the apparatus

only, and not to uncertainties associated with material variability of

insulation samples.

1
2. PRINCIPLE OF MEASUREMENT

The guarded hot plate measures the heat flow through a sample for a particular

temperature boundary condition. Figure 1 shows the basic features of the

apparatus in a two-sided configuration with equivalent specimens on either

side of the hot plate and with both cold plates at the same temperature. There

is a resistance heater in the meter area. A, of the hot plate. The power, Q,

produced by this heater is measured. In order to ensure the most accurate and

repeatable characterization of a sample, it is necessary that this meter heat

flow straight across to the cold plates —that is, that its direction be in one dimension. For this reason, the guard area of the hot plate is maintained

at the same temperature as the meter area. Assuming that the metered heat is

split evenly between the two sides, the average thermal resistance, R, of the

two sides of the sample pair is calculated by

r = ATA

(Qm/2)

( 1 )

That is, R is defined in terms of the basic quantities Qmand A, and of the

boundary conditions, namely the hot plate and the two cold plate temperatures.

The term AT is the average of the temperature differences between the hot

plate and the two cold plates. An additional parameter is the specimen thickness, L, averaged over both sides. The apparent thermal conductivity, X, is

defined by

L » L_

R 2A AT

( 2 )

Note that each of the above parameters represents an average over the meter

area and over the two sides. The procedure is to place specimens in the

apparatus and to monitor the plate temperatures and the power, Qm , to establish

when these become constant—to within the limits of the instrumentation control.

The test is defined to be in steady state when subsequent values of R (or X)

vary randomly about the mean value. Typically, the R-value can be controlled

within a few hundreds of a percent, and when the R-value does not change by more than this amount within ~5 hours there is good confidence that steady

state has been achieved. The time required to achieve steady state varies from

~2 hours for a 75 mm-thick, 10 kg/m^ sample to ~30 hours for a 300 mm-thick,

130 kg/m^ sample. Finally, the apparatus can test a sample on only one side of

the hot plate (referred to as a one-sided configuration) by maintaining the

temperature of the other cold plate approximately equal to the temperature of

the hot plate.

2
3. ERROR ANALYSIS

3.1 SUMMING OF INDIVIDUAL UNCERTAINTIES

The apparent thermal conductivity and the thermal resistance of an insulation

sample are calculated quantities based on several measured parameters. Each of

these measured parameters has an uncertainty. In turn, this uncertainty has a random and systematic (or constant) part. It is possible to estimate the

uncertainty of each parameter by an independent test. For example, the apparatus thickness readout can be compared with an independent thickness detector

placed between the plates. This independent detector offers the advantages

that it is more precise and more easily calibrated and it can be located in

the region of actual interest.

It is not straightforward to estimate the overall uncertainty of the calculated

quantity, because there is usually not sufficient information on the breakdown

between the random and systematic parts for each individual parameter. In

principle, it is possible to gather this information, but in practice it would

be too time consuming.

A simple and practical approach was used in this error analysis. The individual

parameters, such as the thickness or temperature distribution over the meter

area, were measured with an independent detector under test temperature conditions. A comparison of the apparatus readout with these independent measured

values made possible the estimate of an upper bound on the total uncertainty

for each parameter. Since there is not sufficient information to assure that

the measured values are randomly distributed about a "true" mean value, the

upper bounds for each individual parameter are simply added to arrive at the

overall uncertainty. This is different from an alternative approach to treat

the uncertainties as standard deviations with the total uncertainty being

calculated as the square root of the sum of the squares of the individual

standard deviations. This "upper bound" approach results in a somewhat larger

estimate of the overall uncertainty (by as much as 30 percent), but it avoids

the need to make an inordinate number of check-up measurements to assure that

there are no outlier values. This more conservative approach is thought by

the author to be appropriate for a national insulation standards lab.

The following philosophy was used with regard to the estimate of upper bounds.

Even if an uncertainty might have been expected to be smaller, based on theoretical considerations and manufacturer specifications, the uncertainty

value actually used was that of the detector making the independent check.

For example, the plate temperature might very well be known within 5 or 10 mK.

The uncertainty value actually used, of 22 mK, was associated with the thermopile used to independently check the uniformity of the plate temperature. That

is, the claim did not go further than what could be empirically demonstrated.

3.2 SUMMARY LIST OF INDIVIDUAL UNCERTAINTIES

The following is a list of measured parameters that contribute to the major

individual uncertainties which will be discussed in detail in the next sections:

meter area, sample thickness, meter-area heat flow, heat flow across gap

3
between the guard and meter areas, net heat flow to the sample edge, and plate

temperatures.

3.3 METER-AREA

The value of the circular meter area, A, in equation (2) corresponds to the

radius at the geometric center of the gap (refer to figure 2). Since the

measured values are 0.20282 m (7.985 inches) for the inner radius, rp , and

0.20371 m (8.020 inches) for the outer radius, r Q , then the geometric-center

radius, r c , is calculated as follows:

_2 _ 2 1/2

r c = ( r

° + 1

) (3)

rc - 0.20326 m (8.0025 in)

The systematic error in rc was estimated to be not more than 0.012 mm

[0.0005 inches]. This was the machining uncertainty estimate.

This radius corresponds to a temperature of 295. 2K [72°F]. When the hot plate

is at another temperature, the thermal expansion of the plates must be

considered. The radius corresponding to the meter area is calculated by

r£ = r c (l+ a(Th-295.2K)) . (4)

Typically, Th

= 310.8 + 1 K ( 100 F ± 2 F).

a = the coefficient of thermal expansion for the Aluminum hot plate

(type 6061-T6).

a = (24.3 + l)xl 0~6 m/m»K

The meter area is calculated by

A = ir r

c

2

. (5)

The set of independent variables for A is {r c , T^, a}. The values of the

uncertainties in these variables, A^, are as follows:

Ar = 0.012 mm

AT= 1 K

A

Q

= 4 x 10 ^ m/m K

The contributions from the A-p and the A a terms are negligible, so the present

uncertainty in A is

A » 2 A — » = .0001 = .01 percent . ( 6 ) A r_

4
A(37K) = tt r^ 2 (7)

= 0.12990 m

2 ± 0.000016 m

2

= 201.3397 in2

Note, this meter area corresponds to the sample area through which the heat

generated in the meter area heater would flow if the heat flow were one- dimensional.

3.4 THICKNESS

The parameter L in equation (2) refers to the average specimen thickness over

the meter area. For low-density or compressible specimens this is best estimated by the value of the average plate spacing over the meter area. Precaution should be taken to compress such a sample slightly, so as to avoid voids

between the specimen and the plates. For rigid specimens, there are two possible ways to estimate the test thickness. First, if the specimen surfaces and

the plates are sufficiently flat and parallel, then the plate spacing is a good estimate. Second, if any of these surfaces are irregular, then the specimen thickness itself should be measured—with a caliper, for example. There

should then be an estimate of the thermal resistance of the small air gaps at

the irregular surfaces. This should be negligible.

The measurement of the plate spacing is described below. The basic calibration

method is to measure the positions of the four outside "corners" of the cold

plates relative to a known meter-area plate separation. These outside coldplate positions are measured using four thickness transducers on each plate.

Then these thickness transducers measure any change in position relative to

the initial calibration point. This calibration must be done for each plate

orientation and for compressible and rigid samples.

The following is a more detailed discussion of the thickness calibration

referred to as the L-map. Please refer to figure 3. Here is shown the basic

plate support and measurement system. The hot plate is rigidly and permanently

mounted on the four support rods. The cold plates are supported in the center.

This point of support has a load cell to measure the force that the sample

exerts on the plate. It also has a ball joint so that the plate can tilt to

conform to a nonparallel rigid sample. The cold plates are constrained in the

radial direction by steel cables attached to four spring loaded bearings which

are mounted on stainless-steel rods with a diameter of 51 mm (2 inches).

At four points (at 90° intervals) at the edge of each cold plate the positions

(perpendicular to the plates) are measured with thickness transducers (referred

to as Ftt). The manufacturer's stated overall accuracy over a displacement of

150 mm (6 inches) is within 10 ym (0.0004 inches) for a single Ftt. The mean

displacement of the four Ftt's on a single cold plate would then be known

within a factor of 1//4 or 5 ym (0.0002 inches). The Ftt's are mounted on Invar bars to minimize error due to temperature variation of the ambient. A

coefficient of thermal expansion value of 10 _^m/m»K, a length of .25 m

(10 inches) and a temperature range of 50 K correspond to a length change (or

5
error) of 13 pm (0.5 mil). This contribution to error is avoided by the

calibration procedure to be described later, since it is performed at the

ambient temperature appropriate to the test.

The calibration procedure is to correlate an absolute measurement of the

average meter area spacings with average displacements of the Ftt's. The Ftt's

will then provide any displacements from this original calibration point,

provided the plates remain in the same orientation—that is, that they are not

rotated. The absolute measurement of the average meter area spacing is made

with thickness transducers such as that shown in figure 4. This measurement is

made with the plates both horizontal and vertical, since the plate sag is

different in the two cases. These thickness transducers were calibrated with

an uncertainty of 5 micrometers (pm), (0.2 milli-inches or rails), using NBS

gauge blocks accurate within 0.3 pm (0.01 mils) at 293.15 K (20 C). The

uncertainty in the gauge block length, due to temperature difference from that

of calibration when the gauge block was calibrated, is estimated to be 3 pm

(0.1 mils). The measured flatness within the meter area is + 25 pm (1 mil).

In the L-map, the plate spacing is measured at 25 points in the meter area. The

repeatibility of the average of these 25 points was 5 pm (0.2 mils). Thus the

uncertainty is about 1/5 = 1//25 times the flatness.

The next step is to measure the difference between the spacing at the center

of the meter area and the average spacing over the meter area (the average of

the 25 points). The reason is that the calibration procedure can then rely on

the measurement at one point at the center, rather than require 25 points.

Another 2.5 pm (0.1 mil) value is added due to the transducer uncertainty for

the transducer center point measurement.

To understand the final contribution to thickness uncertainty we must look at

the problem of plate bowing deformation. The plates are 1 m (40 inches) in

diameter. The cold plates are 19 mm (0.75 inches) thick and weigh about 50 kg

(110 lb). The hot plate is 16.13 mm (0.635 inches) thick and weighs about 32 kg

(70 lb). Referring to figure 3, the various forces that act to deform the

plate are due to the load cell at the center, the weight of the plates and the

spring-loaded cables at the four corners. As the cold plates open and close,

they tilt, which causes a deformation change. The problem with the bowing

deformation is the following. The Ftt readings correspond to four thickness

points at the plate edges, and the average of these is not necessarily the same as the value at the plate center, when there is a bowing deformation.

Repeatability studies compared the Ftt thickness values with those of a transducer at the plate center, after the cold plates have been opened and closed

back to the original thickness. These gave a repeatability of 5 pm (0.2 mils).

Note that it is important that the stainless steel plate support rods be

straight and parallel to have the best repeatability.

To review, the calibration procedure uses thickness transducers which are calibrated with NBS gauge blocks and placed in the center of the meter area,

between the hot and cold plate, at the test thickness and temperature. The

transducer reading then is used to set the 8 Ftt's (4 on each side). The

plates are then opened, the samples are put in and the plates are closed to the

6
test condition. Below is a summary of the uncertainties, , that contribute

to the thickness uncertainty for compressible (low-density) specimens.

A-[ pm mil

auxiliary thickness transducer 5 0.2

thermal expansion of gauge block 2.5 0.1

A between average and center meter-area spacing 5 0.2

A of center position 2.5 0.1

Ftt (Farrand transducer) 5 0.2

repeatability after opening and closing cold plates 5 0.2

total = Ai 25 1.0

3.5 HEAT FLOW

There are a number of heat flows, designated by 0, involved in the guarded hot

plate apparatus (refer to figure 1). There is the heat generated by the meter

heater, Qm , and the heat generated by any resistive sensors in the meter area, Qr. There is the heat flow between the meter area and the guard area, Q„, which

consists of a part flowing directly across the gap, Q0 , and a part flowing

through the specimen, cX. Here, X is the thermal conductivity of the specimen.

Also, there is some net heat flow between the meter area and the ambient through

the specimen. This will cause the actual heat flow leaving the meter area to

be different from the idealized one-dimensional heat flow. The difference is

the edge heat flow, Q e<j. Qe(j depends on the guard and meter widths, on the

thickness, on the ambient and plate temperatures and on the sample X-value.

The final, "apparatus-independent", meter-area heat flow must take into account

these correction terms.

Even then, it is not entirely apparatus-independent, even in principle,

because part of the heat flow is by radiation. This means that the heat flow

across the sample depends on the plate emittance, e. Intuitively this can be

understood since a smaller emission of thermal radiation from the plates will

result in a smaller sample heat flow. The easiest way to minimize this apparatus dependence is to standardize the values of the plate emittances. The

ASTM Test Method 177 for the guarded hot plate requires that this emittance

value be approximately 0.90. This standardization is intended to minimize

discrepancies among the various apparatuses. Then, if an apparatus has a different emittance value or an uncertainty in emittance value, the adjustment or uncertainty in the measured X could be calculated. This is discussed

in more detail under the section on plate emittance.

7
3.5.1 Qmeter

Let us now look at each heat-flow terra in more detail. The first terra, Qm , is the power produced by the meter heater. Figure 5 shows the schematic for

for this measurement. A known standard shunt resister, Rg , of approximately

0.1 0, that is maintained in an oil bath, is used to determine the current.

Voltage taps across the meter heater lead wires in the center of the gap are used to measure the voltage corresponding to the power that would be expected

to travel one-dimensionally to the cold plate.

Qm ~ heater

R

( 8 )

The value of Rs is calibrated at one year intervals.

_ 0.1000700 ft + 5 ppm at 25.0°C

s

An additional uncertainty of 20 ppm is included since the resistor is in a bath

at 28°C. The uncertainty of 25 ppm corresponds to a 0.0025 percent uncertainty,

which is negligible compared with other uncertainties.

If the heater voltage taps were not in the center of the gap, there would be an

error in the measurement of V^eater . The nichrome wire size is 28 gauge which

has a resistance value of 42 d/305 m (1000 ft). Assuming (conservatively) that

the taps might be misplaced by half the gap width of 6 mm (0.23 in), then the error in R

^ eater is calculated to be 8 x 10 -

^ ft. This is negligible when com- pared with the value of Rheater approximately 56 ft. Next we need to consider

the accuracy of the voltage measurements. The heater voltage, Vyieater , varies

from about 8 volts to 27 volts for one- and two-sided operation at thicknesses

of 25 to 150 mm (1 to 6 inches). The shunt voltage Vg varies from about 14 mV

to 48 mV for the same range of operation. The following equations are needed to

calculate the the uncertainties. First, in order to calculate the approximate

Qm

for a particular specimen in the two-sided mode we use

Qm

=

2 XATA ( 9 )

For low-density material, X « 0.046 W/m*K

A » 0.1299 m

2

, and AT = 27.8 K. For L « 150 mm, Qm

« 2W.

Also Rheater “ 56 ft and Rg « 0.1 ft.

Thus V^ea£ er 560 V g Once Qm

is known, we can calculate Vg with equation (8).

vs = ( Rs <V/2

560 n

» 20 mV for the previous example

8
Thus Vhea£ er

B 10 V.

These order-of-magnitude estimates are necessary to calculate the percentage

errors in V^eater and Vg. As an example, for the 10 V range and for a 90 day

period, the manufacturer's specifications give the following formula to

calculate the voltage uncertainty, Av.

3 Av = ( 0 *0Q23 v + 2 counts x count value)

100

(ID

This is for a 6 digit readout and an average over 10 power line cycles. The

count value is somewhat larger than the minimum detectable signal and has a value of 10 yV. If V = 8V, Av = .000008 = .0008 percent. This is obviously

negligible compared to other voltage errors. Another example is the reading

of Vs on the 0.1 V scale. Let Vs = 14 mV, which is a worst-case value.

3 Av = ( 0»0034 x 20 mV + 24 counts x 0.1 yV)

100

= 3.1 yV

3 Av = 0.00015 = 0.015%

V0

( 12 )

This Av uncertainty is the only contributor to the uncertainty in Qm

that is not

negligible, and it is mostly due to the "24 counts". We will use a worst-case

value of 0.04 percent for AQm/Qm.

3.5.2 Qresistor

The resistive devices in the meter area generate a small amount of power, Qr , which must be added to Qm

. These devices include a platinum resistance thermometer, RTD for plate temperature readout and two thermistors, Rth» one f°r hQt

plate control and one for the control of heat across the gap.

^RTD “ 100 Q; I = 1 ma ; thus, Qr ^ = 0.1 mW

Rth

- 1 K I = 1 ma ; Q r2 = 1 mW

Qr = Q r i + 2Q r2 = 2.1 mW .

Assuming that Qr is known within 20 percent (a conservative estimate), AQr /Qm » .02 percent when Qm = 2W.

3.5.3 Qgap

A crucial point in the design of a guarded hot plate (GHP) is to minimize the

uncertainty due to the heat flow across the gap between the meter and guard

areas in the hot plate. The most straightforward method to estimate this error

is an empirical one. This method for the NBS-GHP will be discussed first, and

a theoretical discussion will be given later.

9
Looking at the empirical method in more detail, remember that the gap

temperature difference is estimated by a thermopile across the gap. In the NBS

GHP this is an 18-stage thermopile. In principle, a zero gap voltage would

correspond to a zero average temperature difference across the gap, which would,

in turn, correspond to a zero net heat flow, across the gap.

The gap is unbalanced by making the average temperature on the meter side of the

gap different from the average "guard-side" temperature. This results in a radial temperature gradient at the gap, which gives a non-zero thermopile voltage value for the gap voltage. Since the gap voltage is normally known or maintained to within several yV of zero, then a reasonable unbalance range

might be 30 yV. The slope of the curve showing the change in Q versus the gap

voltage is a measure of the sensitivity of the gap thermopile (see figure 6).

Values at a larger unbalance provide a better knowledge of the slope, although

points should be taken over the whole range to confirm that the curve is indeed

linear at zero Vg.

The gap sensitivity will be different for various values of specimen thickness

and thermal conductivity. Therefore, sensitivity studies must be performed for

the entire ranges of these specimen parameters. The values with the 18-stage

thermopile of the gap sensitivity for the low-density mineral-fiber insulation

material used for the NBS calibration transfer specimens are 0.00056, 0.00057

and 0.00058 W/ yV for L = 25, 75, and 150 mm (1, 3, and 6 inches), respectively.

The thermocouples comprising the gap thermopile were Chrome 1-Const ant an, Type

E, which have a sensitivity of 62.03 yV/K. Since there are 18 stages, the

actual sensitivity is 18 x 62.03 = 1116 yV/K. The gap sensitivity can be

expressed as 0.636 W/K. As an example, a T

g (temperature difference between

guard and meter side of gap) value of 0.01 K would result in a Qg value of

roughly 6 mW.

The contributions to the uncertainty resulting from the various estimates

of the gap voltage are as follows. The first question is how well does a zero

gap voltage correspond to a zero heat flow across the gap. The question arises

because there is expected to be some angular temperature difference at the gap

due to heater and sensor lead wires and the support pins. (These pins bridge

the gap to support the meter-area part of the hot plate.) The next question

is how accurately can this Tg (or the corresponding gap voltage, Vg) be

measured. This depends on the accuracy of the readout device and on the extent

to which thermals in the lead wires can be eliminated.

Addressing the first question, a calculation of the angular gap temperature

distribution due to heat generated by the heater leads as they come in from the

side was performed in reference [3]

.

Figure 7 shows the results of this calculation for the NBS GHP. For a worst case of a two-sided, low-density mineral

fiber sample at one inch thickness, Qmwould be roughly 12 W, so Q0 would be

about 90 W/m^ (A = 0.1299 m^). In this case, the maximum temperature difference

on the meter side is about 2 mK. (Note that this calculation assumes a zero

gap temperature difference, and that there is no insulation around the incoming

heater leads. In fact, there is wire insulation, and the plates will be more isothermal than what the calculation shows.)

10
In fact, the temperature difference is much larger than 2 mK for the following

interesting reason. The gap has a thermal conductivity roughly equal to that

of air. The thermal conductivity of the aluminum plate is roughly 6000 times

greater than that of air. The heat flow is proportional to the product of the

thermal conductivity and the temperature gradient. Assuming steady state,

then the radial heat flow at the gap (thru the plate and then thru the gap) is

constant. This means that the temperature gradient in the gap is 6000 times

greater than that in the plate. This means that a radial temperature gradient

of 0.2 mK/cm in the plate would give a temperature difference of roughly 100 mK

across the gap (the gap width is ~0.9 mm). In effect, the gap acts as an amplifier of the temperature gradient. As far as the heat flow across the gap

is concerned, however, this effect is very small.

An experimental study was made to measure the temperature difference across the

gap. It used a 16-stage thermopile which was firmly pressed against the plate

by a rigid specimen. Temperature differences of the order of 100 mK were

measured at positions around the gap, even when the gap voltage indicated an average temperature difference of <1 mK. This study showed that heat flowed

into and out of the meter area at various positions around the gap perimeter.

Thus, the net gap heat flow should be zero, even when temperature differences ~ 100 mK are measured at points on the gap perimeter.

It would be Impossible to make a realistic calculation of the angular

distribution of the temperature difference across the gap, because of the aforementioned amplification of temperature difference across the gap and because

the plate gradients that are being amplified are so small. Fortunately, it

is possible to ascertain empirically that the net gap heat flow is zero.

If one could calculate the gap heat flow as a function of angular position, it

would resemble a sinusoidal curve with positive peaks and negative valleys.

The danger, as far as error goes, would be that the thermocouple stages of the

gap thermopile might all lie on peaks. This would result in a systematic error

in the monitored heat flow. Certainly, this unfortunate possibility is less

likely if there is a larger number of evenly spaced thermocouple junctions.

(Note that the adjacent junctions need not be opposite each other, so the

leads can be longer than the gap width.)

A series of calculations was performed on curves of the kind shown in figure 7

,

to estimate the uncertainty in the knowledge of the mean as function of the

number N of thermocouple stages around the gap. (Let us assume, for the moment,

that the temperature difference, AT^, is measured exactly.) This study showed

that the uncertainty of the mean due to the finiteness of the sampling becomes

negligible after N equals 10. The conclusion, then, is that the angular distribution of gap temperature difference is not constant, but this does not

result in any measurable gap error as long as more than 10 thermocouple stages

are used.

Another reason to have a large number N is that the gap voltage increases

proportional to N. Thus the percent uncertainty due to the gap voltage readout

resolution decreases proportional to 1/N. This is true as long as the amount

of heat transferred across the gap through the thermopile wires is small

11
compared to the heat flow through the air. For the NBS GHP, this ratio is less

than 1/1000 for a single wire. As a note to the experimentalist, it is important to thermally anchor the thermocouple wire (for ~5 mm) before it leaves the

plate surface. Also the adjacent thermocouples are staggered to permit the wire

crossing the gap to be long (~10 cm) (see figure 8). This is done since the

heat flow through the wire is inversely proportional to its length. Next we shall consider the readout error of the gap thermopile. Figure 9 shows a graphical representation of the gap measurement situation. As long as it is a null measurement with the meter, gap and guard temperatures at the same value,

there is no readout error in the thermopile signal up to point A.

At point A the 0.25 mm (10 mil) Chromel wire is soldered to the 0.25 mm (10 mil)

copper wire, which is very pure and, hence, should have a very small thermal

signal 00.1 pV) coming out through the guard. The order of magnitude of the

spurious thermal signal due to a temperature difference between the last

Chromel-Constantan junction and the Chromel-Copper junction can be estimated as follows. The two Cr-Cu junctions are located in the same location on the guard

side of the gap, so an overestimate of the temperature difference between their

locations would be 1 mK. The Cr-Cu sensitivity is ~20pV/K. Thus the

spurious signal would be about 0.01 pV, and this is negligible.

The next error source is the thermals generated in the pure copper wire between

points A and B in figure 9. If the wire is handled with care and low-thermal

solder is used, these thermals should be less than 0.1 pV, and is referred to as AVab. We use a value of 0.2 pV for AV^g. (If the Cu wires are connected at

A, the actual thermal signal can be measured within the accuracy of the readout

device.

)

The next error results from any thermal emf’s in the reversing switch B or between B and C. The latter error can be eliminated by reversing the signal at

B. Any residual thermal signal in the reversing switch can be measured by

shorting a copper wire across the switch. This quantity, AV gw^ tc^, is estimated at 0.2 pV. Finally, there is the readout error of the Digital Linear

Amplifier, AV^la* This is estimated to be about 0.1 pV. The sum of AV^la*

AV switch anc* A^AB is 0.5 pV and this is referred to as AV„. Remember that the

other contributions to error were estimated to be negligible.

The resulting heat flow uncertainty, AQg is calculated by AQg = Sg AV

g

, where

Sg is the gap sensitivity previously discussed. The term Sg is equal to the

slope of the curve of the change in heat flow, Qg, as the gap thermopile voltage is unbalanced by an amount equal to Vg. Sg = Qg/Vg. Strictly speaking,

this could be a function of Vg.

The quantity Sg has a value of 0.57 mW/pV, so AQg is estimated to be 0.3 mW.

The percent uncertainty due to the gap is calculated as the ratio, AQg/Qm. In

the two-sided configuration, Qm ~ 1W at 300 mm (12 in), for low-density insulation samples. The corresponding percentage uncertainty value is ~0.03 percent.

3.5.4 Gap Theory

It is possible to model the gap unbalance results with an equation derived by

Woodside [ 2 ]

.

12
Qg (q Q (13)

Here Qg is the heat flow between the meter and the guard sections of the hot

plate as a result of an unbalance in the corresponding temperatures of Tg.

The term q Q represents the heat flow per unit temperature that flows directly

across the gap, and the term "cX" represents the heat flow/unit temperature

in the specimen across the boundary between the part of the specimen next to

the meter area and the part next to the guard area. Here, X is the specimen

apparent thermal conductivity. Just as Sg is the heat flow sensitivity per unit

voltage, Sg is defined as the heat flow sensitivity per unit of temperature

difference.

S

g

V

g

sgstcT

g

S tc = the thermopile sensitivity

= number of stages x 62 yV/K

= 18 x 62 yV/K

=1116 yV/K

(14)

Note that SgS^c = q Q + cX.

The quantity c is a constant which depends on the apparatus dimensions and the

specimen thickness [2],

c «

where 2d = gap width

= (0.89 mm)

= (35 mils)

P = gap perimeter = 2irr

L = specimen thickness

Z = 2TTd/L

r = meter area radius

= 0.2033 ra = 8.002 in

72ir £n(4a)

,

a =

> Z - 1 (15)

For L = 25 mm (1 in), c = 0.021 m (2.93 in). For L = 150 mm (6 in), c = 0.030 m

(4.39 in). The measured value of

0.00057 x 1116 yV

yV K

(16)

» 0.64 W/K

For X = 0.045 W/m-K, cX = 0.0013 W/°K at L = 152 mm (6 in) and cX = 0.0009

at L = 25 mm (1 in).

Thus cX is expected to be smaller than q Q

. Also the change in cX, due to

changes in L or X is much smaller than q Q

.

Looking again at the equation for the gap unbalance.

13
ij# ** SgStc - qo + cX (17)

g

Thus, Sg « q Q / s tc . Since q Q and S tc are constants, independent of the sample,

Sg is expected to be approximately constant for different sample material and

thickness. This conclusion is confirmed by experimental results. The gap

sensitivity, Sg, did not change by more than 5 percent between sample

thicknesses of 25 mm (1 in) to 150 mm (6 in).

Another way to look at this result is that most of the heat flow across the

gap (qQ ) flows directly across the gap, rather than through the sample (cX).

The measured gap sensitivity is slightly larger than what would be expected,

assuming that the heat flows entirely via the conduction mode only through the

air in the gap. This is, of course, what would be expected, since there are support pins and temperature sensors in the gap.

Thus the order of magnitude values predicted by Woodside's [1] equations are roughly confirmed by experiment. It is important to note that the aforementioned method to determine the uncertainty of gap heat flow is strictly

empirical. It is recommended that, for uncertainty analysis, this empirical

method be generally used in place of a theoretical procedure.

3.5.5 Qedge

Looking at the heat flow from the hot-plate, meter area, the difference between

the ideal case with one-dimensional heat flow and the real case is due to the

edge heat flow, Qe(jge

» Referring to figure 1, the ambient temperature, T^^,

is different from that of either the hot or cold plate. Usually, Tam^ is equal

to the average of the hot and cold plate temperatures, TmPan . At first glance,

one would expect the net Q e(jge to be zero when T am^is equal to Tmean . This

would be true if the heat flow were measured midway between the hot and cold

plates. In fact, the heat flow is monitored at the surface of the hot plate.

Some of the heat from the hot surface is going to the ambient, so the measured

heat flow is greater than what would be the case for one-dimensional heat

flow. Therefore, the ambient must be at a temperature higher than Tmean , for

Qedge to zero.

This leads to the essential difficulty in estimating the edge effect, which

is the issue of determining the value of T^^ at which Q e(jge is zero. Figure

10 shows the plot of a theoretically calculated plot [3] or the percentage

change in X as the ambient temperature varies. Since X is proportional to Q,

the edge effect can be characterized by X as well as Q. A value of zero for

the ordinate corresponds to the case where X-edge (or Q e dge) equals zero. Note

that this occurs when T am^, is greater than Tmean . It is possible to calculate

the value of T am^

at which Q e(jge is zero, and one such calculation is given in

reference [3]. The next step is to consider how to check these calculations

with experiment.

It is a simple matter to measure the slope of the curve in figure 10. It is more difficult to determine the zero crossing of the ordinate—at which

(1) Qedge equals zero or, equivalently, at which (2) the Q monitored at the hot

14
plate is equal to what it would be if the heat flow were one-dimensional or,

equivalently, at which (3) one is measuring the "true" apparent thermal conductivity of the material, A Q

. The only way to determine this zero intercept is

to put in a sample with a previously known A-value. Two 150 mm (6-inch)

specimens were measured on the NBS GHP. The edge effect is expected to be

negligibly small at this 150 mm (6-inch) thickness, based on theory and experimental measurements of the slope of the curve in figure 10 (The value of this

slope was less than 0.06 percent/°K). Therefore, the A-value of each 150-mm

(6-inch) sample is known within the experimental uncertainty at this thickness.

The next step in the experiment was to stack the two 150 mm (6-inch) specimens

to make a 12-inch sample. The mean temperature was 23.9 C (75 F), and the

plate temperature difference (T^ot - T co ^^) was 27.8 C (50 F). At this

thickness, for the NBS GHP, the edge effect was expected to be significant

(~0.4 percent change in A-value for a 1°C change). A zero value of the

abscissa in figure 10 corresponds to the ambient temperature being equal to

the mean temperature. A theoretical calculation [3] indicated that the corresponding value of the ordinate [3] would be ~0.07 percent. Preliminary

experimental values were +0.5 percent, about the expected value. The slope

of the experimental curve was about 25 percent greater than that of the

calculated curve.

It should be possible to use an experimental curve similar to that in figure

10 to estimate the value of the abscissa (which is proportional to

f-^mean “ ^ambient^ at which the measured A is equal to A 0

. According to

theory, the ambient temperature should be less than the mean temperature to

measure the true A-value (A - A0 = 0). The uncertainty, as far as the

edge effect is concerned, with which A0 is known might be estimated as follows. The uncertainty in the measured value of A 0 is estimated simply

as the sum of the uncertainty of the measured value of the abscissa and that

of the ordinate. The uncertainty of the ordinate is based on the uncertainties

of the two measured A-values at 150 mm (6 inches). The part of these uncertainties that is systematic will cancel since the ordinate is a ratio. The

remaining part is the repeatability of the measured A-value at 150 mm (6

inches), when the sample is removed from and replaced in the apparatus. This

repeatability was measured to be within 0.2 percent. The uncertainty in the

abscissa is essentially equal to that of the measured ambient temperature,

since the mean temperature is known much more accurately. The ambient temperature was measured with a differential thermocouple in conjunction with a good absolute sensor. Since the ambient temperature was not constant around

the plates, the thermocouple stages were uniformly distributed to measure the

average ambient temperature. A comparison of an average based on 4 points

agreed within 0.3 K (0.6 F) with an average based on 24 points. The uncertainty in the ambient temperature was estimated at 0.3 K (0.6 F). Using the

value of the slope in figure 10, this corresponds to about a 0.2 percent uncertainty in the ordinate. The sum of the two parts was 0.4 percent. A slightly

larger value of 0.5 percent was used in table 2 as the estimate of the upperbound uncertainty of a measured A-value at 300 mm (12 inches), due to edge

effects. Further experimental work must be performed in this area.

15
3 . 6 TEMPERATURE MEASUREMENT

3.6.1 Overview

In order to calculate the apparent thermal conductivity, X, via equation (2), one must measure the plate temperatures. Platinum resistance thermometers

(PRT's) and thermocouples are used in this measurement. The PRT is used to

determine the absolute temperature at a particular plate location, and differential thermocouples are used to measure the relative temperatures between the

PRT and other locations. The discussion of the uncertainties of the average,

meter-area, plate-temperature values follows the discussion of the uncertainties

of the PRT and thermocouples values.

3.6.2 PRT Circuit Rationale

Platinum resistance thermometers are used to measure the absolute temperature

in the meter area of the hot and cold plates. These PRT's were 3 mm in diameter

and 6 mm in length. The circuit in figure 11 is used to measure the PRT resistance values. The standard resistor, Rst(j, is manufactured by the Leeds and

Northrup Company of North Wales, Pennsylvania, Catalog //4030-B, Serial #1875050.

It was calibrated in March, 1980 to have a value of 99.9992 SI at the average

oil bath temperature of 24 C. Assuming that the oil bath, which surrounds

Rgt(j, is maintained within + 2 K, the resistance is constant within + 15 ppm or + 0.001 percent.

The value of the PRT resistance, Rx , is Rx

= (Vx /Vstcj) Rst(j. Note that the DVM

voltage uncertainty due to the zero uncertainty is effectively eliminated by

taking the ratio. Also the DVM uncertainty due to linearity is effectively

eliminated by matching the values of R^ and Rgt(j. Thus, the readout percentage

uncertainty of Rx , ARX , is approximately equal to that of Rs td> which is

+ 0.001 percent. Since a change of 1 corresponds to a change of about 2 K

for these "100 S3" PRT's, the value of 0.001 percent corresponds to 1 milli-ohm

or 2 milli-Kelvin (mK)

.

3.6.3 PRT Calibration

The PRT's were calibrated by the National Bureau of Standards at six temperatures

between 0°C and 50°C. The calibration points were fitted to the equation

T = a + bR + cR2 (18)

where T is the temperature corresponding to the IPTS-68 temperature and R is the

PRT electrical resistance. The equation coefficients were evaluated by a leastsquares fit. The residual standard deviation multiplied by 3 corresponded to a range of 3 to 5 mK, which agreed with the quoted calibration uncertainty of

5 mK. Any systematic error in the NBS bath calibration set-up would be expected

to fall within the above range. The calibration uncertainty is then estimated

at A cai - 5 mK. Any calculational uncertainty using equation (18) is expected

to be negligible.

16
The model number of the PRT’s is S-1059-2. They were produced by the MINCO

Corporation. There is a quoted uncertainty value of 1 mK for 20 temperature

cycles between plus and minus 200°C. This must, of course, be checked by a calibration history. The agreement between calibration values of September

1980 and December 1981 was within 3 mK, which was within the above-mentioned

calibration uncertainty. Since the PRT's are cycled only between -14°C and

+51 °C and since the calibration history does not indicate any detectable

change, the contribution to uncertainty due to re-cycling will be considered

negligible.

3.6.4 PRT Self-Heating Uncertainty

The PRT’s have a current of 1 mA, which heats the medium around them (external

self-heating) and the sensor itself (internal self-heating). The external

self-heating is negligible. That is, the conductivity and the heat capacity

of the aluminum plate surrounding the PRT is large enough to carry away the

PRT-generated heat without raising the nearby plate temperature a measurable

amount. The order of magnitude of the internal self-heating can be estimated

by the "rule of thumb" formula—AT(in mK) = 0.4/1 (in mA) . Thus, for the

PRT, AT would be about 0.5 mK. However, the self-heating in the calibration

would be expected to be about the same as in the measurement, since the current

is 1 mA in each case. Thus, the external self-heating is already taken into

account in the temperature determination, and its contribution to the

temperature uncertainty is considered to be negligible.

3.6.5 Temperature Distribution Over Meter Area

The temperatures are measured at a point near the center of the meter area of

the hot plate and the two cold plates. This measurement is accomplished with

the PRT’s described in the previous section. This next section considers the

question of the difference between this temperature at the center point and

the average temperature over the whole meter area.

The theoretical temperature distribution, T(r, 9), for the hot-plate meter

area is given in reference [3]. The design and construction of the hot and cold

plates are also discussed. The salient features are that the cold plate has

imbedded within it milled channels of a double spiral shape. Ethylene-glycol

fluid is circulated through these channels to maintain the plates at a particular temperature. The double spiral shape of the tubing was chosen to have a more constant (than a single spiral) temperature (T) over the meter area. A

thermopile was used to check the temperature differences between various

points on the surface of the cold plate. This difference was measured to be

less than 10 mK, and this is the T-distribution uncertainty for the cold plates.

The hot plate is heated by a ribbon-shaped heater located at a distance from

the center of about 0.71 times the gap radius. This heater goes around the

center in a circular shape. Thus, the hot plate is relatively hotter at the

heater position than at either the center or gap position (see figure 12). The

location of the heater was chosen so that the gap temperature would be equal

to the average temperature over the whole meter area [3]. The circular geometry

was chosen to have azimuthal symmetry for the temperature distribution in the

17
meter area. This will not be exactly true due to heat sources or leaks which

do not preserve this symmetry—such as the incoming heater leads or the support

pins in the gaps. Thus, there is expected to be some angular distribution

(~5 mK for a worst case of a 1-inch insulation sample).

In order to check the calculations of the temperature distribution it is

desirable to have a measurement precision better than the expected T differences

(~5-30 mK) . Chromel-Constantan thermopiles with 16 stages were chosen. These

had a sensitivity of about 1 raK/pV. Since the voltage measurement accuracy is ~ lpV, there should have been about a 1 mK read-out resolution. There were two

such thermopiles —one with a 2-inch span and another with a 6-inch span. These were constructed so that the thermocouple beads would rest on the hot-plate

surface. It is important to use rigid samples to press the thermopile against

the plate surface. Also, since there is a significant temperature gradient

perpendicular to the plates (i.e., in the z direction), there are two components

to the thermopile voltage. One is due to the plate temperature, and the other

to the bead position. The former changes with the thermopile orientation on the plate, and the latter does not. This is probably due partly to a difference

in the average z-position of the beads on the two sides of the thermopile and

partly to a difference in the contact, and hence, contact thermal resistance

between the thermopile junctions and the plates. To eliminate this constant

bias (it varied between 10-80yV), thermopile readings were taken in the normal

orientation and then with the thermopile rotated 180°. This technique gave a repeatability of + 10 mK. Based on this data the estimate of the upper bound

for the difference between the temperature at these locations and that over

the cold-plate, meter-area is 15 mK. The PRT's in the cold plate are located

2 inches from the center of the plates.

Next, a check was made to ascertain that the temperature of the hot plate at

the gap is equal to the average over the meter area. Referring to figure 13,

the calculation in reference [3] indicated that the ratio of the temperature

difference between the center and the heater location, Ay, and the temperature

difference between the gap and the heater location. Ax, should be 1.6. The

measured value of this ratio was 1.6 + 0.3. Thus, the temperature at the gap

is at the point, between the highest and lowest points, where it should be

according to theory. A value of + 15 mK is ascribed to the uncertainty due to

the average meter-area temperature being approximated by the temperature of

the PRT at the gap. This is essentially the uncertainty of the thermopile

measurement technique described in the previous paragraph. The model may

very well predict the average temperature much better, but it is difficult to

confirm by experiment.

The order of magnitude of the maximum temperature difference in the hot plate,

Ay (between the center and heater positions), was calculated to be ~250 mK

for a value of heat flow per unit area, 0/A, of 90 W/m^ [3]

.

This Q/A corresponds to a low-density, 25 mm (1 inch) insulation specimen measured in the

two-sided configuration. The measured value in this case was ~60 mK. Thus,

the temperature in the meter area was about A times more uniform than expected.

It may be that the models do not give a good prediction for temperature

differences less than ~100 mk.

18
An additional point relates to the high gradient in the gap, which was discussed in the section on the gap. It is important to locate the PRT used

for the hot plate, meter-area temperature measurement on the meter-side of the

gap, before the temperature gradient makes the sharp increase. Remember that

the temperature differences measured above are for the worst case of Q/R = 90

W/m^. These differences decrease inversely with Q/A. The worst case value is

used as an upper-bound estimate of the uncertainty in the following summary.

Another possible source of error is due to the fact that the hot plate PRT is

located in the center of the gap, midway between the plate surfaces upon which

the samples rest. Refer to figure 14. If there is any thermal gradient in

the z-direction, then the temperature at the PRT location would be slightly

different from the average surface temperature over the meter-area surface.

In reference [3] a calculation is performed that indicates that this difference

is negligible (< .01 percent) for the sample conductivities typically measured

(< 0.4 W/m*K).

3.6.6 Summary of Temperature Uncertainties

Let us sum the various contributions, AT, to the uncertainty in the

measurement of the average T over the meter-area.

AT due to PRT resistance _2 mK

AT due to bath calibration _5 mK

AT due to temperature distribution not constant

hot plate JJ> mK

(cold plate) mK

TOTAL-hot plate 22_ mK

-(cold plate) 22 mK

3.7 PLATE EHITTANCE

The amount of heat flow that goes across the sample depends on the emittance

of the hot and cold plates. For the NBS GHP plates a value of normal plate

emittance at room temperature in the visible was measured to be 0.89 + .02.

Admittedly, this could be different from the value of the hemispherical emittance in the infrared (3p-30y) which is appropriate to the following equation. This equation calculates the apparent thermal conductivity, X, as a function of density, D, thickness, L, and plate hemispherical emittance, e, and it is derived in reference [4].

_3

X = a + bD + (19)

3/4 8 + 1

( 2 - 1)

L e

Here, 8 is the extinction coefficient, "a" is approximately equal to the

thermal conductivity of air, ”b" is due to conduction thru the solid fibers, T

is the mean temperature in the sample and a Is the Stef an-Boltzmann constant.

In SI units.

19
a ® .026 W/m«K

T = 297.0 K

b = 0.009 W»m2 /K»kg

3 = 328 m" 1

In English units,

a = .18 Btu*in/hr»f

t

2 »°F

T = 75°F

b = 0.004 Btu»in»f t/lb»hr»°F

8 = 100 ft

-1

The fitted values of "a" and "b" will vary with product. The purpose here is

to evaluate the change in A-value for a change in e. Hence, it is sufficient

that the values have the correct order of magnitude. For a sample with D « 9

kg/m^ (0.6 lb/ft 2 ), if L ® 25 mm (1 inch) the percentage change in A, when e changes from 0.89 to 0.87, is 0.25 percent. For L ® 150 mm (6 inch), the same change is 0.05 percent. For a sample with D » 130 kg/m^ (8 lb/ft^), the percentage change in A is 0.003 percent at L = 1 in. The point of this exercise

is to show that an uncertainty in A-value due to an uncertainty in e is significant for low-density material at L = 25 mm (1 inch). It is not significant

for L = 150 mm (6 in), and it is not at all significant for high density

material.

This point is sufficiently important and subtle to warrant a more detailed

discussion. In the case where there is a significant radiation heat transfer

through a sample, the measured R-value and apparent A-value depend on the

emittance of the plates. For a higher value of e, there is more heat con- ducted thru the sample. Thus, the measured R-value is dependent on an apparatus parameter.

This has a bearing on the matter of comparison among different apparatuses.

Ideally, one measures a quantity that does not depend on the apparatus. Then,

a similar measurement on an identical sample should give an identical result.

Using the previously described model, for the case of a low-density, 1-inch

sample, a difference in e from 0.9 to 0.8 results in a percentage difference

in A-value of 1.2 percent. Clearly, one could have significant errors in a round-robin, apparatus comparison, if the values of e were not considered.

One way to circumvent this problem is to standardize the values of e for the

apparatuses. This is encouraged in the ASTM test methods for the guarded-hotplate and heat-flow-meter apparatus, which require that the value of e be

equal to 0.9. The estimate of e must be based on an actual measurement. Care

must be taken to specify whether the normal or the hemispherical emittance is

20
measured, as these might differ by about 10 percent. It is recommended that

the hemispherical emittance (3-30p) be measured, since it is the quantity

appropriate to equation (19).

Even if the e-values of two apparatuses have been measured to be the same,

there is an uncertainty in their X-value comparison due to the uncertainty in

the measured e. In the case where the e-values for two apparatuses are different, one can use a model like the one presented earlier to estimate the

corresponding adjustments in X-value. A final solution is to use samples of

greater thickness or density, where the uncertainty of X due to that of e is

negligible.

21
4 . TOTAL UNCERTAINTY

In this section, the various uncertainties will be combined to give an overall

uncertainty. Table 1 contains the individual and total uncertainties for a range of test conditions, and this section will explain these values. Each

individual uncertainty is treated as an upper bound on the random plus systematic uncertainties. A value of 3s is used for the random part, where s is

estimate of the standard deviation. The upper bound estimates are simply

added to estimate the upper bound of the total uncertainty. The individual

percent uncertainties in X are the ratio of the uncertainty over the variable,

e.g. , AA/ A

.

_c o The uncertainty in the area determination, AA, is 1.3 x 10 J m , which

corresponds to a 0.01 percent uncertainty in X.

For compressible samples the thickness uncertainty, AL, is 25 pm (1.0 mil).

Thus the percent uncertainty is 0.1 percent for L = 25 mm (1 in) and ~0.01

percent for L = 300 mm (12 in).

The uncertainty in the power generated by the hot-plate, meter-area heater,

AQm , is 0.04 percent. Remember, it is primarily due to the measurement of the

voltage across the 0.1 Q shunt resistor, on the 100 mV range of the digital

voltmeter, and it does not change significantly over the range of power values.

The uncertainty due to the heat generated by resistive devices, AQ r , in the hot

plate meter area is 0.4 mW. This corresponds to 0.04 percent to 0.004 percent

for power values of 1 W to 10 W.

The uncertainty in the heat flow across the gap, AQg, is 0.3 mW. This

corresponds to 0.5 pV for the gap voltage uncertainty. The percent uncertainty

varies from 0.03 to 0.003 as the gap power varies from 1 W to 12 W. The uncertainty due to the edge heat flow, AQ ecjge , is negligible up to thicknesses

150 mm (6 in). At a thickness of 300 mm (12 in), AQ ecjge is estimated to

correspond to 0.5 percent.

The uncertainty of both the hot and cold plate temperature is 22 mK. The

difference between hot and cold plate temperatures is used to caculate the

thermal resistance, so the value of 22 mK for the hot plate is added to the

22 mK for the cold plate to arrive at a value of 44 mK for the temperature

difference uncertainty, A(T^ -T c

). A typical value for (T^ - T c ) is 27.8 K

(50 F), in which case the percent uncertainty is 0.16 percent.

For the purpose of comparison among different apparatuses, the uncertainty in

measured plate emittance of 2 percent corresponds to an uncertainty in thermal

resistance, AR, of 0.25 percent at L = 25 mm (1 in), 0.05 percent at L = 150 mm

(6 in) and 0.02 percent at L = 300 mm (12 in). (Remember that the uncertainty

would be negligible for high-density material.)

The values given in table 1 are for low-densitv samples with an apparent thermal

conductivity of 0.046 W/m»K (0.32 Btu-in/hr»ft^»F) at thicknesses of 25, 75,

150 and 300 mm (1, 3, 6 and 12 in) which correspond to meter power (Qm) values

22
of 12, 4, 2 and 1 W, for two-sided operation. For one-sided operation, Qm is

6,2, 1 and 0.5 W, respectively.

The values of total uncertainty are 0.31 percent for L = 25 mm (1 inch), 0.26

percent for L = 75 mm (3 inches), 0.27 percent for L = 150 mm (6 inches) and

0.79 percent for L = 300 mm (12 inches).

23
5. VERIFICATION OF UNCERTAINTY ESTIMATE

It is appropriate here to discuss the philosophical aspects of the accuracy

issue. The assumption is made that a sample has a particular "true" value for

its thermal resistance, or equivalently for its apparent thermal conductivity.

The question is, "How closely does a data point measured on an apparatus

approximate the "true" value?"

Before addressing this question, let us define two basic kinds of uncertainty

random and systematic. Suppose a large number of X-value data points are measured, and that these points are distributed about an average value in some Gaussian-like distribution. The uncertainty of the knowledge of the average

value is characterized by a standard deviation. Even if the average of this

distribution were known exactly, it would still not necessarily be equal to

the "true" value. This is due to systematic errors in the measured values.

Systematic errors are not known, and one can not reduce them by repeating many

tests. The only way to learn about them is to make an independent measurement

on an identical sample with another apparatus of better or comparable accuracy.

If there is agreement among a number of apparatuses within the calculated

uncertainties, then the uncertainty estimates, and the corresponding model,

can be considered to be verified, in so much as is possible.

We at NBS look forward to participating in comparisons between our apparatus

and other apparatuses, and this will certainly be done in a definitive manner

in the near future. A major difficulty is to ensure that the samples are identical. Important sample requirements are durability and uniformity. The

moisture content should also be the same.

It is, on the other hand, possible to determine the random uncertainty with

many measurements on a single apparatus. Repeated measurements on the same sample give an indication of the overall random uncertainty. These can be

done for a short-term (e.g., 1 week) or a long term (e.g., several years). The

long-term variation might reveal a long-term drift in a systematic error. The

short-term repeatability was measured with 100 mm (4 in) samples of low-density

insulation to be 0.1 percent. Again, this repeatability serves as an estimate

of the random part (or the precision) of the overall uncertainty. The estimates

of uncertainty in the previous sections have been upper bounds on the possible

errors. For the most part they have been systematic in nature although they

include random parts.

Generally speaking, the data on a single test has two parts. The transient

part at the beginning of the test shows a monotonically increasing or decreasing

curve. When there is no monotonic trend, the steady state condition has been

achieved. There is still a scatter of data points due mostly to the cycling of

the bath temperatures. The scatter band is about 1 mK for the hot plate temperature, 6 mK for the cold plate temperatures and 1 mW for the power. The scatter

in the calculated X-value is about 0.01 percent for a two-sided, 1-inch sample

and 0.03 percent for a two-sided, 6-inch sample. The mean value is, of course,

known even better. Clearly, the scatter in the data points, after the steady

state condition is attained, is negligible compared to the estimated systematic

errors in the X-value.

24
6 . REFERENCES

1. Woodside, W. , and Wilson, A. G., "Unbalance Errors in Guarded Hot Plate

Measurements," Symposium on Thermal Conductivity Measurements and Applications of Thermal Insulations, ASTM STP 217, ASTTA, Am. Soc. Testing Mats.,

1957, pp. 32-48.

2. Woodside, W. , "Deviations from One-Dimensional Heat Flow in Guarded Hot

Plate Measurements," Review of Scientific Instruments, RSINA, Vol. 28,

1957, pp. 1933-1937.

3. Hahn, M. H., Robinson, H. E., and Flynn, D. R. , "Robinson Line-Heat-Source

Guarded Hot Plate Apparatus," Heat Transmission Measurements in Thermal

Insulations, ASTM STP 544, American Society Testing and Materials, 1974,

pp. 167-192.

4. Brian G. Rennex, "Thermal Parameters as a Function of Thickness in

Low-Density Insulation," Journal of Thermal Insulation, Vol. 3, pp. 37-62,

1979.

25
Table 1. Percentage Estimate of Uncertainties in the Measured Apparent

Thermal Conductivity for the NBS Guarded Hot Plate

Quantitative

Thickness

Value 25 mm

(1 inch)

75 mm

(3 inches)

150 mm

(6 inches)

300 mm

(12 inches)

Percent Uncertainty

Area

(12 ym or 0.5 mil

in radius

0.01 0.01 0.01 0.01

Thickness

(25 ymor 1.0 mil)

0.1 0.03 0.02 0.01

Meter Power 0.04 0.04 0.04 0.04

Meter Resistive

Device (0.4 mW)

0.00 0.01 0.02 0.04

Gap Heat Flow

(0.3 mW, or 0.5 yV

in gap voltage)

0.00 0.01 0.02 0.03

Edge Heat Flow 0.00 0.00 0.00 0.50

Hot, cold-plate

Temperature

Difference (44 mk)

0.16 0.16 0.16 0.16

TOTAL 0.31 0.26 0.27 0.79

* These values are for compressible, low-density, glass-fiber insulation

measured in the two-sided mode with a plate temperature difference of

28 K. Uncertainty values of less than 0.01 percent are reported as zero.

26
SAMPLE

(ledge

3dniVd3dM3i 1N3I9WV

3dniVM3dW31 1N3I9WV

FIGURE

1

HEAT

FLOW

(Q)

INA

TWO-SIDED

GUARDED

HOT

PLATE
FIGURE

2

SIDE

VIEW

OF

GAP
BEARING,

O

FIGURE 3

PLATE

SUPPORT

AND

THICKNESS

CALIBRATION
FIGURE

A

THICKNESS

TRANSDUCER

FOR

L-MAP
V

heater O

UJ^ 9

- Q_

£ ^

FIGURE

5

CIRCUIT

FOR

METER-AREA

POWER

MEASUREMENT
FIGURE

6

HOT-PLATE,

METERAREA

POWER,

Qm

VERSUS

GAP

VOLTAGE
CM

C\l

CM

CO

CO

CO

CO

CO

oo

in

COS Ao

<^>

QCSI CD

eg

QCH

CO

CD

CO CM

(Oo) 1 I CMI COI

FIGURE 7

CALCULATED

ANGULAR

TEMPERATURE

DISTRIBUTION

AT

GAP
TOP

VIEW

FIGURE 8

DIAGRAM

OF

GAP

THERMOPILE

JUNCTIONS
o

o

FIGURE

9

VOLTAGE

CIRCUIT

AND

WIRE

CONFIGURATION

FOR

THE

GAP

THERMOPILE
EXPERIMENTAL

FIGURE

10

THE

CHANGE

IN

APPARENT

THERMAL

CONDUCTIVITY,

X,

ASA

FUNCTION

OF

AMBIENT

TEMPERATURE*
Vs,

Rs o

00

CD

AAAr

LU

O

O 00

00

CD

FIGURE

11

CIRCUIT

FOR

THE

PLATINUM-RESISTANCE

THERMOMETER

(PRT)

RESISTANCE

MEASUREMENT
FIGURE

12

THEORETICAL

PLOT

OF

THE

HOT-PLATE

METER-AREA

RADIAL

TEMPERATURE

DISTRIBUTION
r gap

'heater

TOP VIEW OF HOT PLATE METER AREA

FIGURE 13 MEASURED HOT-PLATE, METER-AREA

RADIAL TEMPERATURE DISTRIBUTION
TEMPERATURE

AT

SURFACE

OF

HOT

PLATE

METER

AREA

FIGURE

14

FIGURE

SHOWING

TEMPERATURE

UNCERTAINTY

DUE

TO

HOT

PLATE

PRT

LOCATION
NBb-l 1 4A IH6V. 2-80

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1. PUBLICATION OR

REPORT NO.

NBSIR 83-2674

2. Performing Organ. Report No 3. Publ ication Date April 1983

4. TITLE AND SUBTITLE

ERROR ANALYSIS FOR THE NATIONAL BUREAU OF STANDARDS 1016 mm GUARDED HOT PLATE

5. AUTHOR(S)

Brian Rennex

6. PERFORMING ORGANIZATION (If joint or other than NBS. see instructions) 7 . Contract/Grant No. national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9.

SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street. City. State. ZIP)

Department of Energy

National Bureau of Standards

10.

SUPPLEMENTARY NOTES

I | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here)

An error analysis is given for the 1-meter Guarded Hot Plate at the

National Bureau of Standards. This apparatus is used to measure the

thermal resistance of insulation materials. The individual contributions

to uncertainty in thermal resistance are discussed in detail. The total

uncertainty is estimated to be less than 0.5 percent at sample

thicknesses up to 150 mm (6 inches) and less than 1 percent at a

12. thickness of 300 mm (12 inches).

KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) Apparent thermal conductivity; error analysis; guarded hot plate;

thermal insulation; thermal resistance.

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

lCX| Unlimited

| | For Official Distribution. Do Not Release to NTIS 47

[~~] Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402. 15. Price

$X\ Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $8 . 50

USCOMM-DC 6043-P80


;