Historical Review of the Metered Section Area for the Guarded-Hot-Plate Method

Abstract

Results of an extensive literature review and investigation of the metered section area for the guarded-hot-plate method, standardized as ASTM C177, Standard Test Method for Steady-State Heat Flux Measurements and Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparatus, are presented. The guarded-hot-plate apparatus is a primary linear-heat-flow method generally used to determine the thermal conductivity of insulating and building materials. The review examined technical publications from 1885 to 1990 and identified 31 papers of interest. Historical versions of ASTM C177 were also researched as well as test methods from other standard development organizations. The investigation revealed that, over the past 100 years, researchers have independently developed two main approaches for the computation of the metered section area. An assessment of the calculation techniques is presented for round plates with diameters from 250 to 1,000 mm, a guard-to-meter aspect ratio of 2, and guard gap widths of 1–4 mm. The gap effects are not negligible because large gaps (4 mm) on small plates (250 mm) can lead to errors of 10 % or more on the computation of the metered section area, ultimately affecting the uncertainty of the test results of the guarded-hot-plate method. The results of this study are applicable to other thermal conductivity test methods that employ a primary thermal guard to promote 1-D heat flow.

Introduction

The metered section area of the guarded-hot-plate (GHP) method is a fundamental, yet somewhat overlooked, input parameter for the determination of steady-state thermal transmission properties of insulating and building materials. This conduction shape factor represents a theoretical surface area normal to the steady-state heat flux applied across the thickness dimension of a test specimen. It is important that standardized methods apply consistent calculation techniques for the metered section area to provide congruent test results. Recently, it has been brought to the attention of the authors that the supporting technical references for equations defining this primary input quantity, as specified in standardized test methods for the GHP, are absent. In response, the National Institute of Standards and Technology (NIST) has conducted an extensive review of the literature on the GHP method focusing on technical descriptions of the metered section area. The ensuing citations are arranged chronologically.^1–31

This research follows a request from ASTM Subcommittee C16.30 on Thermal Measurement to document the origin and history of the metered section area equations and associated text in Annex A1 of ASTM C177-13, Standard Test Method for Steady-State Heat Flux Measurements and Thermal Transmission Properties by Means of the Guarded-Hot-Plate Apparatus.³² As part of this effort, a working group^* was formed in 2017 to investigate and provide findings with recommendations to the task group responsible for maintaining ASTM C177. The major goal of this investigation is to provide technical support as well as historical context for the establishment of the metering area equations in ASTM C177-13. The literature review surveyed technical papers on the GHP method from 1895 to 1990 and the reports of Committee C16 on Thermal Insulation from 1941 to 1981. Historical documents from other standards development organizations were also reviewed but are not reported here. This article provides background on the GHP test method, summarizes historical definitions for the metered section area used in previous versions of ASTM C177, and examines the mathematical equations utilized to define the metered section area for round plates.

Background

The physical property known as thermal conductivity is an important metric for building material energy performance. Thermal conductivity data of insulating materials measured by means of the GHP method are not only important to engineers and architects but are also essential to thermal insulation manufacturers as part of their quality assurance programs. The impacts on these user communities underscore the need for reliable thermal conductivity data obtained from reproducible test results, preferably from a standard method.

TEST METHOD

Test Method ASTM C177 is internationally recognized as an authoritative measurement technique for accurate determination of steady-state thermal transmission properties of insulating and building materials. Figure 1 illustrates the general features of the main components of the GHP apparatus for operation at or near ambient temperatures. The general arrangement utilizes parallel flat plates as constant temperature heat sources and sinks in contact with the surfaces of homogeneous specimens to establish a steady-state heat flux across the thickness dimension of the specimen. The central heating element of the apparatus is encompassed by a primary guard designed to promote 1-D heat flow (Q) perpendicular to the plate surface in the central portion of the adjoining test specimens. For applications near room temperature, the physical separation between the two components typically consists of an airspace and is commonly designated as the guard gap, or gap for short.

FIG. 1.

GHP apparatus (exploded view). MP: meter plate (also known as the GHP), GP: guard plate (primary guard), CP: cold plate, Q: 1-D heat flow through the metered section area, S: specimen; subscripts₁ and₂ designate upper and lower, respectively. Inset shows a rectangular gap profile between the MP and GP.

In principle, ASTM C177 includes operation of the GHP apparatus in either the double-sided or single-sided modes of measurement. The former requires that the cold plates (CP₁ and CP₂) operate at the same temperature; thus, the heat flow Q represents the measurement for the pair of specimens (fig. 1). The latter is covered explicitly in ASTM C1044, Standard Practice for Using a Guarded-Hot-Plate Apparatus or Thin-Heater Apparatus in the Single-Sided Mode.³³ By implementing independent temperature control of CP₁ and CP₂, 1-D heat flow through either specimen can be obtained; thus, individual testing of either specimen is possible. The GHP can be operated with either vertical or horizontal heat flow through specimens S₁ and S₂—that is, with horizontal plates, as shown in figure 1, or with vertical plates.

Test Method ASTM C177 permits a wide range of apparatus designs to fulfill specific measurement requirements. Historically, plate designs with either square or circular geometries have been utilized. Although only round plates are considered in this analysis, the ensuing results are applicable for square plate geometry. Figure 2 shows a schematic of a hot plate with circular geometry. The lateral width of the guard gap between the meter plate (MP) and guard plate (GP) is purposely exaggerated for visual effect. For determination of the metered section area, A, the outer radius of the MP and inner radius of the GP are taken as r_i and r_o, respectively. The width, w, of the guard gap is defined as the difference between the radii (w = r_o – r_i).

FIG. 2.

Hot plate, round construction (top view). MP (GHP), GP (primary guard), r_i: radius of MP, r_o: inner radius of GP, w: gap width.

GOVERNING EQUATIONS

The thermal transmission calculations based on heat flux measurements from the GHP apparatus are implemented in accordance with ASTM C1045, Standard Practice for Calculating Thermal Transmission Properties Under Steady-State Conditions,³⁴ as shown in equation (1). The governing equation for the GHP method employs an algebraic form of the Fourier heat conduction equation defined in ASTM C1045:

$${\lambda }_{\text{exp}}=\frac{Q{L}_{\text{avg}}}{2A\Delta T_{\text{avg}}}$$ (1)

where:

Q = time rate of 1-D heat flow through the metering area^† of the apparatus, W,

L_avg = mean specimen thickness in the heat flow direction (L₁ + L₂)/2, m,

ΔT_avg = mean temperature difference (ΔT₁ +ΔT₂)/2, K, and

A = specimen area normal to heat flux direction, m²; the 2A term represents heat flow through two surfaces of the metered area (fig. 1).

The metered section area, A, as currently defined by ASTM C177-13, is calculated from either equation (2) or (3) described in Table 1. Equation (2) covers the typical case of a continuous specimen across the guard gap; that is, the radial dimension of the specimen extends to the edge of the GP. In contrast, equation (3) covers the (special) case where the specimen’s lateral dimension extends only to the edge of the MP and a different but similar material is placed over the guard region.

TABLE 1.

Historical review of metered section area in ASTM C177

Version	Description of Metered Section Area, A
ASTM C177–42T,aASTM C177–45	A = actual area normal to the path of heat flow (flat surface)
ASTM C177–63 through ASTM C177–71	Metal-Surfaced Hot Plate	Refractory Hot Plate
	4(c)…“The dimensions of the test area shall be established by measurements to the centers of the separations that surround this area.”	5(c)…“The effective metering area of the refractory plate is determined by the positions of the potential taps used to evaluate the power input to the metering area winding.”
ASTM C177–76	Metal-Surfaced Hot Plate	High-Temperature GHP
	5.3…“The dimensions of the test area shall be established by measurements to the centers of the separations that surround this area unless calculations or tests are used to define the area more precisely.”	6.3…“The effective metering area of the refractory plate shall include one-half the area of this gap unless calculations or tests are used to define the areas more precisely. In designs that do not have definite separation, the effective metering area shall be determined either by the positions of the potential taps used to evaluate this power input to the metering area section of the plate or by analytical means.”
ASTM C177–85 through ASTM C177–13 (9.3)	“8.3 Metered Area - The metered area, A, is obtained from the area, Am, of the metered region plate and the gap area, Ag. If there is no discontinuity in specimen characteristics in the gap region, it is calculated as follows:
	$A=A_{\text{m}}+A_{\text{g}}/2$(2)
	If there is a discontinuity between the specimen in the metered region and the guard region, this equation is modified slightly to include the effects of heat flux distortion in the gap region:
	$A=A_{\text{m}}+\frac{A_{\text{g}}{\lambda }_{\text{g}}}{2\lambda }$(3)
	where:
	λg = thermal conductivity of the material in the guard region; and,
	λ = thermal conductivity of the specimen in the metered region.”

Note:

^aAccepted by the Society⁴⁰ as a tentative method and adopted as standard by letter ballot vote⁴¹ in 1945.

Literature Review

The literature review examined documents from three main areas of interest: historical versions of ASTM C177, associated reports of Committee C16 and technical papers from 1885 to 1990. Similar documentary standards from other standards development organizations, such as BS 874:1973, Methods for Determining Thermal Insulating Properties with Definitions of Thermal Insulating Terms,³⁵ were also reviewed but are not reported here.

ASTM C177

Table 1 summarizes the historical chronology of ASTM C177 definitions for the metered section area from its adoption in 1942 as a tentative method to the 2013 revision. The record delineates four major revisions involving the metered section area and includes pertinent text describing each modification. The original 1942 version did not provide a detailed definition for the metered section area. The 1963, 1971, and 1976 versions described two types of GHP apparatus: metal-surfaced hot plates intended for room or low temperatures and refractory hot plate designs for high-temperature applications. Remarkably, these versions allowed refractory plate designs that did not have a definite separation; hence, the effective metering area was defined in terms of the potential gap locations. The 1976 version introduced the (present-day) concept that the effective metering area shall include half the gap area, albeit only for the high-temperature GHP (Table 1). In 1985, separate treatments for the room- and high-temperature GHP were discontinued, and the metered section area was formally defined by equations (2) and (3) (Table 1).

C16 COMMITTEE REPORTS

The activities of Committee C16, which was formed in 1938, were published annually from 1941 to 1981 in ASTM Committee Reports. The cessation of the printed reports in 1981 precludes finding any information on changes for the 1985 version of ASTM C177. Nonetheless, the reports that are accessible do not address the changes for the metered section area definitions given in Table 1 and instead highlight other items of interest. For instance, in 1968, the task group began revision³⁶ of ASTM C177 to 1) extend the upper and lower temperature range covered by the standard and 2) reference SI units, provided that the necessary conversion tables were included. After approval of the 1971 version, the task group began a comprehensive revision in 1974 to restrict “the use of the conductivity concept” and emphasize the usage of conductance/resistance values,³⁷ as advocated in the C16.30 Subcommittee position paper.¹⁴ These proposed changes were approved for the 1976 version.

Parenthetically, while assembling the chronology of ASTM C177, an unexpected discovery was made of a standard test code³⁸ published in 1928 that is generally not known by Committee C16 members. The early standard test code was prepared 14 years prior to the original publication of ASTM C177-42T by subcommittees under the Committee on Heat Transmission of the National Research Council, which was located at that time in New York (now Washington, DC). The code included technical designs, procedures, and descriptions of several GHP apparatus as well as the guarded hot box and heat flow meter for testing wall systems. Complementing the test code was a report³⁹ by another subcommittee on definitions, nomenclature, symbols, and units for heat transmission. Unfortunately, neither of these precursor standards defined the metered section area in any more detail than the initial definition given in the 1942 version of ASTM C177 (Table 1).

TECHNICAL PUBLICATIONS

An extensive literature review focusing on the metered section area was undertaken for the years 1885 to 1990. The search converged specifically on the 1970s and 1980s, when major changes regarding the metered section area occurred in ASTM C177-85 (Table 1). Starting with an initial pool of 195 documents, approximately 90 articles and papers were reviewed, resulting in 31 papers^1–31 that either mathematically defined the metered section area or provided pertinent information on the meter section area.

Table 2 arranges the 31 references as either experimental, theoretical, or informative. The division between experimental and theoretical papers is approximately equal: 15 experimental versus 13 theoretical papers (9 analytical, 4 numerical, including 3 finite element). Three classic papers providing valuable information on the metered section area are placed in the last column of the table. Jakob¹ reviews several thermal conductivity test methods utilized in Germany in the early 1900s. The ASTM position paper by Subcommittee C16.30¹⁴ discusses philosophical aspects of measuring thermal insulating materials using standard test methods during the 1970s, and De Ponte²⁶ summarizes the research status of the GHP test method in the 1980s.

TABLE 2.

References on metered section area for the GHP method

Experimental	Theoretical		Informative
Heilman,2 Johnston,3 Gilbo,6 Duncan, Bunn, and Henson Jr.,11,12 Pilsworth, Hoge, and Robinson,13 Jackson,17 Moses,18 Bomberg and Solvason,20 Siu and Bulik,21 Smith, Hust, and Van Poolen,22 Rennex,24,27 Hust et al.30,31	Analytical	Numerical	Jakob,1
	Somers and Cyphers,4 Woodside,7,8 Donaldson,9 Pratt,10 De Ponte and Di Filippo,15 Hahn, Robinson, and Flynn,16 Bode,25 Peavy and Rennex29	Dusinberre5	ASTM Subcommittee C16.30,14 De Ponte26
		Finite Element
		De Ponte, Mariotti, and Strada,19 Troussart23,28

Assessment of Calculation Techniques

Inspection of the calculation techniques used in each article listed in Table 2 revealed that when the metered section area, A, was explicitly defined, the investigator used one of four approaches: a) arithmetic mean, b) quadratic mean, c) sum of the MP area plus one-half of the gap area, or d) plate area alone (i.e., without inclusion of the guard gap). It was observed that experimentalists pragmatically defined the conductance shape factor for their specific measurement process using one of these approaches. Generally, theorists defined the boundary of the metered section area using the center dimension of the gap (arithmetic mean approach), particularly for studies on imbalanced heat flow across the guard gap. The finite element analyses by Troussart,^23,28 however, initially utilized approach c) and later b). These two approaches, as shown in the following, yield the same final mathematical equation. Mathematical equations for a metered section area with circular geometry are derived using the radial dimensions given in figure 2.

1. Arithmetic mean: the metered section area was computed using the centerline dimension of the guard gap, that is, the average ($\overline{r}$) of the outer dimension of the MP and the inner dimension of the GP:

   $$A_{\text{a}}=\pi {\left(\frac{r_{\text{i}}+r_{\text{o}}}{2}\right)}^2=\pi {\overline{r}}^2$$ (4)
2. Quadratic mean: the metered section area was computed using a root mean square (r_RMS) approach, that is, the square root of the sum of the outer dimension of the MP squared and the inner dimension of the GP squared divided by 2:

   $$A_{\text{b}}=\pi {\left(\sqrt{\frac{r_{\text{i}}^2+r_{\text{o}}^2}{2}}\right)}^2=\pi \left(\frac{r_{\text{i}}^2+r_{\text{o}}^2}{2}\right)=\pi r_{\text{RMS}}^2$$ (5)
3. Half of the gap area (mid-gap area): following the approach used in equation (3) in Table 1, the metered section area was computed by adding one-half the lateral surface area of the guard gap (circular annulus) to the surface area of the MP:

   $$A_c=\pi r_i^2+\frac{\pi \left(r_o^2-r_i^2\right)}{2}=\pi \left(\frac{r_i^2+r_o^2}{2}\right)=\pi r_{RMS}^2=A_b$$ (6)
4. MP area only: this approach does include any portion of the guard gap area and, therefore, is not included in further analyses:

   $$A_{\text{d}}=\pi r_{\text{i}}^2$$ (7)

The term r_RMS in equations (5) and (6) represents the radius that bisects the guard gap annulus into two equal areas. Because the quadratic mean is always equal to or greater than the arithmetic mean ($r_{\text{RMS}}\ge \overline{r}$), the resultant computation for the metered section area will also be greater (A_b = A_c ≥ A_a >A_d). However, for the narrow-width guard gaps used in the modern GHP apparatus, the results from equations (4) and (6) are practically equivalent (A_b = A_c ≈ A_a), as will be demonstrated later.

Table 3 categorizes the calculation technique utilized by each investigation. Eleven studies computed the metered section area using the arithmetic mean. Of these, a slight majority (6) are theoretical (Table 2). Two of the edge loss studies^4,29 were inferred to use the arithmetic mean. In total, ten papers cited the quadratic mean and bisected gap area (equations (5) and (6), respectively), which, as shown previously, are the same calculation technique. The occurrence of redundant terms for the same mathematical quantity probably bolstered the need for standardization of the terminology used in ASTM C177C177-85 (Table 1). No recorded reference was found for equation (3) in Table 1, i.e., the case of a discontinuous specimen across the guard gap. The investigation by Heilman² (Table 3), which computed the meter area using equation (7), used a refractory hot plate (i.e., no gap present). The papers by Jakob,¹ Troussart,^23,28 De Ponte, Mariotti, and Strada,¹⁹ and De Ponte²⁶ are discussed further in the following sections.

TABLE 3.

Calculation approach used to define metered section area

a) Arithmetic Mean	b) Quadratic Mean	c) Include Half the Gap Area	d) Central Area Only	e) Offset from Arithmetic Mean	f) Unspecified or Inapplicable
Somers and Cyphers,4 Woodside,8 Donaldson,9 Pilsworth, Hoge, and Robinson,13 De Ponte and Di Filippo,15 Jackson,17 Moses,18 Bomberg and Solvason,20 Siu and Bulik,21 Bode,25 Peavy and Rennex29	Rennex,24,27 Troussart28	Jakob,1 Johnston,3 Duncan, Bunn, and Henson Jr.,12 Smith, Hust, and Van Poolen,22 Troussart,23 Hust et al.30,31	Heilman2	De Ponte, Mariotti, and Strada,19 De Ponte26	Dusinberre,5 Gilbo,6 Woodside,7 Pratt,10 Duncan, Bunn, and Henson Jr.,11 ASTM Subcommittee C16.30,14 Hahn, Robinson, and Flynn16

ENERGY BALANCE APPROACH

The finite element analysis by Troussart^23,28 assumes, as does the classic article by Jakob,¹ that one-half the heat flow into the guard gap is supplied by the central MP and the other half is supplied by the GP. Figure 3 models a quarter section of the GHP apparatus from figure1 with the gap width purposely exaggerated. Heat flow through the specimen above the central heater is designated Q_m, and heat flow in the region above the guard gap is Q_gap. Lateral heat flow (not shown) in the gap (r direction) is assumed balanced.

FIG. 3.

One-quarter section model of the circular GHP; MP, GP (primary guard), CP: cold plate, Q: 1-D heat flow in the MP and guard gap region; S: specimen; subscript 1 designates upper; r_i and r_o are the inner and outer radii, respectively, of the gap.

Under the aforementioned assumption, the heat flow through the test portion of the specimen (Q) is equal to the heat flow directly above the central heating plate (MP) plus one-half the heat flow above the guard region (dashed volume in fig. 3). Mathematically, we write the following:

$$Q=Q_{\text{m}}+\frac{Q_{\text{gap}}}{2}$$ (8)

The heat flows Q_m and Q_gap are given by equations (9) and (10), respectively:

$$Q_{\text{m}}=\lambda A\frac{\Delta T}{L}$$ (9)

$$Q_{\text{gap}}=\lambda A_{\text{gap}}\frac{\Delta T}{L}$$ (10)

Substituting into equation (8) and factoring yields the following:

$$Q=\lambda A_{\text{m}}\frac{\Delta T}{L}+\lambda \frac{A_{\text{gap}}}{2}\frac{\Delta T}{L}=\lambda \frac{\Delta T}{L}\left(A_{\text{m}}+\frac{A_{\text{gap}}}{2}\right)$$ (11)

The final term in parentheses in equation (11) is the same as equation (2). Obviously, equation (11) is only valid for the condition of balanced heat flow across the gap. In other words, heat flow from the central MP heater does not penetrate the primary GP and vice versa.

A similar approach for a discontinuous specimen in the guard gap where the insulating material has a thermal conductivity of λ_g corroborates equation (3) (Table 1). In this case, it is assumed that the specimen extends only to the edge of the MP (i.e., the discontinuity is assumed to exist at r_i) and a different insulating material with a similar thermal conductivity, λ_g, covers the GP, including the region above the guard gap.

BOUNDARY DEFINITION

In 1979, De Ponte, Mariotti, and Strada¹⁹ published (and later summarized in De Ponte²⁶) an early finite element analysis of a particular GHP apparatus with square plates. De Ponte investigated the correlation between balancing positions of thermocouples along the gap and the definition of the metering area. He noted that, because of the presence of a guard gap, heat was distributed to the specimen through an area slightly larger than the central heater plate. Current standards of that timeframe (i.e., ASTM C177C177-71 or ASTM C177C177-76) defined the metering area to the centerline of the gap (Table 1). His analysis redefined the metering area by a small distance of Δx from the gap centerline, hence the presence of column e) in Table 3. For values of Δx equal to zero, the metering area was the same as the conventional standards computation, and for negative values, the metering area was smaller. His analysis showed that the imbalance position was related to the definition of the metering area and was strongly influenced by the specimen thermal conductivity.¹⁹

Assessment of Guard Gap Width

The assessment of the metered section area is a multifaceted problem and, paraphrasing De Ponte,²⁶ “unfortunately, only one side of the whole problem has been considered.” The optimum dimensions of the guard gap are dependent on apparatus design, operation range, and specimen characteristics. A large dimension for the width will improve the thermal resistance between the primary guard and the MP, but a small dimension will minimize uncertainty in the definition of the metered section area.²⁶ This section reviews the history of the gap width in ASTM C177 and investigates the effect of the gap width on the metered section area calculations.

REVIEW OF ASTM C177 GUARD GAP WIDTH

Following the same format as Table 1, Table 4 summarizes the historical record for the guard gap design width in ASTM C177. The 1942 to 1976 versions prescribed an upper limit for the gap width dimension. For metal plates, the maximum width has ranged from 3 to 4 mm and, for high-temperature plates in the 1960s and 1970s, the maximum width was limited to 2 mm. The maximum ratio of gap area to metered area has ranged from 5 % (which is the current definition) to 8 % in the 1985 version. Early versions of ASTM C177 utilized nonmandatory notes to define the maximum area ratio. Although ASTM C177-63 and ASTM C177-71 (Table 1) permitted refractory plate designs without the presence of a guard gap, no lower limit for the gap width has been specified for metal plates in any version of ASTM C177C177.

TABLE 4.

Historical review of the guard gap width in ASTM C177

Version	Description of Guard Gap Width, w, or Guard Gap Area
ASTM C177–42T	“Heating units having metallic surfaces shall have a definite separation of air gap not greater than 1/8 in. [3.2 mm] between the measuring area of the central surface plate and the guard surface plates.”
ASTM C177–63	Metal-Surfaced Hot Plate	Refractory Hot Plate
	4(c)…“Heating units shall have a definite separation or gap not greater than 1/8 in. [3.2 mm] between the central surface plates and the guard surface plates.”	5(c)…“In a refractory plate having a gap between the central and guard areas, the gap shall not exceed 0.08 in. [2.0 mm] (see Note 5).”
	“Note 5 - It is recommended that the area of the gap in the plane of the surface plate be not more than 6 per cent of the metering section area on that side.”
ASTM C177–71 (SI units)	Metal-Surfaced Hot Plate	High-Temperature GHP
	4.3…“Heating units shall have a definite separation or gap not greater than 3 mm between the central surface plates and the guard surface plates (Note 5).”	5.3… “In a high-temperature plate having a gap between the central and guard areas, the gap shall not exceed 2 mm (Note 5).”
	“Note 5 - It is recommended that the area of the gap in the plane of the surface plate be not more than 6 percent of the metering section area on that side.”
ASTM C177–76	Metal-surfaced hot plate apparatus	High-temperature GHP apparatus
	5.3…“The heating unit shall have a definite separation or gap not greater than 4 mm between the surface plates of the metering area and the guard. In addition, the area of the gap in the plane of the surface plate shall be not more than 8 % of the metering section area.”	6.3…“…the gap at the surfaces shall not exceed 2 mm. In addition, the area of the surface shall not be more than 8 % of the metering section area.”
ASTM C177–85	“6.2.2 Gap - The gap between the concentric meter/guard heaters shall be uniform, and its width should be optimized to minimize gap heat flow while simultaneously minimizing temperature distortion effects within the specimen near the gap… Most past experience is based on gap areas that are about 5 % of the metered area.”
ASTM C177–13	“6.4 The Gap - The metered section and the primary guard shall be physically separated by a gap.. .The area of the gap in the plane of the surface plates shall not be more than 5 % of the metered section area.”

EFFECT OF GUARD GAP WIDTH

Table 5 summarizes the effect of guard gap width, w, on the metered section areas as calculated from equations (4) and (6) and on the ratio of gap-to-meter areas. The hypothetical plate sizes range from 250 to 1,000 mm in diameter. The ratio of the linear dimension of the primary guard relative to the MP (fig. 1), commonly known as the guard-to-meter aspect ratio, was selected arbitrarily to be 2, which, in turn, established the values for the inner radius, r_i. Values for w ranged from 1 to 4 mm. The upper limit was based on the maximum separation of 4 mm specified in ASTM C177C177-76 (Table 4).

TABLE 5.

Effect of guard gap width, w, on metered section area

Size, mm	ri, mm	ro, mm	$\overline{r}$, mm	w, mm	Agap, cm2	Aa, cm2	Ab = Ac, cm2	$\frac{A_c-A_a}{A_c},\%$	Agap/Ac, %
250	62.5	63.5	63.0	1	3.96	124.69	124.70	0.006	3.2
	62.5	64.5	63.5	2	7.98	126.68	126.71	0.025	6.3
	62.5	65.5	64.0	3	12.06	128.68	128.75	0.055	9.4
	62.5	66.5	64.5	4	16.21	130.70	130.82	0.096	12.4
500	125.0	126.0	125.5	1	7.89	494.81	494.82	0.002	1.6
	125.0	127.0	126.0	2	15.83	498.76	498.79	0.006	3.2
	125.0	128.0	126.5	3	23.84	502.73	502.80	0.014	4.7
	125.0	129.0	127.0	4	31.92	506.71	506.83	0.025	6.3
750	187.5	188.5	188.0	1	11.81	1,110.36	1,110.37	0.001	1.1
	187.5	189.5	188.5	2	23.69	1,116.28	1,116.31	0.003	2.1
	187.5	190.5	189.0	3	35.63	1,122.21	1,122.28	0.006	3.2
	187.5	191.5	189.5	4	47.63	1,128.15	1,128.28	0.011	4.2
1,000	250.0	251.0	250.5	1	15.74	1,971.36	1,971.37	0.000	0.8
	250.0	252.0	251.0	2	31.54	1,979.23	1,979.27	0.002	1.6
	250.0	253.0	251.5	3	47.41	1,987.13	1,987.20	0.004	2.4
	250.0	254.0	252.0	4	63.33	1,995.04	1,995.16	0.006	3.2

In general, the percent differences ([A_c – A_a]/A_c) between equation (4) and equation (6) in Table 5 are practically equivalent and increase with w. The differences are largest for small plates and large gap widths. For a plate 250 mm in diameter with a gap width of 4 mm, the difference is approximately 0.1 % (Table 5). Similarly, the relative ratios (A_gap/A_c) of the guard gap and metered section areas are largest for small plates and large gaps. For a 4-mm gap in a 250-mm diameter plate, the area ratio exceeds 12 %. In fact, several relative ratios for A_gap/A_c in Table 5 exceed the value of 5 % specified in ASTM C177C177-13 (Table 4). The results in Table 5 show that the implementation of small gaps decreases the uncertainty in the definition of the metered section area.²⁶

Summary

From the simple, but vague, ASTM C177-42T definition of metered section area as “the actual area normal to the path of heat flow” to more formal definitions in subsequent ASTM C177 versions, it is evident from the results of this study that our understanding and, consequently, the definition of the metered section area have evolved over time. Presently, there are two main approaches for the calculation of the theoretical conduction shape factor known as the metered section area. One approach defines the boundary of the metered section area as the centerline of the guard gap surrounding the central heater. The other approach defines the metered section area to include one-half the lateral surface area of the gap. Under the second scenario, the effective boundary of the metered section area is marginally larger. This latter approach is also corroborated by an energy balance of the guard gap that assumes equal heat input at the gap from the central and guard heater elements.

For a modern GHP apparatus with a small gap width, the calculated values using either the arithmetic mean or the mid-gap area are practically equivalent. Nonetheless, formal standards, such as ASTM C177, are established to normalize technical requirements and ensure reproducibility of test results. It is the intention of the authors that the results of this study will assist standards committees to obtain consensus for the calculation of the metered section area used for the GHP or for other test methods utilizing a thermal guard. Furthermore, it is important for users of ASTM C177 to realize that the two main approaches are very good first-order approximations. When necessary, however, more rigorous calculations, such as finite element analysis, other calculation techniques, or auxiliary experiments may be needed to define the metered section boundary more precisely.