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5 September 2017 Report #: ANM-112N-17-03

EQUIVALENT ROUND, EQUIVALENT DIAMETER & RULING SECTION

ABSTRACT

In heat treatment, it is important that any given object hardens throughout the section, to develop uniform properties in all locations. However, much of the data available in the literature regarding the effect of section size are based on testing of round bars. To use these data successfully, it is necessary to “convert” the dimensions of the object in question to a hypothetical equivalent round bar of uniform cross section. In the US, the diameter of the hypothetical equivalent bar is referred to as the “Equivalent Round”

(𝐸𝑅). In the UK, it is recognized as the “Equivalent Diameter” or the “Ruling Section.” US heat treatment

specifications and the MMPDS contain graphics (the US Graphics) that list 𝐸𝑅 values for various shapes.

There is a notion in the technical community that the US 𝐸𝑅 and the UK Equivalent Diameter / Ruling Section are one and the same.

This report is a review of the US 𝐸𝑅 and the UK Equivalent Diameter / Ruling Section information available in the literature. Illustrated usage examples of the UK data are provided. It is determined that the notion

that “the US 𝐸𝑅 and the UK Equivalent Diameter / Ruling Section are one and the same” is invalid. It is

shown that the 𝐸𝑅 values listed in the US Graphics were variedly assigned based on: (a) equivalent surface areas; (b) empirical formulae or some other arbitrary method, the validity of which cannot be proven or disproven, or; (c) special cases in the UK Equivalent Diameter / Ruling Section specifications, presented as general cases.

1.0 INTRODUCTION

In heat treatment, it is important that any given object hardens throughout the section, to develop uniform properties in all locations. However, much of the data available in the literature regarding the effect of section size are based on testing of round bars. To use these data successfully, it is necessary to “convert” the dimensions of the object in question to a hypothetical equivalent round bar of uniform cross section. In the US,

the diameter of the hypothetical equivalent bar is referred to as the “Equivalent Round” (𝑬𝑹). In the UK, it is recognized as the “Equivalent Diameter” or the “Ruling Section.”

US specifications Mil-H-6875, AMS-H-6875, Mil-H-2100, AMS-H-81200 and AS 1260, and the MMPDS a contain

identical graphics that list 𝑬𝑹 values for various simple shapes. In addition, AS 1260 deals with some complex shapes; none of the other documents does so. Mil-H-6875, AMS-H-6875, AS 1260, and the MMPDS pertain to steels, whereas Mil-H-81200 and AMS-H-81200 pertain to titanium alloys. Said graphics existed in the 1970s, but they were probably introduced earlier. They, hereinafter collectively referred to as the “US Graphics,” are reproduced in Illustration 1.

The terms “Equivalent Diameter” and “Ruling Section” originated in the United Kingdom (UK) in the 1930s, specifically for the heat treatment of steel. There are three (3) UK documents that are quoted regarding Equivalent Diameter: Special Report No. 14, (1) BS 971, (2) and BS 5046. (3) The undersigned is aware of only one US publication by the Climax Molybdenum Company (4) that refers to Equivalent Diameter. This publication makes no mention of the US ER or the US Graphics.

There is a notion in the technical community that the US 𝑬𝑹 and the UK Equivalent Diameter / Ruling Section are one and the same.

2.0 PURPOSE

The purpose of this report is threefold. First, to introduce the US 𝑬𝑹 and the UK Equivalent Diameter / Ruling Section information available in the literature. Second, to provide illustrated usage examples of the UK data.

Third, to critically examine the notion that the US 𝑬𝑹 and the UK Equivalent Diameter / Ruling Section are one and the same.

3.0 REPORT LAY-OUT

Section 4 introduces the US 𝑬𝑹 , and the UK Equivalent Diameter / Ruling Section information available in the

literature. Section 5.0 presents analyses of the 𝑬𝑹 values listed in the US Graphics, to determine a basis for these values. These analyses involve comparisons to the predictions of BS 971 and BS 5046, as well as to simple computations based on equivalent surface areas: that is, equivalency of the surface areas of the equivalent round bar and the shape in question, both being of the same length. These computations are presented in separate

a MMPDS: Metallic Materials Properties Development and Standardization. Formerly, Mil-HDBK-5.

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Appendices (A-D) that appear at the end of the main text, so as not to interfere with the flow of the main text. Section 6 presents a summary of the contents of this report.

Footnotes are indicated in the text as superscript letters, and the footnote content appears at the bottom of the same page. References are cited as superscript numerals between brackets, and they are listed at the end of the main text, just before the Appendices. b Illustrations are presented as close as possible to where they are first referenced in the text, not at the end of the text.

4.0 BACKGROUND

This section introduces the US 𝑬𝑹 , and the UK Equivalent Diameter / Ruling section information available in the literature.

4.1 Equivalent Round (𝑬𝑹) in the US:

As indicated earlier, Mil-H-6875, AMS-H-6875, Mil-H-81200, AMS-H-81200, AS 1260, and the MMPDS contain

graphics (the US Graphics) that list 𝑬𝑹 values for simple and complex shapes, Illustration 1.

b E-copies and hard copies of the references are available upon request.

Illustration 1: The US Graphics.

(a) Mil-H-6875 H. Steel.

(b) MMPDS 10. Steel.

(c) AS 1260. Steel.

(d) AMS-H-81200. Titanium Alloys.

(a)

(c) (d)

(b)

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Originally, Mil-H-6875 (steel) and Mil-H-81200 (titanium) covered heat treatment of both raw materials and parts made of the respective alloys. However, the superseding AMS specifications, AMS-H-6875 and AMS-H-81200,

only cover heat treatment of raw materials, and they both contain graphics and 𝑬𝑹 values that are identical to those shown in Illustration 1 (a)-(d). Strangely, these graphics are not included where they are needed the most; viz., for the heat treatment of parts, which is covered by AMS 2759 (steel) and AMS 2801 (titanium). c

It is important to note here that the documents that contain the US Graphics do not indicate how the listed values were derived for the simple shapes depicted in these graphics. This issue is examined in detail in section 5.0. It is also important to note that the data in these graphics neither take account of the properties of the material being cooled nor those of the cooling medium.

4.2 Equivalent Diameter / Ruling Section in the UK:

The term “Equivalent Diameter” originated in the UK in the 1930s, specifically for the heat treatment of steel. The Equivalent Diameter of any object (or part of an object) is the diameter of a hypothetical infinitely long bar of uniform circular cross section which, if subjected to the same cooling conditions as the object (i.e., the same initial and final temperatures and same cooling medium), would have a cooling rate at its axis (center) equivalent to that at the slowest-cooling position (center of thickest section) in the object in question. Thus, for an actual long bar of uniform diameter, the Equivalent Diameter is the same as the actual bar diameter. For other shapes, or short round bars, the Equivalent Diameter must be calculated.

As indicated in section 1, there are three (3) UK documents that are quoted regarding Equivalent Diameter: Special Report No. 14, (1) BS 971, (2) and BS 5046. (3)

BS 971 and BS 5046 introduced the term “Ruling Section,” which is used interchangeably with Equivalent Diameter. BS 5046 offers the following two definitions:

Ruling Section: The ruling section is the equivalent diameter of that portion of the object at the time of heat treatment that is important in relation to mechanical properties;

Limiting Ruling Section- The limiting ruling section for any given steel composition is the largest diameter in which certain specified mechanical properties can be expected after a specified heat-treatment.

Also, as indicated in section 1, the undersigned is aware of only one US publication by the Climax Molybdenum Company (4) that refers to Equivalent Diameter. This publication makes no mention of the US ER or the US Graphics of Illustration 1. BS 971, BS 5046 and the Climax publication indicate that the origin of the Equivalent Diameter concept is Special Report No. 14.

4.2.1 Special Report No. 14: (1)

The report is a generalized treatment of heating and cooling of steel objects, involving solutions of differential equations for heat transfer in terms of temperatures and time. The temperatures in the computations are: the

constant initial temperature of the object (𝑻𝟏), the constant temperature of the heating or cooling medium (𝑻𝟐), the

variable temperature at the surface of the object (𝑻𝑺), and the variable temperature at any given point in the object

(𝑻). The inputted data include the following physical properties of specific steel grades: thermal conductivity (k),

specific heat (s), density (), and thermal diffusivity (k / s). In what follows, a brief outline of the report contents and the computational sequence are presented.

General Theory

If heat is passing along a bar in one direction, then the rate at which heat passes over unit area 𝒅𝑸

𝒅𝒕 at temperature

T is directly proportional to the temperature gradient (𝒅𝑻

𝒅𝑺) at that surface, or:

𝒅𝑸

𝒅𝒕= 𝒌 .

𝒅𝑻

𝒅𝑺 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏

Where: is the thermal conductivity at temperature 𝑻 ; 𝒕 is the time, measured from the commencement of

heating or cooling, and; 𝑸 is the quantity of heat given out per unit area, in time 𝒕 .

From 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏 , the following fundamental partial differential equation is deduced for the flow of heat in three directions at right angles to one another:

c AMS 2759 and AMS 2801 existed when Mil-H-6875 and Mil-H-81200 were active.

Illustration 1, cont’d: The US Graphics.

(e) The Complex shapes in AS 1260.

(e)

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𝝏𝟐𝑻

𝝏𝒙𝟐+ 𝝏𝟐𝑻

𝝏𝒚𝟐+ 𝝏𝟐𝑻

𝝏𝒛𝟐= 𝒔𝝆

𝒌 .𝝏𝑻

𝝏𝒕 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐

where s, and , respectively, are the specific heat, density and thermal conductivity.

Considering a cylinder heated symmetrically about its axis and converting 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟐 from cartesian to

cylindrical coordinates gives the following equation, after omitting end (𝒛) effects:

𝝏𝟐 𝑻

𝝏𝒓𝟐+𝟏

𝒓 .𝝏𝑻

𝝏𝒓 = 𝒔𝝆

𝒌 𝝏𝑻

𝝏𝒕 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟑

where 𝒓 is the radius of the cylinder.

Case 1

The body to be heated or cooled is initially at a uniform temperature (𝑻𝟏), and is then placed in

a medium at the temperature that the body is to ultimately attain (𝑻𝟐) .

The assumed theoretical conditions are:

(1) The numerical values of the physical properties, k, , s, and the thermal diffusivity (a2 = k / s ) do not vary with

temperature.

(2) Newton's law of heating or cooling holds; i.e., the rate of absorption or emission of heat by the body is

proportional to the difference in temperature between the surface of the body (𝑻𝑺) and its surroundings (𝑻𝟐).

For cooling,

𝒅𝑸

𝒅𝒕= 𝑪 (𝑻𝒔 − 𝑻𝟐 ) 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟒

where C is a constant.

From 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏 and 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟒 :

𝒅𝑻

𝒅𝑺 =

𝑪

𝒌 (𝑻𝒔 − 𝑻𝟐)

Define (𝝏𝑻

𝝏𝒏 )𝒔 as being the temperature gradient at a point on the surface of the body, where 𝒏 is in the direction of

an outward drawn normal at that point; positive for heating and negative for cooling. Then,

− (𝝏𝑻

𝝏𝒏 )𝒔= =

𝑪

𝒌 . (𝑻𝒔 − 𝑻𝟐) 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟓

Define 𝑼 = 𝑻− 𝑻𝟐

𝑻𝟏−𝑻𝟐

Then,

(𝝏𝑼

𝝏𝒏 ) 𝒔 =

𝟏

𝑻𝟏 − 𝑻𝟐 . (

𝝏𝑻

𝝏𝒏 ) 𝒔 = −

𝑪

𝒌 .𝑻𝒔−𝑻𝟐

𝑻𝟏−𝑻𝟐= −𝒉𝑼𝒔 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟔 ,

i.e.,

(𝝏𝑻

𝝏𝒏 ) 𝒔 = −𝒉 𝑼𝒔 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟕

where 𝒉 (𝒔𝒖𝒓𝒇𝒂𝒄𝒆 𝒉𝒆𝒂𝒕𝒊𝒏𝒈 𝒓𝒂𝒕𝒆) = 𝑪

𝒌

Note

The second condition is often stated in terms of Equation 5 or Equation 7.

(3) For the particular shapes considered (cylinder, plate, and rectangular block), the following applies: for the cylinder, there will be no heat flow across the cylinder axis; for the plate and rectangular block, there will be no heat flow across the planes of symmetry parallel to the faces. d

(4) The body is initially at a uniform temperature; i.e., U = 1 for all values of r, at t = 0.

d These describe unidirectional heat flow.

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The solutions of the appropriate differentials for different shapes are as follows:

Case 1 (a). Infinite Plate Heated from Both Sides.

𝑷𝒍𝒂𝒕𝒆 𝑻𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔 = 𝟐𝑳

𝑫𝒊𝒔𝒕𝒂𝒏𝒄𝒆 𝒇𝒓𝒐𝒎 𝒕𝒉𝒆 𝒄𝒆𝒏𝒕𝒆𝒓 = 𝒙

Since heat flow is in one direction only, Equation 2 becomes:

𝝏𝟐𝑻

𝝏𝒙𝟐 =

𝒔𝝆

𝒌 .𝝏𝑻

𝝏𝒕 = 𝑲

𝝏𝑻

𝝏𝒕 =

𝟏

𝒂𝟐 𝝏𝑻

𝝏𝒕

and the solution which satisfies the above four conditions is:

𝑼 = 𝚺𝒏=𝟏𝒏=∞ 𝑩𝒏 𝒆

− 𝒄𝒏𝟐𝒂𝟐𝒕 𝒄𝒐𝒔 (𝒄𝒏𝒙) 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟖

where c is defined by the following equation:

𝒄 𝒕𝒂𝒏(𝒄𝑳) = 𝒉 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟗

and where

𝑩𝒏 =𝟐𝒉 𝒔𝒊𝒏 (𝒄𝒏𝑳)

𝒄𝒏 {𝒉𝑳 + 𝒔𝒊𝒏𝟐 (𝒄𝒏𝑳)}

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟎

Now we define the following constants:

𝑴 = 𝒄𝑳, 𝝉 = 𝒂𝟐

𝑳𝟐 . 𝒕,

𝑹 = 𝒙

𝑳, and

𝑯 = 𝒉𝑳

𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔 𝟖, 𝟗, 𝒂𝒏𝒅 𝟏𝟎, 𝒓𝒆𝒔𝒑𝒆𝒄𝒕𝒊𝒗𝒆𝒍𝒚, 𝒏𝒐𝒘 𝒃𝒆𝒄𝒐𝒎𝒆 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏𝒔 𝟏𝟏, 𝟏𝟐, 𝒂𝒏𝒅 𝟏𝟑:

𝑼 = 𝚺𝒏=𝟏𝒏=∞ 𝑩𝒏 𝒆

−𝝉𝑴𝒏𝟐 𝒄𝒐𝒔 (𝑴𝒏 𝑹) 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟏

𝑴 𝒕𝒂𝒏 𝑴 = 𝑯 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟐

𝑩𝒏 =𝟐𝑯 𝒔𝒊𝒏 𝑴𝒏

𝑴𝒏 (𝑯 + 𝒔𝒊𝒏𝟐𝑴𝒏) 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟑

At this point, Special Report No. 14 provides Tables that give values of (hence, the temperature) at points 𝑹 =

𝟎. 𝟓, 𝟎. 𝟔, 𝟎. 𝟕, 𝟎. 𝟖, 𝟎. 𝟗, 𝒂𝒏𝒅 𝟏, for an infinite plate, at selected 𝑯 values in the 0.2-∞ range, and for different values of

; an example is shown in Illustration 2. By suitably graphing and interpolating the data provided, Tables for other

values of 𝑯 may be obtained. Similarly, values of 𝑼 for intermediate values of or 𝒙

𝑳 may be obtained by

interpolation from graphs. The Report, however, does not show examples of the graphing or interpolation.

Special Report No. 14 presents solutions for other Case 1 shapes: (Case 1b) infinite cylindrical bar, (Case 1c) infinite rectangular rod, and (Case 1d) rectangular block, as well as solutions for Case 2 (body temperature at 0 C), and for Case 3 (constant quantity of heat in or out of the body per unit surface area per unit time). These are just solutions to differential equations, following a similar pattern to the one just described, with numerical Tables, similar to that in Illustration 2, listing the results.

The Report makes no mention of 𝑬𝑹 , Equivalent Diameter or Ruling Section, and the Tables presented therein (e.g., Illustration 2) are of no use to the practical metallurgist. Yet, the other references (2-4) indicate that that Report is the origin of Equivalent Diameter concept. The preceding summary of the contents of the Report is offered here for the sake of completeness.

4.2.2 BS 971-1944: (2)

The information pertinent to equivalent sections is contained in the section titled “Ruling Section, Equivalent Section and So-Called Mass Effect,” pp. 15-21. The following is based on the contents of that section.

In steel selection for a particular part, one of the most important considerations is whether the desired mechanical properties can be obtained after suitable heat treatment in the size and shape of that part. It is the rate of cooling that governs mechanical properties. Cooling rate is a function, not only of part size and shape, but also of the physical properties of the steel (e.g., thermal conductivity, specific heat, density, and the incidence of phase changes). One of the chief objectives of BS 971, however, is to draw attention to the influence of size and shape (Mass Effect) in determining the properties which may be obtained by heat treatment of steel. Accordingly, for the

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purposes of this work, these physical properties must necessarily be left out, so as to focus on the effect of size and shape upon the rate of cooling of steels of similar physical properties. e A mathematical approach has been adopted for finding the diameters of round bars, the centers of which would cool through a given range of temperature in the same time as the centers of rectangular or square bars of different sizes; the diameter of that equivalent round is referred to as the “Ruling Section.”

The methods employed are fully described in the first report of the Alloy Steels Research Committee, Iron and Steel Institute, Special Report No. 14, 1936, p. 149; (1) see 4.2.1. The mathematical calculations were made for cooling from 820 to 300 C (approx. 1510 to 570 F) in air or oil at 20 C (68 F). The results, however, are equally applicable to any other temperature range, where the temperature drop at the center has the same ratio to the original temperature (relative to the surrounding medium) as for the conditions assumed; i.e., where:

𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒂𝒕 𝒕𝒉𝒆 𝒄𝒆𝒏𝒕𝒆𝒓 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒊𝒆𝒄𝒆−𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒐𝒊𝒍 (𝑨𝒊𝒓)

𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒊𝒆𝒄𝒆−𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝑶𝒊𝒍 (𝑨𝒊𝒓)= 𝟎. 𝟑𝟓 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝟏𝟒

Note

Special Report No. 14, however, makes no mention of the ratio expressed in the above Equation.

e Special Report No. 14 (4.2.1), by contrast, considers the thermal conductivity (k), the specific heat (s), and the density () of the steel;

they just are considered not to vary with temperature.

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BS-971-1944 does not delve into the details of the calculations, except to point out, without elaboration, that the

heat transfer factor, “𝒉”, f for oil quenching is taken to be 0.8 cm-1, whereas for air cooling, it is taken to be 0.03 cm-1.

The results of the calculations are presented in the form of tables and graphs, for oil quenching and air cooling, Illustration 3. The tables provide direct conversion from plates / rectangular sections to equivalent rounds. The

graphs are provided for those who prefer to use a factor, 𝒇, where the diameter of the equivalent round = 𝒇 𝑻 (thickness of plate or side of square section). The following 2 examples show how the graphs and tables work for the case of oil quench (a) in Illustration 3:

Example 1: Square Section, 1 ½x 1 ½, blue entries. 𝑩

𝑻= 𝟏, 𝑻 = 𝟏 ½ . The graph indicates an 𝒇 value of

approximately 1.08. Thus, the diameter of the equivalent round = 1.08 x 1 ½ = 1.62. In the table, the diameter

of the equivalent round is read directly at the intersection of 𝑩

𝑻= 𝟏, 𝑻 = 1 ½, in this case 1.620.

Example 2: Rectangular section (plate) 3 x 1 ½, green entries. 𝑩 = 3, 𝑻 = 1 ½ and 𝑩

𝑻 = 2. The graph

indicates an 𝒇 value of approximately 1.42. Thus, the diameter of the equivalent round = 1.42 x 1 ½ = 2.13. In

the table, the diameter of the equivalent round is read directly at the intersection of 𝑩

𝑻= 𝟐 𝒂𝒏𝒅 𝑻 = 1 ½ , in this

case 2.124, in close agreement with the value (2.13) determined from the graph.

BS-971-1944 states that the tables and graphs must only be used for comparing sections of the same steel, not sections of different steels. The reason for this is, as indicated earlier, that the physical properties of the steel

(𝒌, 𝒔, 𝝆) were left out of the calculations. BS-971-1944 further states that sections other than rounds or rectangles are difficult to treat mathematically, but close approximations can usually be obtained. Octagonal and hexagonal parts are intermediate in cooling rate between round and square, the order of increasing time of cooling being:

round, octagonal, hexagonal, square. Thus, for a hexagonal section with a distance 𝑻 between flats, 𝒇 should be

between those of a round section (𝒇 = 𝟏) and a square section, 𝑩

𝑻= 𝟏 (𝒇 = 𝟏 𝒕𝒐 𝟏. 𝟎𝟖𝟓). For oval sections with

major and minor axes of "𝒂" and "𝒃", accurate results may be obtained by first converting the section to its

equivalent rectangular section having sides "𝑨" and "𝑩", such that the area of the rectangle equals the area of the

oval and 𝑨

𝑩=

𝒂

𝒃 . This equivalent rectangular section may then be converted to an equivalent round bar using the

graphs or tables. Of course, for a round bar, the diameter of the equivalent bar is the same as that of the bar itself.

4.2.3 BS 5046-1974: (3)

The basis for Equivalent Diameter calculations is explained in Appendix A of BS 5046-1974. The BS also offers the following definitions: Ruling Section- The ruling section is the equivalent diameter of that portion of the object at the time of heat treatment that is important in relation to mechanical properties; Limiting Ruling Section- The limiting ruling section for any given steel composition is the largest diameter in which certain specified mechanical properties can be expected after a specified heat-treatment.

The following is based on the contents of Appendix A.

Over the years, several criteria were used to calculate equivalent diameters. The equivalent diameter is defined as the diameter of the round bar, the center of which would cool through a given range of temperature in the same time as the slowest cooling point of the shape in question (rectangle, square, etc.). The criterion used in BS 5046-1974 is the one presented in Appendix E of the now obsolete 1955 edition of BS 970. g This criterion is based on the equality of times for the temperature to fall to a chosen fraction (0.35) of the overall fall in temperature:

𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒂𝒕 𝒔𝒍𝒐𝒘𝒆𝒔𝒕 𝒄𝒐𝒐𝒍𝒊𝒏𝒈 𝒑𝒐𝒊𝒏𝒕 𝒊𝒏 𝒕𝒉𝒆 𝒔𝒉𝒂𝒑𝒆 𝒂𝒏𝒅 𝒃𝒂𝒓 − 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒐𝒐𝒍𝒊𝒏𝒈 𝒎𝒆𝒅𝒊𝒖𝒎

𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒉𝒂𝒑𝒆 𝒂𝒏𝒅 𝒃𝒂𝒓 − 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒐𝒐𝒍𝒊𝒏𝒈 𝒎𝒆𝒅𝒊𝒖𝒎= 𝟎. 𝟑𝟓

Note

This criterion is the same as that used in BS 971-1944 (4.2.2), Equation 14.

f The heat transfer factor is the quantity of heat transferred per unit area of the surface per unit time per degree of temperature difference

between the body and the surrounding medium, divided by the thermal conductivity of the material. Here, the material is the cooling medium, air or oil (steel properties were left out). Note that BS 971-1944 does not explicitly refer to “h” as the heat transfer factor; this term is found in BS 5046 (4.2.3). Note also that this “h” is different from the one used in Special Report No. 14. g A copy of BS 970-1955, published in 1966, was purchased and reviewed. It is just a listing of some steel grades and their properties,

and it contains no Appendix E (only Appendices I-VI). It makes no mention of Equivalent diameter, Ruling Section or 𝐸𝑅, and it presents no calculations of the type discussed here.

8

Illu

str

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on

3: T

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s a

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s o

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S 9

71

-194

4. 𝑩 a

nd

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dic

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ickn

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on, 𝑩𝑻

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(a)

Oil

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. T

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(b)

Air

co

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.

(a)

𝒇=

1.0

8

𝒇=

1.4

2

2.1

24

1.6

20

𝑩 𝑻= ∞

( P

late

) 𝑩 𝑻=𝟐𝟏 𝟐

𝑩 𝑻=𝟐

𝑩 𝑻=𝟏𝟏 𝟐

𝑩 𝑻= 1

(Sq

uar

e S

ect

ion

)

(b)

𝑩 𝑻= ∞

( P

late

)

𝑩 𝑻=

4

𝑩 𝑻=𝟐

𝑩 𝑻=𝟑𝟏 𝟐

𝑩 𝑻= 1

(Sq

uar

e S

ect

ion

)

𝑩 𝑻=𝟑

𝑩 𝑻=𝟏𝟏 𝟐

𝑩 𝑻=𝟐𝟏 𝟐

9

The BS indicates that, for most hardening steels, 0.35 corresponds to a temperature around MS. It further indicates that the calculations were based on the methods described in Special Report No. 14, (1) discussed in 4.2.1. BS 5046-1974, however, does not delve into the details of the calculations, except to point out that the

heat transfer factor “𝒉” values were selected based on experience, being 0.003, 0.08, and, 0.2 mm-1

, for air, oil, and water, respectively. h

The results of the calculations are presented in Tables 1-12 and Figures 1-6 in the main text, together with illustrating examples. The BS states that: “for an actual long round bar, the equivalent diameter is the same as the actual diameter. For other shapes or shorter round bars, the equivalent diameter may be obtained from Tables 1-12 or Figures 1-6.” The Tables list the equivalent diameters associated with air cooling, oil quenching, and water quenching various shapes of different dimensions. The equivalent diameter is read directly from these Tables at the intersection of the column and row that correspond to certain dimensions of the shape in question.

The Figures are provided to facilitate interpolation, or where different 𝒉 values may be chosen; the BS, however, does not show how interpolation may be performed. The text of the BS does not explicitly indicate whether or not

the properties of the steels (𝒌, 𝒔, 𝝆) were factored in the calculations.

BS 5046-1974 covers 3 basic shapes: (a) cylinders and discs; (b) rectangular sections, and; (c) tubes and rings. In addition, Appendix B and Figures 7 and 8 cover Two-and three- cylinder problems (d). Appendix C covers Miscellaneous shapes (e). Other pertinent information is also presented in the BS (f).

(a) Cylinders and Discs:

For cylinders and discs, the relevant data are contained in Tables 1 (air cooling), 2 (oil quench), and 3 (water quench), and Figures 4 and 6. Illustration 4 shows usage examples of Table 2 for oil quenched cylinders and discs. Illustration 5 shows usage examples of Figure 4 (cylinders) and Figure 6 (discs). Figure 4 shows the

variation of 𝒇 vs. 𝒉𝑫 and 𝑫

𝑳 for cylinders (Equivalent Diameter = 𝒇 x cylinder diameter, 𝑫); 𝑳 is cylinder length.

Figure 6 shows the variation of 𝒇 vs. 𝒉 𝑻 and 𝑻

𝑫 for discs (Equivalent Diameter = 𝒇 𝒙 𝒅𝒊𝒔𝒄 𝒕𝒉𝒊𝒄𝒌𝒏𝒆𝒔𝒔 𝑻).

h The definition of “h” is: the quantity of heat transferred per unit area per unit time per degree (of difference between the body and the

medium) / the thermal conductivity of the cooling medium; dimensions, (L-1). Note that, h = 0 indicates an infinitely slow quench and h = ∞ an infinitely fast quench (temperature of shape falls instantaneously to the temperature of the cooling medium). The common cooling media arranged in order of increasing “h” are: air, oil, and water. Note that this is not the same “h” used in Special Report No. 14, but it is the same as that used in BS 971-1944.

Illustration 4: Usage examples, BS 5046-1974. Table 2, cylinders and discs, oil quench.

Cylinder 50 mm in diameter and 100 mm in length: Equivalent diameter = 48 mm.

Disc 80 mm in diameter and 40 mm in thickness: Equivalent diameter = 50 mm

10

(b) Rectangular Sections-Including Plates:

The relevant data are contained in Tables 4 (air cooling), 5 (oil quench), and 6 (water quench) and Figure 5 of the

BS. The Tables list the equivalent diameters of infinitely long rectangular sections in terms of the thickness 𝑻 and breadth 𝑩 of the section. The equivalent diameter is read directly from these tables at the intersection of the

𝑻 column and the 𝑩 row. Figure 5 shows the variation of 𝒇 vs. 𝒉𝒕 and 𝑻

𝑩 for rectangular shapes

(𝑬𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 = 𝒇 𝑻).

For sections of finite length, including plates, the two shortest dimensions are taken and converted to an equivalent diameter as though the length dimension were infinite, using Tables 4-6. This equivalent diameter and the original finite length are then used to obtain a new equivalent diameter from Tables 1-3.

Example: Consider a slab 100 mm (thickness) x 300 mm (breadth) x 500 mm (length) oil quenched.

Step 1: The 100 mm x 300 mm dimensions (thickness and breadth, respectively) are used to find the equivalent diameter of an infinite bar from Table 5 (oil quench) of the BS, in this case 150 mm, as shown in Illustration 6 (a).; Figure 5 of the BS, Illustration 7, indicates 148 mm, in reasonable agreement with Table 5 prediction.

Step 2: Table 2 (oil quench) is used to find a new equivalent diameter for a round bar 500 mm long with a 150-mm diameter, in this case 150 mm (by interpolation), as shown in Illustration 6 (b); the BS provides no graphics for Table 2.

(c) Tubes & Rings:

For tubes and rings, 𝒛, 𝒚 and 𝒙, respectively, denote the axial length, wall thickness, and bore diameter,

Illustration 8. The following distinction is made between tubes and rings:

For a tube,

𝒛

𝒚 ≥ 𝟏

whereas for a ring (which is always supposed to be of rectangular cross section), i

i The BS, however, does not explain why this is so.

Illustration 5: Graphs to calculate 𝒇 for:

(a) cylinders; (b) discs. BS 5046-1974.

(a)

In reasonable agreement with Table prediction(Illustration 4)

Oil Quench: 𝒉 = 𝟎. 𝟎𝟖 𝒎𝒎−𝟏

Cylinder: 𝑳 = 𝟏𝟎𝟎 𝒎𝒎, 𝑫 = 𝟓𝟎 𝒎𝒎

𝒉𝑫 = 𝟒,𝑫

𝑳= 0.5

𝑬𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 = 𝟒𝟖.𝟓 𝒎𝒎

𝒇 = 𝟎. 𝟗𝟕

𝑬𝒒𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 = 𝒇 𝑫

(b)

In agreement with Table prediction(Illustration 4)

Oil Quench: 𝒉 = 𝟎. 𝟎𝟖 𝒎𝒎−𝟏

DISC: 𝑫 = 𝟖𝟎 𝒎𝒎, 𝑻 = 𝟒𝟎 𝒎𝒎

𝒉𝑻 = 𝟑.𝟐, 𝑻

𝑫= 0.5

𝑬𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 = 𝟓𝟎 𝒎𝒎

𝒇 = 𝟏. 𝟐𝟓

𝑬𝒒𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 = 𝒇 𝑻

11

𝒛

𝒚 < 𝟏

The relevant data for tubes are contained in Tables 7 (air cooling), 8 (oil quench), and 9 (water quench) of the BS, while those for rings in Tables 10 (air cooling), 11 (oil quench), and 12 (water quench) of the BS. Because of the

large number of parameters involved in specifying tube and ring sizes, these Tables quote multiplying factors (𝒇):

in terms of 𝒛

𝒚 and

𝒙

𝒚 for tubes (𝑬𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 = 𝒇 𝒚), and; in terms of

𝒚

𝒛 and

𝒙

𝒛 for rings

(𝑬𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 = 𝒇 𝒛).

Rectangular Sections-Oil Quenched

(a) Step 1: From Table 5 of the BS, find equivalent diameter of an infinite bar 100 mm x 300 mm, in this case 150 mm.

(b) Step 2: From Table 2 of the BS, find new equivalent diameter of an 500 mm long round bar with a diameter of 150 mm,

in this case 150 mm (by interpolation).

Illustration 6: Finding the equivalent diameter of a rectangular bar: 100 mm (thickness) x 300 mm

(breadth) x 500 mm (length). BS 5046-1974.

(a) Step 1: From Table 5 of the BS.

(b) Step 2: From Table 2 of the BS.

12

Example 1: Consider a tube 900 mm long (𝒛), 700 mm OD, and 200 mm wall thickness (𝒚), oil quenched. Thus, the bore diameter

𝒙 = 𝟕𝟎𝟎 − (𝟐 × 𝟐𝟎𝟎) = 𝟑𝟎𝟎 𝒎𝒎,

then,

𝒙

𝒚 =

𝟑𝟎𝟎

𝟐𝟎𝟎= 𝟏. 𝟓 and

𝒛

𝒚 =

𝟗𝟎𝟎

𝟐𝟎𝟎= 𝟒. 𝟓

From Table 8 (oil quench) of the BS, the appropriate

𝒇 factor is 1.51 (by interpolation), Illustration 9 (a).

Thus, equivalent diameter = 1.51 x 200 = 302 mm.

Example 2: Consider a ring 150 mm axial thickness

(𝒛), 270 mm radial thickness (𝒚), and 400 mm bore

diameter (𝒙), oil quenched. Thus,

𝒙

𝒛=

𝟒𝟎𝟎

𝟏𝟓𝟎= 𝟐.𝟔𝟕 and

𝒚

𝒛=

𝟐𝟕𝟎

𝟏𝟓𝟎= 𝟏. 𝟖

From Table 11 (oil quench) of the BS, the appropriate f factor is 1.34 (by interpolation), Illustration 9 (b).

Thus, Equivalent Diameter = 1.34 x 150 = 201 mm.

(d) Shapes Consisting of Combinations of 2 or 3 Cylinders with Common Axis:

BS 5046-1974 presents Figures 1-3 to be used, in conjunction with Appendix B and Figures 7 and 8, to calculate equivalent diameters for some 2- and 3-cylinder combinations with common axis and finite lengths; examples of the calculations are also presented. This portion of the BS will not be considered any further here, since cylinder combinations of these types are not addressed in the US Graphics or in any other document reviewed herein.

Illustration 7: Graph to calculate 𝒇 for a rectangular bar with finite length. BS 5046-1974.

In reasonable agreement with Table prediction(Illustration 6 a)

Oil Quench: 𝒉 = 𝟎. 𝟎𝟖 𝒎𝒎−𝟏

Bar: 𝑩 = 𝟑𝟎𝟎 𝒎𝒎, 𝑻 = 𝟏𝟎𝟎 𝒎𝒎

𝑬𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝑫𝒊𝒂𝒎𝒆𝒕𝒆𝒓 = 𝟏𝟒𝟖 𝒙 𝟏𝟎𝟎 = 𝟏𝟒𝟖 𝒎𝒎

𝒇 = 𝟏. 𝟒𝟖

𝑻

𝑩= 𝟎. 𝟑𝟑

𝒉𝑻 = 𝟖

Illustration 8: Dimensions of tubes and rings according to BS

5046-1974.

13

(e) Miscellaneous Shapes:

Appendix C of BS 5046-1974 deals with shapes other than those discussed above. The Appendix consists of two sections, C.1 and C.2.

C.1 General: This section states that sections other than rounds or rectangles are difficult to treatmathematically, but close approximations may usually be obtained. Octagonal and hexagonal parts areintermediate in cooling rate between round and square, the order of increasing time of cooling being: round,

octagonal, hexagonal, and square. Oval sections with major axis "𝒂" and minor axis "𝒃" will cool more slowly

than a round bar of diameter "𝒃" but faster than a rectangle 𝒂 x 𝒃. More accurate results may be obtained by

first converting the section to its equivalent rectangular section having sides "𝑨" and "𝑩", such that the area of

the rectangle equals the area of the oval and 𝑨

𝑩=

𝒂

𝒃 . This information is similar to that presented in BS 971-

1944 (last paragraph in 4.2.2); note, however, that BS 971-1944 includes graphics that cover rectangular sections, including plate and square sections.

C.2 Three-cylinder objects where the middle cylinder has the smallest diameter: This configuration is not ofinterest here, since it is not addressed in the US Graphics or in any other document reviewed herein.

Illustration 9: Finding the appropriate 𝒇 factors for tubes and rings, oil quenched. BS 5046-1974.

(a) Tube: 800 mm length (𝒛), 500 mm OD, and 200 mm wall thickness (𝒚).Table 8 of the BS.

(b) Ring: 150 mm axial thickness (𝒛), 270 mm radial thickness (𝒚), and 400mm bore diameter (𝒙). Table 11 of the BS.

(a)

(b)

14

(f) Other Information in BS 5046-1974:

To provide further information on the effect of different quenching conditions (i.e., varying 𝒉) on the 𝒇 factors, the

BS presents Table 13. This Table compares the values of 𝒇 associated with 𝒉 values between zero (an infinitely slow quench) and infinity (surface temperature of object instantaneously falling to the temperature of the cooling medium), for rectangular sections, discs, and cylinders. Table 13, however, is not reproduced here.

4.2.4 The Climax Molybdenum Company Publication: (4)

The Climax publication is the only US document that the undersigned is aware of, which presents the Equivalent Diameter concept as it is recognized in the UK. The publication is basically the same as BS 971-1944, (2) except that only the Tables, not the graphs, are presented. The Tables are identical to those shown at the bottom part of Illustration3.

5.0 ANALYSIS OF THE 𝑬𝑹 DATA IN THE US GRAPHICS

As pointed out in section 4.1, the US Graphics do not indicate how the listed 𝑬𝑹 values were derived for the simple shapes depicted: square, hexagon, rectangle, and tube (open at both ends, and closed at one or both ends). In what follows, the formulae in the graphics will be compared to the predictions of BS 5046-1974 and BS 971-1944, for oil quenched shapes; oil was selected because it is the most widely used cooling medium. Occasionally, however, water quenching and air cooling data will be checked as well; note that that the US Graphics do not address cooling methods. The formulae in the graphics will also be compared to the results of simple computations based on equivalent surface areas: that is, based on equivalency of the surface areas of the equivalent round bar and that of the particular shape in question, both being of the same length. The subject computations are presented in separate Appendices (A-D) at the end, so as not to interfere with the flow of the main text. These Appendices should not be confused with those of BS 5046.

5.1 Square Section:

For a square section with a side length of 𝑻, the US Graphics list the formula 𝑬𝑹 = 𝟏. 𝟐𝟓 𝑻. To enable comparison to BS 5046-1974 and BS 971-1944, a square section is defined as a rectangular section with a breadth to thickness

ratio 𝑩

𝑻= 𝟏 . Illustration 10 depicts Table 5 (oil quench) of BS 5046. There, listed in red font, are the

𝑫𝑬𝒒

𝑻 ratios ( 𝒇

factors) corresponding to 𝑩

𝑻= 𝟏 , displayed as a function of square side length, 𝑻. As can be seen, the 𝒇 values

are in the 1.0-1.1 range. j As to BS 971, Figure 1 and Table 1 of that BS, depicted here as Illustration 11, indicate factors in the 1.052-1.084 range, in reasonable agreement with the range indicated by of BS 5046. The fact that the

graphics indicate an f value of 1.25, for a square section, indicates that the formula 𝑬𝑹 = 𝟏. 𝟐𝟓 𝑻, listed in those

graphics, was not based on BS 5946 or BS 971. Appendix A computations indicate an 𝒇 factor of 1.27, suggesting

the that the formula 𝑬𝑹 = 𝟏. 𝟐𝟓 𝑻, listed in the graphics for a square section, was likely derived based upon the equivalency of surface areas (that is the equivalence of the surface area of the equivalent round bar and that of a square); the 1.27 value was just rounded to 1.25.

5.2 Hexagonal Section:

For a hexagonal section with a distance 𝑻 between flats, the US Graphics list the formula 𝑬𝑹 = 𝟏. 𝟏 𝑻. BS 971-1944 and BS 5046-1974 do not offer any equivalent round numerical values for hexagonal sections. All they offer is the following vogue statement: “Sections other than rounds or rectangles are difficult to treat mathematically, but close approximations may usually be obtained. Octagonal and hexagonal parts are intermediate in cooling rate between round and square, the order of increasing time of cooling being round, octagonal, hexagonal,

square.” In other words, all these BSs indicate is that, for a hexagonal section, 𝒇𝒉𝒆𝒙 should be between those of a

round section (𝒇 = 𝟏) and a square section, 𝑩

𝑻= 𝟏 , (𝒇 = 𝟏 𝒕𝒐 𝟏. 𝟏) ; refer to Illustration 11 for the square section.

Appendix B computations indicate a factor of 1.1 for a hexagonal section, which is the same as that listed in the

US Graphics. This indicates that the formula (𝑬𝑹 = 𝟏. 𝟏 𝑻) listed in those graphics for a hexagonal section was derived based upon the equivalency of surface areas; that is the equivalence of the surface area of the equivalent round bar and that of a hexagonal section.

5.3 Rectangular Section / Plate:

For a rectangular section / plate with thickness T, the US Graphics list the formula 𝑬𝑹 = 𝟏. 𝟓 𝑻. Figure 1 of BS

971-1944 (Illustration 12), however, indicates that an 𝒇 factor of 1.5 applies only for specific combinations: 𝑩

𝑻=

∞,𝑻 = 𝟒 ; 𝑩

𝑻= 𝟖, 𝑻 = 𝟑;

𝑩

𝑻= 𝟐. 𝟓, 𝑻 = 𝟏. 𝟐, and;

𝑩

𝑻= 𝟐. 𝟓, 𝑻 = 𝟎. 𝟐. Other

𝑩

𝑻 between 2.5 and 8 and between 8 and ∞

may also produce an 𝒇 value of 1.5. Illustration 13 depicts an example for 𝑩

𝑻= 𝟑 (which is between 2.5 and 8),

where an 𝒇 value (1.48) close to 1.5 is obtained.

j BS 5046 has tables for oil and water quenching (Tables 5 and 6, respectively) as well as for air cooling (Table 4).

15

Per BS 971-1944, therefore, the 𝒇 value (1.5) listed in the US Graphics for rectangular section / plate represents special cases, not the general case.

Illustration 14 depicts Table 5 (oil quenched rectangular bar / plate) of BS 5046-1974, together with some

computations. The 𝒇 computations are shown in blue font, whereas the corresponding 𝑩

𝑻 computations are

shown in green font; 𝑩

𝑻 ratios range from 2.5 (rectangular section) to ∞ (plate). It is seen that

only a few combinations of 𝑩 and 𝑻 produce an 𝒇 factor of exactly 1.5. A somewhat larger number of

combinations produce 𝒇 factors (in the 1.47-1.48 range) that are close to 1.5. k

Per BS 5046-1974, therefore, the 𝒇 value (1.5) listed in the US Graphics for rectangular section / plate represents special cases, not the general case. The same conclusion was reached above in conjunction with BS 971-1944.

Appendix C computations indicate that 𝒇 cannot be determined without knowing 𝑩. As such, the formula listed in

the US Graphics (𝑬𝑹 = 𝟏. 𝟓 𝑻) for a rectangular section / plate was not derived based on equivalent surface areas,

since these graphics do not list a value for 𝑩 or a 𝑩

𝑻 ratio.

k In Table 5 of BS 5046, f values range from 1 to 1.8.

130/120= 1.08

175/160= 1.09

215/200= 1.08

270/250= 1.08320/300= 1.06

430/400= 1.08

540/500= 1.08

650/600= 1.08860/800= 1.08

1080/1000= 1.08

T

B

10/10=1

13/12= 1.06

21/20= 1.0517/16= 1.08

27/25= 1.0832/30= 1.07

43/40= 1.08

54/50= 1.0865/60= 1.08

87/80= 1.09110/100= 1.1

𝒇 = 𝑫𝑬𝒒

𝑻

Illustration 10: Table 5, BS 5046-1974. The “𝒇” factors for an oil

quenched square section ). (𝑩

𝑻= 𝟏

(𝒇 = 𝑫𝑬𝒒

𝑻)

16

Illustration 11: Figure 1 and Table 1, BS 971-1944. The 𝒇 factors for oil

quenched square section .

𝑩

𝑻= 𝟏 𝑺𝒒𝒖𝒂𝒓𝒆

𝒇 = 𝑫𝑬𝒒

𝑻 𝟏. 𝟎𝟓𝟐− 𝟏. 𝟎𝟖𝟒

𝒇 = 𝑫𝑬𝒒

𝑻

𝑫𝑬𝒒 = 𝒇𝑻

𝑩

𝑻= 𝟏, 𝑺𝒒𝒖𝒂𝒓𝒆

𝒇 = 𝑫𝑬𝒒

𝑻

𝑫𝑬𝒒 = 𝒇𝑻

(𝒇 = 𝑫𝑬𝒒

𝑻 )

𝑩

𝑻= 𝟏

17

𝑩

𝑻= ∞, 𝑷𝒍𝒂𝒕𝒆

𝑩

𝑻= 𝟖,𝑹𝒆𝒄𝒕𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝑩𝒂𝒓

𝑩

𝑻= 𝟐.𝟓, 𝑹𝒆𝒄𝒕𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝑩𝒂𝒓

0.2 1.2 3 4

Illustration 12: Figure 1 BS 971-1944 depicting 𝑻 and 𝑩

𝑻combinations

associated with an 𝒇 value of 1.5. Oil quenched plate and rectangular bar.

𝒇 = 𝑫𝑬𝒒

𝑻

𝒇 = 𝟓.𝟗𝟎𝟒

𝟒= 𝟏. 𝟒𝟖 𝒇 =

𝟓.𝟏𝟖𝟔

𝟑.𝟓= 𝟏. 𝟒𝟖 𝒇 =

𝟒.𝟒𝟖𝟓

𝟑= 𝟏. 𝟒𝟖

Illustration 13: Table 1 BS 971-1944 depicting 𝒇 values close to 1.5 for 𝑩

𝑻 = 3.

Oil quenched plate and rectangular bar.

18

Illu

str

ati

on

14:

Tab

le 5

, B

S 5

046 (o

il q

uen

ch

ed

recta

ng

ula

r b

ar

/ p

late

). 𝑫

𝑬𝒒 v

alu

es th

at p

rod

uce 𝒇

valu

es o

f 1.5

are

en

clo

sed

in c

ircle

s o

r e

llip

ses, w

here

as r

ec

tan

gle

s e

nclo

se 𝑫𝑬𝒒 v

alu

es p

rod

ucin

g 𝒇

valu

es in

th

e 1

.47

-1.4

8 r

an

ge (i.

e.,

clo

se to

1.5

);

𝒇c

om

pu

tati

on

s in

blu

e, 𝑩 𝑻

co

mp

uta

tio

ns

in

gre

en

.

30

/12

=

2.5

30

0/1

00

to ∞

/10

0

= 3

to

∞

16

0/6

0

=

2.7

10

0/4

0

=

2.5

40

/16

=

2.5

25

/10

=

2.5

15

/10

=

1.5

18

/12

= 1.5

24

/16

= 1.5

30

/20

= 1.5

60

/40

= 1.5

90

/60

= 1.5

12

0/8

0

=

1.5

15

0/1

00

= 1.5

18

0/1

20

= 1.5

15

0/1

00

= 1.5

50

/20

=

2.5

25

0/8

0

To

∞/8

0

= 3

.1-∞

40

0/1

00

to ∞

/10

0

= 4

to

∞

40

0/1

20

to ∞

/12

0

= 3

.3 t

o ∞

𝑫𝑬𝒒

𝑫𝑬𝒒

𝒇= 𝑫

𝑬𝒒

𝑻

𝑩 𝑻

𝑻𝑩

𝑻𝑩

19

5.4 Tube, Open Both Ends:

For an open-ended tube (outside diameter 𝑫, wall thickness 𝑻, and length 𝑳), the US Graphics list three cases: the

general case; the case where 𝑳 < 𝑫 and; the case where 𝑳 < 𝑻 .

BS 971-1944 does not deal with tubes at all. BS 5046-1974, however, does: it indicates that for tubes, the ratio of

the axial length 𝒛 to wall thickness 𝑻 is greater than or equal to unity, whereas for a ring (which is always

supposed to be of rectangular cross section), l this ratio is less than unity; that is, 𝒛

𝒚 ≥ 𝟏 for a tube and

𝒛

𝒚 < 𝟏 for

a ring.

Illustration 15 depicts the dimensions used in BS 5046-1974 for an open-ended tube and a ring, and their counterparts (between brackets) in the US Graphics; clearly, (b) and (c) in this Illustration represent rings, rather

than tubes. In what follows, the 𝑬𝑹 data presented in the graphics will be evaluated based on the information presented in BS 5046-1974 for oil quenching, as was done in the previous cases. As an added measure of

assurance, the 𝑬𝑹 data will also be evaluated based on the information presented in BS 5046-1974 for water quenching and air cooling.

5.4.1 The General Case:

For the general case, Illustration 15 (a), the US Graphics list the formula 𝑬𝑹 = 𝟐 𝑻 .

5.4.1.1 BS 5046 Oil Quench Data:

Illustration 16 depicts Table 8 (oil quench) of BS 5046-1974. It is seen that 𝒇 = 𝟐 only when two conditions are satisfied:

Condition 1:

𝒛

𝒚 𝒐𝒓 (

𝑳

𝑻) 𝒊𝒔 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝟒 𝒂𝒏𝒅 ∞

Condition 2:

𝒙

𝒚 𝒐𝒓 (

𝑫 − 𝟐𝑻

𝑻) = 𝟎

Condition 2 can be rewritten as:

𝑫 = 𝟐 𝑻 (𝒊. 𝒆. , 𝒕𝒉𝒊𝒄𝒌 𝒘𝒂𝒍𝒍 𝒕𝒖𝒃𝒊𝒏𝒈)

5.4.1.2 BS 5046 Water Quench Data:

Illustration 17, depicts Table 9 (water quench) of BS 5046-1974. Again, 𝒇 = 𝟐 only when two conditions are satisfied:

Condition 1:

𝒛

𝒚 𝒐𝒓 (

𝑳

𝑻)𝒓𝒂𝒕𝒊𝒐 𝒕𝒉𝒂𝒕 𝒊𝒔 𝒂 𝒍𝒊𝒕𝒕𝒍𝒆 𝒍𝒆𝒔𝒔 𝒕𝒉𝒂𝒏 𝟑

Condition 2:

The same as in oil quench:

𝑫 = 𝟐 𝑻 (𝒊. 𝒆. , 𝒕𝒉𝒊𝒄𝒌 𝒘𝒂𝒍𝒍 𝒕𝒖𝒃𝒊𝒏𝒈)

5.4.1.3 BS 5046 Air Cooling Data:

Illustration 18 depicts Table 7 (air cooling) of BS 5046. As was the case with oil and water quench, 𝒇 = 𝟐 only when 2 conditions are satisfied:

Condition 1:

𝒛

𝒚 𝒐𝒓 (

𝑳

𝑻) 𝒃𝒆𝒕𝒘𝒆𝒆𝒏 𝟏𝟎𝟎 𝒂𝒏𝒅 ∞

Condition 2:

The same as in oil and water quenches

𝑫 = 𝟐 𝑻 (𝒊. 𝒆. , 𝒕𝒉𝒊𝒄𝒌 𝒘𝒂𝒍𝒍 𝒕𝒖𝒃𝒊𝒏𝒈)

l The BS, however, does not explain why this is so.

20

Thus, BS 5046-1974 indicates that that 𝒇 = 𝟐 only for thick wall tubing (𝑫 = 𝟐𝑻; 𝒊. 𝒆. , 𝒙 = 𝟎) and only for specific

values / ranges of 𝑳 that depend on the cooling method (oil or water quenching or air cooling). Thus, the formula

𝑬𝑹 = 𝟐 𝑻 listed in the US Graphics as the general case for open ended tubing, in fact, represents the special case of thick wall tubing.

Appendix D computations, based on equivalent surface areas, indicate that 𝒇 = 𝟐 only for thick wall tubing, in

agreement with BS 5046 data for thick-walled tubing; 𝑳 does not enter in these computations.

Thus, both BS 5046-1974 and Appendix D computations indicate that, for open ended tubing, 𝒇 = 𝟐 represents a special case for thick wall tubing. The US Graphics, however, present it as the general case.

Illustration 15: The 𝒙,𝒚 and 𝒛 dimensions used in BS 5046-1974, and their

counterparts 𝑫, 𝑻 and 𝑳 used in the US Graphics (between brackets).

(a) The general case. Tube, 𝒛

𝒚 ≥ 𝟏.

(b) The case where 𝑳 < 𝑫,𝒛

𝒚 < 𝟏.

(c) The case where 𝑳 < 𝑻,𝒛

𝒚 < 𝟏.

𝒛 (𝑳)

𝒙 (𝑫 − 𝟐𝑻)(𝑫)

𝒚 (𝑻)

𝒚 (𝑻)

𝒄 𝑳 < 𝑻

𝒚 (𝑻)

𝒚 (𝑻)

𝒛 (𝑳)

(𝑫) 𝒙 (𝑫 − 𝟐𝑻)

𝒂 𝑻𝒉𝒆 𝒆𝒏𝒆𝒓𝒂𝒍 𝑪𝒂𝒔𝒆 𝒃 𝑳 < 𝑫

𝒚 (𝑻)

𝒛 (𝑳)

(𝑫)

𝒚 (𝑻)

𝒙 (𝑫 − 𝟐𝑻)

Illustration 16: Open-ended tube data in BS 5046-1974. Oil quenched.

21

Illustration 17: Open-ended tube data in BS 5046-1974. Water quenched.

Illustration 18: Open-ended tube data in BS 5046-1974. Air cooled.

22

5.4.2 Special Cases:

In addition to the “general case” described above for open ended tubing, the US Graphics present two “special” cases:

5.4.2.1 The Case where 𝑳 < 𝑫: Where 𝑳 < 𝑫 , Illustration 15 (b), the graphics instruct the user to consider the

section as a plate of thickness 𝑻. Per section 5.3, this means 𝑬𝑹 = 𝟏. 𝟓 𝑻.

5.4.2.1 The Case where 𝑳 < 𝑻: Where 𝑳 < 𝑻 , Illustration 15 (c), the graphics instruct the user to consider the

section as a plate of thickness L. Per section 5.3, this means 𝑬𝑹 = 𝟏. 𝟓 𝑻.

Thus, the value of 𝑳 relative to 𝑻 or 𝑫 distinguishes the two cases from one another. Yet, 𝑳 is not part of either formula. m It is not clear as to what these cases signify. Furthermore, neither BS 5046 nor equivalent surface areas computations can predict these formulae. The formulae appear to be Empirical in nature; that is, their validity cannot be proven or disproven.

5.5 Tube, Closed at One or Both Ends:

Referring to Illustration 1 (a)-(d), this case is presented in the lower right of each graphic. The user is instructed

to use 𝑬𝑹 = 𝟐. 𝟓 𝑻 when 𝑫 < 𝟐. 𝟓 𝒊𝒏𝒄𝒉𝒔 and 𝑬𝑹 = 𝟑. 𝟓 𝑻 when 𝑫 > 𝟐. 𝟓 𝒊𝒏𝒄𝒉𝒔 . This is one of only two cases in the US Graphics where an actual dimension of a shape is listed; the other being the complex shapes of AS 1260 (section 5.6). Accordingly, it appears that the subject formulae are Empirical in nature; that is, that is, their validity cannot be proven or disproven. This said, it is important to note that the title of the case “Tube, Closed at One or Both Ends” indicates the possibility of heat treating a tube closed at both ends. This is an unrealistic case that involves an unsafe practice.

5.6 The Complex Shapes in AS 1260:

AS 1260, in addition to listing formulae for simple shapes, Illustration 1 (a)-(d), includes a graphic for some

complex shapes, Illustration 1 (e). This graphic shows a method that can be used to calculate the 𝑬𝑹 for such shapes. The method utilizes sample dimensions to show how to perform the calculations. There appears to be

no technical basis for the method, and the calculated 𝑬𝑹 values cannot be derived from simple principles. In other words, the validity of the method cannot be proven or disproven.

6.0 SUMMARY

6.1 In heat treatment, it is important that any given body hardens throughout the section, to develop uniform properties in all locations. However, much of the data available in the literature regarding the effect of section size are based on testing of round bars. To use these data successfully, it is necessary to “convert” the dimensions of the body in question to a hypothetical equivalent round bar of uniform cross section. In the US, the

diameter of the hypothetical equivalent bar is referred to as the “Equivalent Round” (𝑬𝑹). In the UK, it is recognized as the “Equivalent Diameter” or the “Ruling Section.”

6.2 US heat treatment specifications Mil-H6875, AMS-H-6875, Mil-H-81200, AMS-H-81200, and AS 1260, and the

MMPDS contain identical graphics (the US Graphics) that list 𝑬𝑹 values for various simple shapes: square, hexagon, rectangular section / plate, open ended tubing, and restricted tubing. In addition, AS 1260 deals with some complex shapes; none of the other documents does so. Mil-H-6875, AMS-H-6875, AS 1260, and the MMPDS pertain to steels, whereas Mil-H-81200 and AMS-H-81200 pertain to titanium alloys. Said graphics existed in the 1970s, but they were probably introduced earlier. The documents that contain the US Graphics do not indicate how the listed values were derived for the simple shapes depicted in these graphics, and the data in the graphics do not take account of the properties of the material being cooled or those of the cooling medium.

6.3 The terms “Equivalent Diameter” and “Ruling Section” originated in the United Kingdom (UK) in the 1930s, specifically for the heat treatment of steel. There are three (3) UK documents that are quoted regarding Equivalent Diameter: Special Report No. 14, BS 971, and BS 5046. BS 971 and BS 5046 introduced the term “Ruling Section,” which is used interchangeably with Equivalent Diameter. The undersigned is aware of only one US publication,

by the Climax Molybdenum Company, that refers to Equivalent Diameter, and it makes no mention of the US 𝑬𝑹 or the US Graphics.

6.4 There is a notion in the technical community that the US 𝑬𝑹 and the UK Equivalent Diameter are one and the same.

6.5 BS 971, BS 5046 and the Climax Molybdenum Company publication indicate that the origin of the Equivalent Diameter / Ruling Section concept is Special Report No. 14.

6.5.1 Special Report No. 14:

m In fact, L is neither a part of any of the formulae, listed in the US Graphics, for all the other shapes, nor does it enter into the equivalent

surface area computations presented in the Appendices.

23

This Report is a generalized treatment of heating and cooling of steel objects, involving solutions of differential equations for heat transfer in terms of temperatures and time. The temperatures in the equations are: the

constant initial temperature of the object (𝑻𝟏), the constant temperature of the heating or cooling medium (𝑻𝟐), the variable temperature at the surface of the object (𝑻𝑺), and the variable temperature at any given point in the

object (𝑻). The inputted data include the following physical properties of specific steel grades: thermal

conductivity (k), specific heat (s), density (), and thermal diffusivity (k / s), which are considered not to vary with temperature. The Report presents solutions to heat flow equations for several shapes (cylinder, plate, and rectangular block) in the form of numerical Tables that are of no use to the practical metallurgist. The Report

makes no mention of 𝑬𝑹 , Equivalent Diameter or Ruling Section.

6.5.2 BS 971-1944:

The mathematical calculations were made for cooling from 820 to 300 C (approx. 1510 to 570 F) in air or oil at 20 C (68 F). The results, however, are equally applicable to any other temperature range, where the temperature drop at the center has the same ratio to the original temperature (relative to the surrounding medium) as for the conditions assumed; i.e., where:

𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒂𝒕 𝒕𝒉𝒆 𝒄𝒆𝒏𝒕𝒆𝒓 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒊𝒆𝒄𝒆−𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒐𝒊𝒍 (𝑨𝒊𝒓)

𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒑𝒊𝒆𝒄𝒆−𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝑶𝒊𝒍 (𝑨𝒊𝒓)= 𝟎. 𝟑𝟓

This BS does not delve into the details of the calculations, except to point out, without elaboration, that the

heat transfer factor, “𝒉”, for oil quenching is taken to be 0.8 cm-1, whereas for air cooling, it is taken to be 0.03 cm-1; the heat transfer factor is the quantity of heat transferred per unit area of the surface per unit time per degree of temperature difference between the body and the surrounding medium, divided by the thermal conductivity of the cooling medium, air or oil (steel properties were left out). The results of the calculations are presented in the form of tables and graphs, for oil quenching and air cooling of rectangular sections, including plate and square sections.

The BS indicates that the data should be used for comparing sections of the same steel, not sections of

different steels, since the physical properties of the steel (𝒌, 𝒔, 𝝆) were left out of the calculations.

6.5.3 BS 5046-1974:

The basis for equivalent diameter mathematical calculations is explained in Appendix A of the BS. The equivalent diameter is defined as the diameter of the round bar, the center of which would cool through a given range of temperature in the same time as the slowest cooling point of the shape in question (rectangle, square, etc.). The criterion used in this BS is the same as that used in BS 971-1944. This criterion is based on the equality of times for the temperature to fall to a chosen fraction (0.35) of the overall fall in temperature:

𝑻𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒂𝒕 𝒔𝒍𝒐𝒘𝒆𝒔𝒕 𝒄𝒐𝒐𝒍𝒊𝒏𝒈 𝒑𝒐𝒊𝒏𝒕 𝒊𝒏 𝒕𝒉𝒆 𝒔𝒉𝒂𝒑𝒆 𝒂𝒏𝒅 𝒃𝒂𝒓 − 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒐𝒐𝒍𝒊𝒏𝒈 𝒎𝒆𝒅𝒊𝒖𝒎

𝑰𝒏𝒊𝒕𝒊𝒂𝒍 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒉𝒂𝒑𝒆 𝒂𝒏𝒅 𝒃𝒂𝒓 − 𝒕𝒆𝒎𝒑𝒆𝒓𝒂𝒕𝒖𝒓𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒄𝒐𝒐𝒍𝒊𝒏𝒈 𝒎𝒆𝒅𝒊𝒖𝒎= 𝟎. 𝟑𝟓

BS 5046-1974 does not delve into the details of the calculations, except to point out: that the heat transfer

factor “𝒉” values were selected based on experience, being 0.003, 0.08, and, 0.2 mm-1

, for air, oil, and water, respectively, and; that for most hardening steels, 0.35 corresponds to a temperature around MS. The results of the calculations are presented in the form of Tables and graphs for oil and water quenching and air cooling of: cylinders and discs; rectangular sections, and; tubes and rings. The text of the BS does not explicitly indicate

whether or not the properties of the steels (𝒌, 𝒔, 𝝆) were factored in the calculations.

6.5.4 The Climax Molybdenum Company Publication:

The Climax publication is the only US document that the undersigned is aware of, which presents the

Equivalent Diameter concept as it is recognized in the UK; it makes no mention of 𝑬𝑹 or the US Graphics. The publication is basically the same as BS 971-1944, except that only the Tables, not the graphs, are presented.

6.6 The formulae in the US Graphics were compared to the predictions of BS 5046-1974 and BS 971-1944, as well as to the results of simple computations based on equivalent surface areas: that is, based on equivalency of the surface areas of the equivalent round bar and that of the particular shape in question, both being of the same length:

6.6.1 Square Section:

The formula 𝑬𝑹 = 𝟏. 𝟐𝟓 𝑻, listed in the US Graphics, was not based on BS 5946 or BS 971. The formula was likely derived based upon the equivalency of surface areas; that is, based on the equivalence of the surface area of the equivalent round bar and that of a square section, both being of the same length.

6.6.2 Hexagonal Section:

BS 971 and BS 5946 do not offer any equivalent round numerical values for hexagonal sections. The formula

𝑬𝑹 = 𝟏. 𝟏 𝑻, listed in the US Graphics, was derived based upon the equivalency of surface areas; that is, based

24

on the equivalence of the surface area of the equivalent round bar and that of a hexagonal section, both being of the same length.

6.6.3 Rectangular Section / Plate:

Per BS 971 and BS 5046, the formula 𝑬𝑹 = 𝟏. 𝟓 𝑻, listed in the US Graphics, represents special cases, not the general case. The formula cannot be derived based on equivalent surface areas; that is, it cannot be derived based on the equivalence of the surface area of the equivalent round bar and that of a rectangular section / plate, both being of the same length. This is because the US Graphics list only thickness of the section, whereas equivalent surface areas computations require knowledge of both the breadth and thickness of the section.

6.6.4 Open-Ended Tubing-General Case:

BS 971 does not deal with tubing. BS 5046 indicates that the formula 𝑬𝑹 = 𝟐 𝑻 , listed in the US Graphics,

applies only to thick wall tubing (𝑫 = 𝟐𝑻), and specific values / ranges of 𝑳 that depend on the cooling method

(oil or water quenching or air cooling). The formula 𝑬𝑹 = 𝟐 𝑻 can be derived based on equivalent surface areas (that is, based on the equivalence of the surface area of the equivalent round bar and that of a tube, both

being of the same length) only for thick wall tubing (𝑫 = 𝟐𝑻), in agreement with BS 5046. Thus, the formula

𝑬𝑹 = 𝟐 𝑻, listed in the US Graphics, represents a special case (thick wall tubing) rather than the general case.

6.6.5 Open-Ended Tubing-Special Cases:

The US Graphics list formulae for 2 special cases of open ended tubing: 𝑬𝑹 = 𝟏. 𝟓 𝑻 for the case where 𝑳 < 𝑫

and 𝑬𝑹 = 𝟏. 𝟓 𝑻 for the case where 𝑳 < 𝑻 . It is not clear as to what these cases signify. Furthermore, neither BS 5046 nor equivalent surface areas computations can predict these formulae. The formulae appear to be Empirical in nature; that is, their validity cannot be proven or disproven.

6.6.6 Tube, Closed at One or Both Ends:

The US Graphics list formulae for 2 cases: 𝑬𝑹 = 𝟐. 𝟓 𝑻 when 𝑫 < 𝟐. 𝟓 𝒊𝒏𝒄𝒉𝒔 and 𝑬𝑹 = 𝟑. 𝟓 𝑻 when 𝑫 >𝟐. 𝟓 𝒊𝒏𝒄𝒉𝒔 . This is one of only two cases in the US Graphics where an actual dimension of a shape is listed; the other being the complex shapes of AS 1260 (6.6.7). Accordingly, it appears that the subject formulae are Empirical in nature; that is, their validity cannot be proven or disproven.

6.6.7 The Complex Shapes in AS 1260:

AS 1260, in addition to listing formulae for simple shapes, includes a graphic for complex shapes. This

graphic shows a method that can be used to calculate the 𝑬𝑹 for such shapes. The method utilizes sample dimensions to show how to perform the calculations. There appears to be no technical basis for the method,

and the calculated 𝑬𝑹 values appear to be arbitrary, and they cannot be derived from basic principles. In other words, the validity of the method cannot be proven or disproven.

The above analyses indicate that the 𝑬𝑹 values listed in the US Graphics were variedly assigned based on: (a) equivalent surface areas; (b) empirical formulae or some other arbitrary method, the validity of which cannot be proven or disproven, or; (c) special cases in BS 971 and / or BS 5046, presented as general cases. It can be said, therefore, that the US ER and the UK Equivalent Diameter / Ruling Section are not one and the same.

Terry Khaled, Ph.D. Chief Scientific / Technical Advisor, Metallurgy Federal Aviation Administration (562) 627-5267 terry.khaled@faa.gov

References:

1- Special Report No. 14, First Report of the Alloy Steel Research Committee, Iron and Steel Institute (UK), 1936. The relevant information is contained in Section IX (pp. 125-187).

2- BS 971-1944 (Complementary to BS 970-1942), “Wrought Steels (Carbon and Alloy Steels), T.A.C 1-33, Steels En. 1-58,”

Prepared by the Technical Advisory Committee of the Special and Alloy Steels Committee (Steel Control of the Ministry of

Supply), September 1944. The applicable information can be found in pages 15-21: “Ruling Section,” “Equivalent” Section and

So-Called “Mass Effect.”

3- BS 5046, “Method for the estimation of equivalent diameters in the heat treatment of steel”, March 1974.

4- “Molybdenum,” Climax Molybdenum Company, The Hudson Press, 1948, Appendix D, Determination of “Equivalent Rounds”,

pp. 344-347.

25

APPENDIX A

Square Section Computations-Equivalent Surface Areas

𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒆𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒓𝒐𝒖𝒏𝒅 𝒃𝒂𝒓 𝒐𝒇 𝒍𝒆𝒏𝒈𝒕𝒉 𝑳 = 𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒔𝒒𝒖𝒂𝒓𝒆 𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒍𝒆𝒏𝒈𝒕𝒉 𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝟏

For the bar with a square cross section,

𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒒𝒖𝒂𝒓𝒆 = 𝑨𝑺 = 𝑪𝒔 𝑳,𝒘𝒉𝒆𝒓𝒆

𝑪𝑺 = 𝑪𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒔𝒒𝒖𝒂𝒓𝒆 = 𝟒 𝑻, 𝒂𝒏𝒅 𝑳 𝒊𝒔 𝒕𝒉𝒆 𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒂𝒓

Then,

𝑨𝑺 = 𝟒 𝑻 𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝟐

For the corresponding equivalent round bar, 𝑨𝑬𝑹−𝑺 = 𝝅 𝑳 𝑫𝑬𝑹−𝑺 , where 𝑨𝑬𝑹−𝑺 is the surface area, 𝑫𝑬𝑹−𝑺 is the

diameter, and 𝑳 is the length. Then,

𝑨𝑬𝑹−𝑺 = 𝝅 𝑳 𝑫𝑬𝑹−𝑺 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝟑

Per 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝟏 , 𝒔𝒆𝒕 𝑨𝑺 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝟐) = 𝑨𝑬𝑹−𝑺 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝟑)

Then,

𝑨𝑬𝑹−𝑺 = 𝝅 𝑳 𝑫𝑬𝑹−𝑺 = 𝟒 𝑻 𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝟒

Thus,

𝑫𝑬𝑹−𝑺 =𝟒𝑻

𝝅= 𝟏. 𝟐𝟕 𝑻 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑨𝟓

The 1.27 factor in Equation A5 is basically the same as that (1.25) listed in the US Graphics, suggesting that the

formula for a square section presented in those graphics (𝑬𝑹 == 𝟏. 𝟐𝟓 𝑻) was likely derived based on equivalent surface areas (that is, based on the equivalence of the surface area of the equivalent round bar and that of a square section); the 1.27 value was just rounded up to 1.25.

Square, 𝑳𝒆𝒏𝒈𝒕𝒉, 𝑳

𝑻

𝑻

Equivalent Round

𝑳𝒆𝒏𝒈𝒕𝒉,𝑳

26

APPENDIX B

Hexagonal Section Computations-Equivalent Surface Areas

𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒆𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒓𝒐𝒖𝒏𝒅 𝒃𝒂𝒓 𝒐𝒇 𝒍𝒆𝒏𝒈𝒕𝒉 𝑳 = 𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒉𝒆𝒙𝒂𝒈𝒐𝒏𝒂𝒍 𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒍𝒆𝒏𝒈𝒕𝒉 𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟏

For the bar with hexagonal cross section,

𝑺𝒊𝒏 𝟔𝟎 = 𝟎. 𝟖𝟕 =𝑻

𝟐𝒂

Then,

𝒂 = 𝟎. 𝟓𝟖 𝑻

𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒉𝒆𝒙𝒂𝒈𝒐𝒏 = 𝑨𝑯 = 𝑪𝑯 𝑳,𝒘𝒉𝒆𝒓𝒆 𝑪𝑯 𝒊𝒔 𝒕𝒉𝒆 𝒄𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒉𝒆𝒙𝒂𝒈𝒐𝒏 𝒂𝒏𝒅 𝑳 𝒊𝒔 𝒕𝒉𝒆 𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒂𝒓

Thus,

𝑪𝑯 = 𝟔𝒂 = 𝟑. 𝟒𝟖 𝑻

𝑨𝑯 = 𝟑. 𝟒𝟖 𝑻 𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟐

For the corresponding equivalent round bar, 𝑨𝑬𝑹−𝑯 𝒊𝒔 𝒕𝒉𝒆 𝒔𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂,𝑫𝑬𝑹−𝑯 𝒊𝒔 𝒕𝒉𝒆 𝒅𝒊𝒂𝒎𝒕𝒆𝒓, 𝒂𝒏𝒅 𝑳 𝒊𝒔 𝒕𝒉𝒆 𝒍𝒆𝒏𝒈𝒕𝒉

Then,

𝑨𝑬𝑹−𝑯 = 𝑳𝑫𝑬𝑹−𝑯 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟑

Per 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟏 , 𝒔𝒆𝒕 𝑨𝑯 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟐) = 𝑨𝑬𝑹−𝑺 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟑) , then,

𝑨𝑬𝑹−𝑯 = 𝝅𝑳𝑫𝑹𝑹−𝑯 = 𝟑. 𝟒𝟖 𝑻𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟒

and,

𝑫𝑬𝑹−𝑯 =𝟑. 𝟒𝟖 𝑻

𝝅 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟓

Thus,

𝑫𝑬𝑹−𝑯 = 𝟏. 𝟏 𝑻 = 𝒇𝒉𝒆𝒙 𝑻 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟔

𝒇𝒉𝒆𝒙 in 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑩𝟔 is the same as that listed in the US Graphics formula (𝑬𝑹 = 𝟏. 𝟏 𝑻) for a hexagonal section, indicating that that formula was derived based on equivalent surface areas (that is, based on the equivalence of the surface area of the equivalent round bar and that of a hexagonal section).

Equivalent Round

𝑳𝒆𝒏𝒈𝒕𝒉,𝑳

Hexagon, 𝑳𝒆𝒏𝒈𝒕𝒉, 𝑳

𝒂

𝑻

𝟐

𝑻

𝟐𝑻

27

APPENDIX C

Rectangular Section / Plate Computations-Equivalent Surface Areas

𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒆𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒓𝒐𝒖𝒏𝒅 𝒃𝒂𝒓 𝒐𝒇 𝒍𝒆𝒏𝒈𝒕𝒉 𝑳 = 𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒓𝒆𝒄𝒕𝒂𝒏𝒈𝒖𝒍𝒂𝒓 𝒔𝒆𝒄𝒕𝒊𝒐𝒏 /𝒑𝒍𝒂𝒕𝒆 𝒐𝒇 𝒍𝒆𝒏𝒈𝒕𝒉 𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟏

For the bar with a rectangular cross section,

𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒕𝒉𝒆 𝒓𝒆𝒄𝒕𝒂𝒏𝒈𝒍𝒆 = 𝑨𝑹 = 𝑪𝑹 𝑳,𝒘𝒉𝒆𝒓𝒆

𝑪𝑹 = 𝑪𝒊𝒓𝒄𝒖𝒎𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒐𝒇 𝒕𝒉𝒆 𝒓𝒆𝒄𝒕𝒂𝒏𝒈𝒍𝒆 = 𝟐(𝑩 + 𝑻), 𝒂𝒏𝒅 𝑳 𝒊𝒔 𝒕𝒉𝒆 𝒍𝒆𝒏𝒈𝒕𝒉 𝒐𝒇 𝒕𝒉𝒆 𝒃𝒂𝒓

Then,

𝑨𝑹 = 𝟐 (𝑩 + 𝑻) 𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟐 For the corresponding equivalent round bar,

𝑨𝑬𝑹−𝑹 𝒊𝒔 𝒕𝒉𝒆 𝒔𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂, 𝑫𝑬𝑹−𝑹 𝒊𝒔 𝒕𝒉𝒆 𝒅𝒊𝒂𝒎𝒆𝒕𝒆𝒓, 𝒂𝒏𝒅 𝑳 𝒊𝒔 𝒕𝒉𝒆 𝒍𝒆𝒏𝒈𝒕𝒉

Then,

𝑨𝑬𝑹−𝑹 = 𝝅 𝑳 𝑫𝑬𝑹−𝑹 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟑

Per 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟏, 𝒔𝒆𝒕

𝑨𝑹 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟐) = 𝑨𝑬𝑹−𝑹(𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟑)

𝟐 (𝑩 + 𝑻) 𝑳 = 𝝅𝑳 𝑫𝑬𝑹−𝑹 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟒

𝟐 (𝑩 + 𝑻) 𝑳 = 𝝅 𝑳 𝑫𝑬𝑹−𝑹 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟓

Thus,

𝑫𝑬𝑹−𝑹 =𝟐 (𝑩+𝑻)

𝝅 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟔

Clearly, 𝑫𝑬𝑹−𝑹 cannot be determined without knowing 𝑩 and 𝑻 or making some assumptions. One obvious

assumption is that 𝑩 >>>> 𝑻 (𝒑𝒍𝒂𝒕𝒆); 𝒊. 𝒆. ,𝑩

𝑻= ∞ 𝒐𝒓

𝑻

𝑩= 𝟎 . Even with this assumption, 𝑫𝑬𝑹−𝑹 cannot be

determined without knowing 𝑩. The conclusion to be made here is that the formula 𝑬𝑹 = 𝟏. 𝟓 𝑻, listed in the US Graphics for a rectangular section / plate, was not derived based on equivalent surface areas, since these

graphics do not list a value for 𝑩 or a 𝑩

𝑻 ratio.

It was of interest, however, to find out the 𝑩

𝑻 ratio that corresponds to the 𝒇 = 𝟏. 𝟓 in the formula listed in the US

Graphics.

Rearranging the terms of 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑪𝟔 ,

𝑫𝑬𝑹−𝑹

𝑻=

𝟐

𝝅 (𝑩

𝑻+ 𝟏) .

But, 𝒇 = 𝑫𝑬𝑹−𝑹

𝑻= 𝟏. 𝟓 .

Then, 𝑩

𝑻=

𝟏.𝟓 𝝅

𝟐− 𝟏 , and

𝑩

𝑻= 𝟏. 𝟑𝟔 , or

𝑻

𝑩 = 𝟎. 𝟕𝟒

It is not clear as to why such an odd ratio was selected to represent the general case for rectangular sections.

Rectangle, 𝑳𝒆𝒏𝒈𝒕𝒉, 𝑳Equivalent Round

𝑳𝒆𝒏𝒈𝒕𝒉, 𝑳

𝑻

𝑩

28

APPENDIX D

Open-Ended Tube Section Computations-Equivalent Surface Areas

𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒆𝒒𝒖𝒊𝒗𝒂𝒍𝒆𝒏𝒕 𝒓𝒐𝒖𝒏𝒅 𝒃𝒂𝒓 𝒐𝒇 𝒍𝒆𝒏𝒈𝒕𝒉 𝑳 = 𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒐𝒑𝒆𝒏 − 𝒆𝒏𝒅𝒆𝒅 𝒕𝒖𝒃𝒆 𝒔𝒆𝒄𝒕𝒊𝒐𝒏 𝒐𝒇 𝒍𝒆𝒏𝒈𝒕𝒉 𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟏

For the open-ended tube,

𝑪𝑶𝑬𝑻 is the circumference, 𝑫 is the outside diameter, 𝑻 is the wall thickness, and 𝑳 is the length. Then,

𝑺𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂 𝒐𝒇 𝒐𝒑𝒆𝒏 𝒆𝒏𝒅𝒆𝒅 𝒕𝒖𝒃𝒆 = 𝑨𝑶𝑬𝑻 = 𝑪𝑶𝑬𝑻 𝑳

𝑨𝑶𝑬𝑻 = [𝝅 𝑫 + 𝝅 (𝑫 − 𝟐 𝑻)] 𝑳 = 𝝅 𝑳 (𝟐𝑫 − 𝟐𝑻) 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟐

For the corresponding equivalent round bar,

𝑨𝑬𝑹−𝑶𝑬𝑻 𝒊𝒔 𝒕𝒉𝒆 𝒔𝒖𝒓𝒇𝒂𝒄𝒆 𝒂𝒓𝒆𝒂,𝑫𝑬𝑹−𝑶𝑬𝑻 𝒊𝒔 𝒕𝒉𝒆 𝒅𝒊𝒂𝒎𝒆𝒕𝒆𝒓, 𝒂𝒏𝒅 𝑳 𝒊𝒔 𝒕𝒉𝒆 𝒍𝒆𝒏𝒈𝒕𝒉

Then,

𝑨𝑬𝑹−𝑶𝑬𝑻 = 𝝅 𝑳 𝑫𝑬𝑹−𝑶𝑬𝑳 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟑

𝑷𝒆𝒓 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟏, 𝒔𝒆𝒕 𝑨𝑶𝑬𝑻 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟐) = 𝑨𝑬𝑹−𝑶𝑬𝑻 (𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟑)

𝝅 𝑳 (𝟐 𝑫 − 𝟐 𝑻) = 𝝅 𝑳 𝑫𝑬𝑹−𝑶𝑬𝑻 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟒

𝑫𝑬𝑹−𝑶𝑬𝑻 = 𝟐 (𝑫 − 𝑻) 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟓

Clearly, DER-OET cannot be determined without knowing 𝑻 and 𝑫 or making some assumptions.

One obvious assumption is that 𝑫 >>> 𝑻; i.e., 𝑫

𝑻= ∞ or

𝑻

𝑫= 𝟎, as would be the case for thin-wall tubing.

𝑫𝑬𝑹−𝑶𝑬𝑻 = 𝟐 𝑫 (𝟏 −𝑻

𝑫) = 𝟐𝑫

Still, 𝑫𝑬𝑹−𝑶𝑬𝑻 cannot be determined without knowing 𝑫.

Another possible assumption is that 𝑫 = 𝟐 𝑻, as would be the case for thick-wall tubing having a vanishingly small

bore diameter 𝑫 = 𝑻.

That is,

(𝑫 − 𝟐 𝑻) = 𝟎 𝒐𝒓 𝑫 = 𝟐𝑻

Thus,

𝑫𝑬𝑹−𝑶𝑬𝑻 = 𝟐 𝑫 − 𝟐𝑻 = 𝟒 𝑻 − 𝟐 𝑻 = 𝟐 𝑻

𝑫𝑬𝑹−𝑶𝑬𝑻 = 𝟐 𝑻 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟔

The coefficient (2) in 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 𝑫𝟔 is the same as that in the formula (𝑬𝑹 = 𝟐 𝑻) listed in the US Graphics, indicating that the formula in these graphics was derived based on equivalent surface areas. The problem is that the graphics present that formula as the general case, whereas it only represents a special case for thick-walled

tubing, 𝑫 = 𝟐 𝑻 .

Equivalent Round

𝑳𝒆𝒏𝒈𝒕𝒉,𝑳

Open-Ended Tube, 𝑳𝒆𝒏𝒈𝒕𝒉, 𝑳

𝑫

𝑻

𝑻