ASTM C680-23a

Standard Practice for Estimate of the Heat Gain or Loss and the Surface Temperatures of Insulated Flat, Cylindrical, and Spherical Systems by Use of Computer Programs

SIGNIFICANCE AND USE
5.1 Manufacturers of thermal insulation express the performance of their products in charts and tables showing heat gain or loss per unit surface area or unit length of pipe. This data is presented for typical insulation thicknesses, operating temperatures, surface orientations (facing up, down, horizontal, vertical), and in the case of pipes, different pipe sizes. The exterior surface temperature of the insulation is often shown to provide information on personnel protection or surface condensation. However, additional information on effects of wind velocity, jacket emittance, ambient conditions and other influential parameters may also be required to properly select an insulation system. Due to the large number of combinations of size, temperature, humidity, thickness, jacket properties, surface emittance, orientation, and ambient conditions, it is not practical to publish data for each possible case, Refs (7,8).  
5.2 Users of thermal insulation faced with the problem of designing large thermal insulation systems encounter substantial engineering cost to obtain the required information. This cost can be substantially reduced by the use of accurate engineering data tables, or available computer analysis tools, or both. The use of this practice by both manufacturers and users of thermal insulation will provide standardized engineering data of sufficient accuracy for predicting thermal insulation system performance. However, it is important to note that the accuracy of results is extremely dependent on the accuracy of the input data. Certain applications may need specific data to produce meaningful results.  
5.3 The use of analysis procedures described in this practice can also apply to designed or existing systems. In the rectangular coordinate system, Practice C680 can be applied to heat flows normal to flat, horizontal or vertical surfaces for all types of enclosures, such as boilers, furnaces, refrigerated chambers and building envelopes. In the cylindrical c...
SCOPE
1.1 This practice provides the algorithms and calculation methodologies for predicting the heat loss or gain and surface temperatures of certain thermal insulation systems that can attain one dimensional, steady- or quasi-steady-state heat transfer conditions in field operations.  
1.2 This practice is based on the assumption that the thermal insulation systems can be well defined in rectangular, cylindrical or spherical coordinate systems and that the insulation systems are composed of homogeneous, uniformly dimensioned materials that reduce heat flow between two different temperature conditions.  
1.3 Qualified personnel familiar with insulation-systems design and analysis should resolve the applicability of the methodologies to real systems. The range and quality of the physical and thermal property data of the materials comprising the thermal insulation system limit the calculation accuracy. Persons using this practice must have a knowledge of the practical application of heat transfer theory relating to thermal insulation materials and systems.  
1.4 The computer program that can be generated from the algorithms and computational methodologies defined in this practice is described in Section 7 of this practice. The computer program is intended for flat slab, pipe and hollow sphere insulation systems.  
1.5 The values stated in inch-pound units are to be regarded as standard. The values given in parentheses are mathematical conversions to SI units that are provided for information only and are not considered standard.  
1.6 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of regulatory limitations prior to use.  
1.7 This international standard was developed in accordance with internationally recognized pr...

General Information

• Status: Published
• Publication Date: 31-Oct-2023
• ICS: 27.220 - Heat recovery. Thermal insulation
• Technical Committee: C16 - Thermal Insulation
• Drafting Committee: C16.30 - Thermal Measurement

Relations

• Replaces: ASTM C680-23 - Standard Practice for Estimate of the Heat Gain or Loss and the Surface Temperatures of Insulated Flat, Cylindrical, and Spherical Systems by Use of Computer Programs
• Effective Date: 01-Nov-2023
• Refers: ASTM C168-24 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 15-Apr-2024
• Refers: ASTM C168-22 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-May-2022
• Referred By: ASTM C335/C335M-23 - Standard Test Method for Steady-State Heat Transfer Properties of Pipe Insulation
• Effective Date: 01-Nov-2023
• Referred By: ASTM C592-22a - Standard Specification for Mineral Fiber Blanket Insulation and Blanket-Type Pipe Insulation (Metal-Mesh Covered) (Industrial Type)
• Effective Date: 01-Nov-2023
• Referred By: ASTM C892-19 - Standard Specification for High-Temperature Fiber Blanket Thermal Insulation
• Effective Date: 01-Nov-2023
• Referred By: ASTM C612-14(2019) - Standard Specification for Mineral Fiber Block and Board Thermal Insulation
• Effective Date: 01-Nov-2023
• Referred By: ASTM C1055-20 - Standard Guide for Heated System Surface Conditions that Produce Contact Burn Injuries
• Effective Date: 01-Nov-2023
• Referred By: ASTM C1045-19 - Standard Practice for Calculating Thermal Transmission Properties Under Steady-State Conditions
• Effective Date: 01-Nov-2023
• Referred By: ASTM F3319-20 - Standard Specification for Selection and Application of Field-Installed Cryogenic Pipe and Equipment Insulation Systems on Liquefied Natural Gas (LNG)-Fueled Ships
• Effective Date: 01-Nov-2023
• Referred By: ASTM F683-23a - Standard Practice for Selection and Application of Thermal Insulation for Piping and Machinery
• Effective Date: 01-Nov-2023
• Referred By: ASTM C553-13(2019) - Standard Specification for Mineral Fiber Blanket Thermal Insulation for Commercial and Industrial Applications
• Effective Date: 01-Nov-2023
• Referred By: ASTM C1057-22 - Standard Practice for Determination of Skin Contact Temperature from Heated Surfaces Using a Mathematical Model and Thermesthesiometer
• Effective Date: 01-Nov-2023
• Referred By: ASTM C1371-15(2022) - Standard Test Method for Determination of Emittance of Materials Near Room Temperature Using Portable Emissometers
• Effective Date: 01-Nov-2023
• Referred By: ASTM C1774-13(2019) - Standard Guide for Thermal Performance Testing of Cryogenic Insulation Systems
• Effective Date: 01-Nov-2023

Overview

ASTM C680-23a is the recognized standard for estimating the heat loss or gain and the surface temperatures of insulated flat, cylindrical (pipe), and spherical systems using computer programs. Published by ASTM International, this standard practice provides robust computational methodologies and algorithms to predict the thermal performance of insulation systems under steady-state and quasi-steady-state conditions. These predictions are crucial for engineers, manufacturers, and insulation system designers who need highly accurate data to optimize energy efficiency, ensure personnel safety, and prevent issues such as condensation or heat loss.

Key to this practice is the use of computer analysis for thermal calculations, greatly simplifying the complexity arising from the many variables inherent in insulation system design, including material properties, geometry, ambient conditions, and installation parameters.

Key Topics

• Calculation Methodologies: Provides detailed algorithms for predicting heat transfer (gain or loss) in flat, cylindrical, and spherical geometries.
• Thermal Resistance and Conductivity: Emphasizes the use of temperature-dependent thermal conductivity, allowing for realistic, accurate modeling of insulation performance.
• Surface Temperatures: Calculates inner and outer surface temperatures of insulation systems, supporting assessments related to personnel protection and condensation risks.
• Data Input and Analysis: Relies on accurate input data, including material properties, insulation thicknesses, system orientation, ambient temperature, wind velocity, and surface emittance.
• Interactive Computer Programs: Specifies that calculations are best performed using interactive computer software, reducing labor and the potential for human error.
• Customization: Users can modify input and output routines to meet project-specific or regulatory requirements, provided the accuracy of modifications is validated.

Applications

ASTM C680-23a is applicable across a wide range of industries such as energy, building construction, industrial processing, and cold storage. Its practical applications include:

• Design of Thermal Insulation Systems: Helps engineers optimize insulation thickness and material selection for industrial equipment, pipelines, storage tanks, cylindrical vessels, and building envelopes.
• Performance Validation: Enables manufacturers to present the thermal performance of their products under various operating conditions by simulating real-world variables.
• Facility Upgrades and Retrofitting: Assists in evaluating existing insulation performance for process plants, refineries, and power stations, supporting decisions on upgrades or maintenance.
• Safety and Compliance: Supports assessments for surface temperatures to prevent contact burn injuries and to ensure compliance with applicable health and safety regulations.
• Process Optimization: Aids in minimizing energy losses in temperature-sensitive facilities such as boilers, furnaces, refrigeration chambers, and LNG storage, contributing to operational efficiency.

Related Standards

• ASTM C168: Terminology Relating to Thermal Insulation.
• ASTM C177: Test Method for Steady-State Heat Flux Measurements using the Guarded-Hot-Plate Apparatus.
• ASTM C335: Test Method for Steady-State Heat Transfer Properties of Pipe Insulation.
• ASTM C518: Test Method for Heat Flow Meter Apparatus.
• ASTM C585: Practice for Pipe Insulation Dimensions.
• ASTM C1055: Guide for Surface Conditions Producing Contact Burn Injuries.
• ASTM C1057: Practice for Skin Contact Temperature Determination.

Practical Value

ASTM C680-23a offers standardized, scientifically grounded tools for evaluating and specifying insulation systems in all common forms. By leveraging computer-based methodologies, the standard delivers accurate, repeatable results, enabling stakeholders to reduce engineering costs, enhance energy efficiency, safeguard personnel, and comply with industry requirements. The reliance on standardized input and calculation procedures also promotes transparency and comparability across insulation products and systems, ensuring confident decision-making in both new design and retrofit applications.

For optimal results, users should ensure high-quality input data and possess a fundamental understanding of heat transfer theory as it relates to insulation materials and system engineering.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: C680 − 23a
Standard Practice for
Estimate of the Heat Gain or Loss and the Surface
Temperatures of Insulated Flat, Cylindrical, and Spherical
Systems by Use of Computer Programs
This standard is issued under the fixed designation C680; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope priate safety, health, and environmental practices and deter-
mine the applicability of regulatory limitations prior to use.
1.1 This practice provides the algorithms and calculation
1.7 This international standard was developed in accor-
methodologies for predicting the heat loss or gain and surface
dance with internationally recognized principles on standard-
temperatures of certain thermal insulation systems that can
ization established in the Decision on Principles for the
attain one dimensional, steady- or quasi-steady-state heat
Development of International Standards, Guides and Recom-
transfer conditions in field operations.
mendations issued by the World Trade Organization Technical
1.2 This practice is based on the assumption that the thermal
Barriers to Trade (TBT) Committee.
insulation systems can be well defined in rectangular, cylindri-
cal or spherical coordinate systems and that the insulation 2. Referenced Documents
systems are composed of homogeneous, uniformly dimen- 2
2.1 ASTM Standards:
sioned materials that reduce heat flow between two different
C168 Terminology Relating to Thermal Insulation
temperature conditions.
C177 Test Method for Steady-State Heat Flux Measure-
1.3 Qualified personnel familiar with insulation-systems ments and Thermal Transmission Properties by Means of
design and analysis should resolve the applicability of the
the Guarded-Hot-Plate Apparatus
methodologies to real systems. The range and quality of the C335 Test Method for Steady-State Heat Transfer Properties
physical and thermal property data of the materials comprising of Pipe Insulation
the thermal insulation system limit the calculation accuracy. C518 Test Method for Steady-State Thermal Transmission
Persons using this practice must have a knowledge of the Properties by Means of the Heat Flow Meter Apparatus
practical application of heat transfer theory relating to thermal C585 Practice for Inner and Outer Diameters of Thermal
insulation materials and systems. Insulation for Nominal Sizes of Pipe and Tubing
C1055 Guide for Heated System Surface Conditions that
1.4 The computer program that can be generated from the
Produce Contact Burn Injuries
algorithms and computational methodologies defined in this
C1057 Practice for Determination of Skin Contact Tempera-
practice is described in Section 7 of this practice. The computer
ture from Heated Surfaces Using a Mathematical Model
program is intended for flat slab, pipe and hollow sphere
and Thermesthesiometer
insulation systems.
2.2 Other Document:
1.5 The values stated in inch-pound units are to be regarded
NBS Circular 564 Tables of Thermodynamic and Transport
as standard. The values given in parentheses are mathematical
Properties of Air, U.S. Dept of Commerce
conversions to SI units that are provided for information only
and are not considered standard.
3. Terminology
1.6 This standard does not purport to address all of the
3.1 Definitions:
safety concerns, if any, associated with its use. It is the
3.1.1 For definitions of terms used in this practice, refer to
responsibility of the user of this standard to establish appro-
Terminology C168.
3.1.2 thermal insulation system—for this practice, a thermal
insulation system is a system comprised of a single layer or
This practice is under the jurisdiction of ASTM Committee C16 on Thermal
Insulation and is the direct responsibility of Subcommittee C16.30 on Thermal
Measurement. For referenced ASTM standards, visit the ASTM website, www.astm.org, or
Current edition approved Nov. 1, 2023. Published December 2023. Originally contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
approved in 1971. Last previous edition approved in 2023 as C680 – 23. DOI: Standards volume information, refer to the standard’s Document Summary page on
10.1520/C0680-23A. the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
C680 − 23a
layers of homogeneous, uniformly dimensioned material(s) fer theory as outlined in textbooks and handbooks, Refs
intended for reduction of heat transfer between two different (1,2,3,4,5,6). Heat flux solutions are derived for temperature
temperature conditions. Heat transfer in the system is steady- dependent thermal conductivity in a material. Algorithms and
state. Heat flow for a flat system is normal to the flat surface, computational methodologies for predicting heat loss or gain of
and heat flow for cylindrical and spherical systems is radial. single or multi-layer thermal insulation systems are provided
by this practice for implementation in a computer program. In
3.2 Symbols:
addition, interested parties can develop computer programs
3.2.1 The following symbols are used in the development of
from the computational procedures for specific applications
the equations for this practice. Other symbols will be intro-
and for one or more of the three coordinate systems considered
duced and defined in the detailed description of the develop-
in Section 6.
ment.
4.1.1 The computer program combines functions of data
where:
input, analysis and data output into an easy to use, interactive
h = surface transfer conductance, Btu/(h·ft ·°F) (W/ computer program. By making the program interactive, little
(m ·K)) h at inside surface; h at outside surface
training for operators is needed to perform accurate calcula-
i o
k = apparent thermal conductivity, Btu·in./(h·ft ·°F) (W/
tions.
(m·K))
4.2 The operation of the computer program follows the
k = effective thermal conductivity over a prescribed tem-
e
2 procedure listed below:
perature range, Btu·in./(h·ft ·°F) (W/(m·K))
2 2
4.2.1 Data Input—The computer requests and the operator
q = heat flux, Btu/(h·ft ) (W/m )
inputs information that describes the system and operating
q = time rate of heat flow per unit length of pipe, Btu/(h·ft)
p
environment. The data includes:
(W/m)
2 2
R = thermal resistance, °F·h·ft /Btu (K·m /W) 4.2.1.1 Analysis identification.
r = radius, in. (m); r − r = thickness
m+1 m 4.2.1.2 Date.
t = local temperature, °F (K)
4.2.1.3 Ambient temperature.
t = inner surface temperature of the insulation, °F (K)
i
4.2.1.4 Surface transfer conductance or ambient wind
t = inner surface temperature of the system
speed, system surface emittance and system orientation.
t = temperature of ambient fluid and surroundings, °F (K)
o
4.2.1.5 System Description—Material and thickness for
x = distance, in. (m); x − x = thickness
m+1 m
each layer (define sequence from inside out).
ε = effective surface emittance between outside surface
4.2.2 Analysis—Once input data is entered, the program
and the ambient surroundings, dimensionless
-8
σ = Stefan-Boltzmann constant, 0.1714 × 10 Btu/
calculates the surface transfer conductances (if not entered
2 4 -8 2 4
(h·ft ·°R ) (5.6697 × 10 W/(m ·K )) directly) and layer thermal resistances. The program then uses
T = absolute surface temperature, °R (K)
this information to calculate the heat transfer and surface
s
T = absolute surroundings (ambient air if assumed the
temperature. The program continues to repeat the analysis
o
same) temperature, °R (K)
using the previous temperature data to update the estimates of
T = (T + T )/2
m s o layer thermal resistance until the temperatures at each surface
L = characteristic dimension for horizontal and vertical
repeat within 0.1°F between the previous and present tempera-
flat surfaces, and vertical cylinders
tures at the various surface locations in the system.
D = characteristic dimension for horizontal cylinders and
4.2.3 Program Output—Once convergence of the tempera-
spheres
tures is reached, the program prints a table that presents the
c = specific heat of ambient fluid, Btu/(lb·°R) (J/(kg·K))
p
2 input data, calculated thermal resistance of the system, heat
h = average convection conductance, Btu/(h·ft ·°F) (W/
c
flux and the inner surface and external surface temperatures.
(m ·K))
k = thermal conductivity of ambient fluid, Btu/(h·ft·°F)
f
5. Significance and Use
(W/(m·K))
V = free stream velocity of ambient fluid, ft/h (m/s)
5.1 Manufacturers of thermal insulation express the perfor-
2 2
υ = kinematic viscosity of ambient fluid, ft /h (m /s)
mance of their products in charts and tables showing heat gain
2 2
g = acceleration due to gravity, ft/h (m ⁄s )
or loss per unit surface area or unit length of pipe. This data is
β = volumetric thermal expansion coefficient of ambient
presented for typical insulation thicknesses, operating
-1 -1
fluid, °R (K )
temperatures, surface orientations (facing up, down, horizontal,
3 3
ρ = density of ambient fluid, lb/ft (kg ⁄m )
vertical), and in the case of pipes, different pipe sizes. The
ΔT = absolute value of temperature difference between
exterior surface temperature of the insulation is often shown to
surface and ambient fluid, °R (K)
provide information on personnel protection or surface con-
Nu = Nusselt number, dimensionless
densation. However, additional information on effects of wind
Ra = Rayleith number, dimensionless
velocity, jacket emittance, ambient conditions and other influ-
Re = Reynolds number, dimensionless
ential parameters may also be required to properly select an
Pr = Prandtl number, dimensionless
insulation system. Due to the large number of combinations of
4. Summary of Practice
size, temperature, humidity, thickness, jacket properties, sur-
4.1 The procedures used in this practice are based on face emittance, orientation, and ambient conditions, it is not
standard, steady-state, one dimensional, conduction heat trans- practical to publish data for each possible case, Refs (7,8).
C680 − 23a
5.2 Users of thermal insulation faced with the problem of 5.6 The calculation of surface temperature and heat loss or
designing large thermal insulation systems encounter substan- gain of an insulated system is mathematically complex, and
tial engineering cost to obtain the required information. This because of the iterative nature of the method, computers best
cost can be substantially reduced by the use of accurate handle the calculation. Computers are readily available to most
producers and consumers of thermal insulation to permit the
engineering data tables, or available computer analysis tools, or
both. The use of this practice by both manufacturers and users use of this practice.
of thermal insulation will provide standardized engineering
5.7 Computer programs are described in this practice as a
data of sufficient accuracy for predicting thermal insulation
guide for calculation of the heat loss or gain and surface
system performance. However, it is important to note that the
temperatures of insulation systems. The range of application of
accuracy of results is extremely dependent on the accuracy of
these programs and the reliability of the output is a primary
the input data. Certain applications may need specific data to
function of the range and quality of the input data. The
produce meaningful results.
programs are intended for use with an “interactive” terminal.
Under this system, intermediate output guides the user to make
5.3 The use of analysis procedures described in this practice
programming adjustments to the input parameters as necessary.
can also apply to designed or existing systems. In the rectan-
The computer controls the terminal interactively with program-
gular coordinate system, Practice C680 can be applied to heat
generated instructions and questions, which prompts user
flows normal to flat, horizontal or vertical surfaces for all types
response. This facilitates problem solution and increases the
of enclosures, such as boilers, furnaces, refrigerated chambers
probability of successful computer runs.
and building envelopes. In the cylindrical coordinate system,
Practice C680 can be applied to radial heat flows for all types
5.8 The user of this practice may wish to modify the data
of piping circuits. In the spherical coordinate system, Practice
input and report sections of the computer programs presented
C680 can be applied to radial heat flows to or from stored fluids
in this practice to fit individual needs. Also, additional calcu-
such as liquefied natural gas (LNG).
lations may be desired to include other data such as system
costs or economic thickness. No conflict exists with such
5.4 Practice C680 is referenced for use with Guide C1055
modifications as long as the user verifies the modifications
and Practice C1057 for burn hazard evaluation for heated
using a series of test cases that cover the range for which the
surfaces. Infrared inspection, in-situ heat flux measurements,
new method is to be used. For each test case, the results for
or both are often used in conjunction with Practice C680 to
heat flow and surface temperature must be identical (within
evaluate insulation system performance and durability of
resolution of the method) to those obtained using the practice
operating systems. This type of analysis is often made prior to
described herein.
system upgrades or replacements.
5.9 This practice has been prepared to provide input and
5.5 All porous and non-porous solids of natural or man-
output data that conforms to the system of units commonly
made origin have temperature dependent thermal conductivi-
used by United States industry. Although modification of the
ties. The change in thermal conductivity with temperature is
input/output routines could provide an SI equivalent of the heat
different for different materials, and for operation at a relatively
flow results, no such “metric” equivalent is available for some
small temperature difference, an average thermal conductivity
portions of this practice. To date, there is no accepted system of
may suffice. Thermal insulating materials (k < 0.85 {Btu·in}/
metric dimensions for pipe and insulation systems for cylin-
{h·ft ·°F}) are porous solids where the heat transfer modes
drical shapes. The dimensions used in Europe are the SI
include conduction in series and parallel flow through the
equivalents of American sizes (based on Practice C585), and
matrix of solid and gaseous portions, radiant heat exchange
each has a different designation in each country. Therefore, no
between the surfaces of the pores or interstices, as well as
SI version of the practice has been prepared, because a
transmission through non-opaque surfaces, and to a lesser
standard SI equivalent of this practice would be complex.
extent, convection within and between the gaseous portions.
When an international standard for piping and insulation sizing
With the existence of radiation and convection modes of heat
occurs, this practice can be rewritten to meet those needs. In
transfer, the measured value should be called apparent thermal
addition, it has been demonstrated that this practice can be used
conductivity as described in Terminology C168. The main
to calculate heat transfer for circumstances other than insulated
reason for this is that the premise for pure heat conduction is no
systems; however, these calculations are beyond the scope of
longer valid, because the other modes of heat transfer obey
this practice.
different laws. Also, phase change of a gas, liquid, or solid
within a solid matrix or phase change by other mechanisms
6. Method of Calculation
will provide abrupt changes in the temperature dependence of
6.1 Approach:
thermal conductivity. For example, the condensation of the
gaseous portions of thermal insulation in extremely cold 6.1.1 The calculation of heat gain or loss and surface
conditions will have an extremely influential effect on the temperature requires: (1) The thermal insulation is homoge-
apparent thermal conductivity of the insulation. With all of this neous as outlined by the definition of thermal conductivity in
considered, the use of a single value of thermal conductivity at Terminology C168; (2) the system operating temperature is
an arithmetic mean temperature will provide less accurate known; (3) the insulation thickness is known; (4) the surface
predictions, especially when bridging temperature regions transfer conductance of the system is known, reasonably
where strong temperature dependence occurs. estimated or estimated from algorithms defined in this practice
C680 − 23a
x t
m+1 m+1
based on sufficient information; and, (5) the thermal conduc-
q dx 5 2 k t dt (4)
* * ~ !
tivity as a function of temperature for each system layer is
x t
m m
known in detail.
6.1.2 The solution is a procedure calling for (1) estimation
t 2 t
m m11
q 5 k
e,m
of the system temperature distribution; (2) calculation of the x 2 x
m11 m
thermal resistances throughout the system based on that
For heat flow in the hollow cylinder, let p = r, q = Q/(2πrl)
distribution; (3) calculation of heat flux; and (4) reestimation of
and integrate Eq 2:
the system temperature distribution. The iterative process
r t
continues until a calculated distribution is in reasonable agree-
m+1 m+1
Q dr
ment with the previous distribution. This is shown diagram- 5 2 k t dt (5)
* * ~ !
2πl r
r t
m m
matically in Fig. 1. The layer thermal resistance is calculated
each time with the effective thermal conductivity being ob-
t 2 t
m m11
Q 5 k 2πl
tained by integration of the thermal conductivity curve for the
e,m
ln r /r
~ !
m11 m
layer being considered. This practice uses the temperature
dependence of the thermal conductivity of any insulation or
Divide both sides by 2πrl
multiple layer combination of insulations to calculate heat
t 2 t
flow.
m m11
q 5 k
e,m
rln~r /r !
m11 m
6.2 Development of Equations—The development of the
mathematical equations is for conduction heat transfer through
For radial heat flow in the hollow sphere, let p = r, q =
homogeneous solids having temperature dependent thermal
Q/(4πr ) and integrate Eq 2:
conductivities. To proceed with the development, several
r t
m+1 m+1
Q dr
precepts or guidelines must be cited:
5 k t dt (6)
* * ~ !
4π r
6.2.1 Steady-state Heat Transfer—For all the equations it is r t
m m
assumed that the temperature at any point or position in the
t 2 t
m m11
solid is invariant with time. Thus, heat is transferred solely by
Q 5 k 4π
e,m
1 1
temperature difference from point to point in the solid.
r r
m m11
6.2.2 One-dimensional Heat Transfer—For all equations it
is assumed there is heat flow in only one dimension of the 2
Divide both sides by 4πr and multiply both sides by r r /r r
m m11 m m11
particular coordinate system being considered. Heat transfer in
the other dimensions of the particular coordinate system is
r r t 2 t
m m11 m m11
q 5 k
e,m
considered to be zero.
r r 2 r
m11 m
6.2.3 Conduction Heat Transfer—The premise here is that
Note that the effective thermal conductivity over the tem-
the heat flux normal to any surface is directly proportional to
perature range is:
the temperature gradient in the direction of heat flow, or
t
m+1
dt
q 5 2k (1) k t dt
* ~ !
dp
t
m
k 5 (7)
e,m
t 2 t
m11 m
where the thermal conductivity, k, is the proportionality
constant, and p is the space variable through which heat is
6.3 Case 1, Flat Slab Systems:
flowing. For steady-state conditions, one-dimensional heat
6.3.1 From Eq 4, the temperature difference across the mth
flow, and temperature dependent thermal conductivity, the
layer material is:
equation becomes
t 2 t 5 qR (8)
m m11 m
dt
q 5 2k~t! (2)
dp
x 2 x
~ !
m11 m
where R 5
m
k
e,m
where at all surfaces normal to the heat flux, the total heat
flow through these surfaces is the same and changes in the
Note that R is defined as the thermal resistance of the mth
m
thermal conductivity must dictate changes in the temperature
layer of material. Also, for a thermal insulation system of n
gradient. This will ensure that the total heat passing through a
layers, m = 1,2.n, it is assumed that perfect contact exists
given surface does not change from that surface to the next.
between layers. This is essential so that continuity of tempera-
6.2.4 Solutions from Temperature Boundary Conditions—
ture between layers can be assumed.
The temperature boundary conditions on a uniformly thick,
6.3.2 Heat is transferred between the inside and outside
homogeneous mth layer material are:
surfaces of the system and ambient fluids and surrounding
t 5 t at x 5 x ~r 5 r !; (3)
m m m surfaces by the relationships:
q 5 h ~t 2 t ! (9)
t 5 t at x 5 x r 5 r
~ ! i i 1
m11 m11 m
For heat flow in the flat slab, let p = x and integrate Eq 2: q 5 h t 2 t
~ !
o n11 o
C680 − 23a
FIG. 1 Flow Chart
C680 − 23a
where h and h are the inside and outside surface transfer The temperature difference can be defined by Eq 8, where:
i o
conductances. Methods for estimating these conductances are 2
r ~r 2 r !
n11 m11 m
R 5 (17)
found in 6.7. Eq 9 can be rewritten as:
m
k r r
e,m m m11
t 2 t 5 qR (10)
i 1 i
Again, utilizing the methodology presented in case 1 (6.3),
the heat flux, q , and the surface temperature, t , can be found
t 2 t 5 qR
n n+1
n11 o o
by successive iterations. However, one should note that the
1 1
definition of R found in Eq 17 must be substituted for the one
m
where R 5 , R 5
i o
h h
i o presented in Eq 8.
For the computer program, the inside surface transfer
6.6 Calculation of Effective Thermal Conductivity:
conductance, h , is assumed to be very large such that R = 0,
i i
6.6.1 In the calculational methodologies of 6.3, 6.4, and 6.5,
and t = t is the given surface temperature.
1 i
it is necessary to evaluate k as a function of the two surface
e,m
6.3.3 Adding Eq 8 and Eq 10 yields the following equation:
temperatures of each layer comprising the thermal insulating
t 2 t 5 q~R 1R 1…1R 1R 1R ! (11) system. This is accomplished by use of Eq 7 where k(t) is
i o 1 2 n i o
defined as a polynomial function or a piecewise continuous
From the previous equation a value for q can be calculated
function comprised of individual, integrable functions over
from estimated values of the resistances, R. Then, by rewriting
specific temperature ranges. It is important to note that tem-
Eq 8 to the following:
perature can either be in °F (°C) or absolute temperature,
t 5 t 2 qR (12)
m11 m m
because the thermal conductivity versus temperature relation-
ship is regression dependent. It is assumed for the programs in
t 5 t 2 qR , for R .0
1 i i i
this practice that the user regresses the k versus t functions
The temperature at the interface(s) and the outside surface
using °F.
can be calculated starting with m = 1. Next, from the calculated
6.6.1.1 When k(t) is defined as a polynomial function, such
2 3
temperatures, values of k (Eq 7) and R (Eq 8) can be
e,m m
as k(t) = a + bt + ct + dt , the expression for the effective
calculated as well as R and R . Then, by substituting the
o i thermal conductivity is:
calculated R-values back into Eq 11, a new value for q can be
t
m11
calculated. Finally, desired (correct) values can be obtained by 2 3
a1bt1ct 1dt dt
* ~ !
repeating this calculation methodology until all values agree
m
k 5 (18)
e,m
t 2 t
with previous values. ~ !
m11 m
6.4 Case 2, Cylindrical (Pipe) Systems:
b c d
2 2 3 3 4 4
6.4.1 From Eq 5, the heat flux through any layer of material a~t 2 t !1 ~t 2 t !1 ~t 2 t !1 ~t 2 t !
m11 m m11 m m11 m m11 m
2 3 4
k 5
is referenced to the outer radius by the relationship:
e,m
t 2 t
~ !
m11 m
r t 2 t
m m11
q 5 q 5 k (13)
n m e,m b c d
r r ln~r /r ! 2 2 3 2
n11 n11 m11 m
k 5 a1 ~t 1t !1 ~t 1t t 1t !1 ~t 1t t
e,m m m11 m m m11 m11 m m m11
2 3 4
and, the temperature difference can be defined by Eq 8,
2 3
1t t 1t
!
m m11 m11
where:
It should be noted here that for the linear case, c = d = 0, and
r ln~r /r !
n11 m11 m
R 5 (14)
m for the quadratic case, d = 0.
k
e,m
6.6.1.2 When k(t) is defined as an exponential function,
Utilizing the methodology presented in case 1 (6.3), the heat
a+bt
such as k(t) = e , the expression for the effective thermal
flux, q , and the surface temperature, t , can be found by
n n+1
conductivity is:
successive iterations. However, one should note that the
t
m11
definition of R found in Eq 14 must be substituted for the one
m
a1bt
e dt
*
presented in Eq 8.
m
k 5 (19)
6.4.2 For radial heat transfer in pipes, it is customary to e,m
~t 2 t !
m11 m
define the heat flux in terms of the pipe length:
q 5 2πr q (15) a1bt a1bt
p n11 n m11 m
e 2 e
~ !
b
k 5
where q is the time rate of heat flow per unit length of pipe. e,m
p
~t 2 t !
m11 m
If one chooses not to do this, then heat flux based on the
a1bt a1bt
m11 m
e 2 e
interior radius must be reported to avoid the influence of ~ !
k 5
e,m
outer-diameter differences. b t 2 t
~ !
m11 m
6.5 Case 3, Spherical Systems:
6.6.1.3 The piece-wise continuous function may be defined
6.5.1 From Eq 6, the flux through any layer of material is
as:
referenced to the outer radius by the relationship:
k t 5 k t t # t # t (20)
~ ! ~ !
1 bl l
r r r ~t 2 t !
m m11 m m11
q 5 q 5 k (16)
n m 2 e,m 2
r r r 2 r 5 k t t # t # t t # t and t # t
~ ! ~ !
n11 n11 m11 m 2 l u bl m m11 bu
C680 − 23a
5 k t t # t # t where:
~ !
3 u bu
ε = effective surface emittance between outside surface
where t and t are the experimental lower and upper
bl bu
and the ambient surroundings, dimensionless,
boundaries for the function. Also, each function is integrable,
-8
σ = Stefan-Boltzman constant, 0.1714 × 10 Btu/
and k (t ) = k (t ) and k (t ) = k (t ). In terms of the effective
1 l 2 l 2 u 3 u
2 4 -8 2 4
(h·ft ·°R ) (5.6697 × 10 W/(m ·K )),
thermal conductivity, some items must be considered before
T = absolute surface temperature, °R (K),
s
performing the integration in Eq 8. First, it is necessary to
T = absolute surroundings (ambient air if assumed the
o
determine if t is greater than or equal to t . Next, it is
m+1 m
same) temperature, °R (K), and
necessary to determine which temperature range t and t fit
m m+1
T = (T + T )/2
m s o
into. Once these two parameters are decided, the effective
thermal conductivity can be determined using simple calculus. 6.7.3 Convective Heat Transfer Conductance—Certain con-
For example, if t ≤ t ≤ t and t ≤ t ≤ t then the effective ditions need to be identified for proper calculation of this
bl m l u m+1 bu
thermal conductivity would be: component. The conditions are: (a) Surface geometry—plane,
T T t cylinder or sphere; (b) Surface orientation—from vertical to
1 u m+1
horizontal including flow dependency; (c) Nature of heat
k t dt1 k t 1 k t
* ~ ! * ~ ! * ~ !
1 2 3
t T T
m l u
transfer in fluid—from free (natural) convection to forced
k 5 (21)
e,m
t 2 t
~ !
m11 m
convection with variation in the direction and magnitude of
fluid flow; (d) Condition of the surface—from smooth to
It should be noted that other piece-wise functions exist, but
various degrees of roughness (primarily a concern for forced
for brevity, the previous is the only function presented.
convection).
6.6.2 It should also be noted that when the relationship of k
with t is more complex and does not lend itself to simple 6.7.3.1 Modern correlation of the surface transfer conduc-
tances are presented in terms of dimensionless groups, which
mathematical treatment, a numerical method might be used. It
is in these cases that the power of the computer is particularly are defined for fluids in contact with solid surfaces. These
groups are:
useful. There are a wide variety of numerical techniques
available. The most suitable will depend of the particular
H H
h L h D
c c
H H
situation, and the details of the factors affecting the choice are
Nusselt, Nu 5 or Nu 5 (24)
L D
k k
f f
beyond the scope of this practice.
3 3
g·β·ρ·c ~ΔT!L g·β·ρ·c ~ΔT!D
p p
6.7 Surface Transfer Conductance:
Rayleigh, Ra 5 or Ra 5
L D
ν·k ν·k
f f
6.7.1 The surface transfer conductance, h, as defined in
(25)
Terminology C168, assumes that the principal surface is at a
VL VD
uniform temperature and that the ambient fluid and other
Reynolds, Re 5 or Re 5 (26)
L D
ν ν
visible surfaces are at a different uniform temperature. The
conductance includes the combined effects of radiant,
ν·ρ·c
p
Prandtl, Pr 5 (27)
convective, and conductive heat transfer. The conductance is
k
f
defined by:
where:
h 5 h 1h (22)
r c
L = characteristic dimension for horizontal and vertical
where h is the component due to radiation and h is the flat surfaces, and vertical cylinders feet (m), in
r c
component due to convection and conduction. In subsequent
general, denotes height of vertical surface or length of
sections, algorithms for these components will be presented. horizontal surface,
D = characteristic dimension for horizontal cylinders and
6.7.1.1 The algorithms presented in this practice for calcu-
spheres feet (m), in general, denotes the diameter,
lating surface transfer conductances are used in the computer
c = specific heat of ambient fluid, Btu/(lb·°R) (J/(kg·K)),
program; however, surface transfer conductances may be p
¯
h = average convection conductance, Btu/(h·ft ·°F) (W/
estimated from published values or separately calculated from c
(m ·K)),
algorithms other than the ones presented in this practice. One
k = thermal conductivity of ambient fluid, Btu/(h·ft·°F)
f
special note, care must be exercised at low or high surface
(W/(m·K)),
temperatures to ensure reasonable values.
V = free stream velocity of ambient fluid, ft/h (m/s),
6.7.2 Radiant Heat Transfer Conductance—The radiation
2 2
ν = kinematic viscosity of ambient fluid, ft /h (m /s),
conductance is simply based on radiant heat transfer and is
2 2
g = acceleration due to gravity, ft/h (m/s ),
calculated from the Stefan-Boltzmann Law divided by the
β = volumetric thermal expansion coefficient of ambient
average difference between the surface temperature and the air
-1 -1
fluid, °R (K ),
temperature. In other words: 3 3
ρ = density of ambient fluid, lb/ft (kg/m ), and
4 4
σε T 2 T ΔT = absolute value of temperature difference between
~ !
s o
h 5 or (23)
r
T 2 T surface and ambient fluid, °R (K).
s o
3 2 2 3 It needs to be noted here that (except for spheres–forced
h 5 σε·~T 1T T 1T T 1T ! or
r s s o s o o
convection) the above fluid properties must be calculated at the
T 2 T
film temperature, T , which is the average of surface and
s o f
h 5 σε·4T 11
F S D G
r m
T 1T ambient fluid temperatures. For this practice, it is assumed that
s o
C680 − 23a
the ambient fluid is dry air at atmospheric pressure. The 0. Also, it is important to note that the free convection
properties of air can be found in references such as Ref (9). correlations apply to vertical cylinders in most cases.
This reference contains equations for some of the properties
6.7.4.3 For natural convection on horizontal flat surfaces,
and polynomial fits for others, and the equations are summa-
Incropera and Dewitt (p. 498) cite Heat Transmission by
rized in Table A1.1.
McAdams, “Natural Convection Mass Transfer Adjacent to
6.7.3.2 When a heated surface is exposed to flowing fluid, Horizontal Plates” by Goldstein, Sparrow and Jones, and
the convective heat transfer will be a combination of forced “Natural Convection Adjacent to Horizontal Surfaces of Vari-
and free convection. For this mixed convection condition, ous Platforms” for the following correlations:
Churchill (10) recommends the following equation. For each
1/4 4 7
H
Heat flow up: Nu 5 0.54 Ra 10 ,Ra ,10 (34)
n,L L L
geometric shape and surface orientation the overall average
Nusselt number is to be computed from the average Nusselt
H 1/3 7 11
Nu 5 0.15 Ra 10 ,Ra ,10
n,L L L
number for forced convection and the average Nusselt number
for natural convection. The film conductance, h, is then
1/4 5 10
H
Heat flow down: Nu 5 0.27 Ra 10 ,Ra ,10
n,L L L
computed from Eq 24. The relationship is:
j j j
In the case of horizontal flat surfaces, the characteristic
H H H
~Nu 2 δ! 5 ~Nu 2 δ! 1~Nu 2 δ! (28)
f n
dimension, L, is the area of the surface divided by the perimeter
where the exponent, j, and the constant, δ, are defined based of the surface (ft). To compute the overall Nusselt number (Eq
on the geometry and orientation. 28), set j = 3.5 and δ = 0.
6.7.3.3 Once the Nusselt number has been calculated, the 6.7.5 Convection Conductances for Horizontal Cylinders:
surface transfer conductance is calculated from a rearrange-
6.7.5.1 For forced convection with fluid flow normal to a
ment of Eq 24:
circular cylinder, Incropera and Dewitt (p. 370) cite Heat
Transfer by Churchill and Bernstein for the following correla-
H
h 5 Nu ·k /L (29)
c L f
tion:
H H 1/2 1/3 5/8 4/5
h 5 Nu ·k /D 0.62Re Pr Re
c D f D D
H
Nu 5 0.31 11 (35)
F S D G
f,D 2/3 1/4
11 0.4/Pr 282 000
@ ~ ! #
where L and D are the characteristic dimension of the
system. The term k is the thermal conductivity of air deter-
All Re ·Pr.0.2
a
D
mined at the film temperature using the equation in Table A1.1.
In the case of horizontal cylinders, the characteristic
6.7.4 Convection Conductances for Flat Surfaces:
dimension, D, is the diameter of the cylinder, (ft). In addition,
6.7.4.1 From Heat Transfer by Churchill and Ozoe as cited
this correlation should be used for forced convection from
in Fundamentals of Heat and Mass Transfer by Incropera and
vertical pipes.
Dewitt, the relation for forced convection by laminar flow over
6.7.5.2 For natural convection on horizontal cylinders, In-
an isothermal flat surface is:
cropera and Dewitt (p. 502) cite “Correlating Equations for
1/2 1/3
0.6774 Re Pr
L
Laminar and Turbulent Free Convection from a Horizontal
H 5
Nu 5 Re ,5 × 10 (30)
f,L 2/3 1/4 L
@11~0.0468/Pr! #
Cylinder” by Churchill and Chu for the following correlation:
1/6 2
For forced convection by turbulent flow over an isothermal 0.387Ra
D
H 12
Nu 5 0.601 Ra ,10 (36)
H J
n,D 9/16 8/27 D
flat surface, Incropera and Dewitt suggest the following: @11~0.559/Pr! #
H 4/5 1/3 5 8
Nu 5 ~0.037 Re 2 871! Pr 5 × 10 ,Re ,10 (31) To compute the overall Nusselt number using Eq 28, set j =
f,L L L
It should be noted that the upper bound for Re is an
4 and δ = 0.3.
L
approximate value, and the user of the above equation must be
6.7.6 Convection Conductances for Spheres:
aware of this.
6.7.6.1 For forced convection on spheres, Incropera and
DeWitt cite S. Whitaker in AIChE J. for the following
6.7.4.2 In “Correlating Equations for Laminar and Turbu-
correlation:
lent Free Convection from a Vertical Plate” by Churchill and
1/4
Chu, as cited by Incropera and Dewitt, it is suggested for
μ
H 1/2 2/3 0.4
Nu 5 21 0.4 Re 10.06 Re Pr (37)
~ ! S D
f,D D D
natural convection on isothermal, vertical flat surfaces that:
μ
s
1/6 2
0.387 Ra
L
H
0.71,Pr,380
Nu 5 0.8251 All Ra (32)
H J
n,L 9/16 8/27 L
@11~0.492/Pr! #
3.5,Re ,7.6 × 10
D
For slightly better accuracy in the laminar range, it is
suggested by the same source (p. 493) that:
1.0, μ/μ ,3.2
~ !
s
1/4
0.670 Ra
L
H 9
Nu 5 0.681 Ra ,10 (33) where μ and μ are the free stream and surface viscosities of
n,L 9/16 4/9 L s
@11~0.492/Pr! #
the ambient fluid respectively. It is extremely important to note
In the case of both vertical flat and cylindrical surfaces the that all properties need to be evaluated based on the free stream
characteristic dimension, L or D, is the vertical height (ft). To temperature of the ambient fluid, except for μ , which needs to
s
compute the overall Nusselt number (Eq 28), set j = 3 and δ = be evaluated based on the surface temperature.
C680 − 23a
6.7.6.2 For natural convection on spheres, Incropera and 7.4.2 The input for the thermal conductivity versus mean
DeWitt cite “Free Convection Around Immersed Bodies” by S. temperature parameters must be obtained as outlined in 6.6.
W. Churchill in Heat Exchange Design Handbook (Schlunder) The type code determines the thermal conductivity versus
for the following correlation: temperature relationship applying to the insulation. The same
type code may be used for more than one insulation. As
1/4
0.589 Ra
D
H
Nu 5 21 (38)
9/16 4/9 presented, the programs will operate on three functional
n,D
@11~0.469/Pr! #
relationships:
0.7 # Pr
Type Functional Relationship
Quadratic k = a + bt + ct
Ra ,10
D
where a, b, and c are constants
where all properties are evaluated at the film temperature. To
Linear k = a + b t; t < t
1 1 L
compute the overall Nusselt number for spheres (Eq 28) set j =
k = a + b t; t < t < t
2 2 L U
4 and δ = 2.
k = a + b t; t > t
3 3 U
where a1, a2, a3, b1, b2, b3 are constants, and
t and t are, respectively, the lower and upper
L U
7. Computer Program
inflection points of an S-shaped curve
7.1 General:
Additional or different relationships may be used, but the main
7.1.1 The computer program(s) can be written in any
program must be modified.
language, but Appendix X3 is written in C#.
7.1.2 The program consists of a main program that utilizes
8. Report
several subroutines. Other subroutines may be added to make
the program more applicable to the specific problems of
8.1 The results of calculations performed in accordance
individual users. with this practice may be used as design data for specific job
conditions, or may be used in general form to represent the
7.2 Functional Description of Program—The flow chart
performance of a particular product or system. When the
shown in Fig. 1 is a schematic representations of the opera-
results will be used for comparison of performance of similar
tional procedures for each coordinate system covered by the
products, it is recommended that reference be made to the
program. The flow chart presents the logic path for entering
specific constants used in the calculations. These references
data, calculating and recalculating system thermal resistances
should include:
and temperatures, relaxing the successive errors in the tem-
8.1.1 Name and other identification of products or
perature to within 0.1° of the temperature, calculating heat loss
components,
or gain for the system and printing the parameters and solution
8.1.2 Identification of the nominal pipe size or surface
in tabular form.
insulated, and its geometric orientation,
7.3 Computer Program Variable Descriptions—The de-
8.1.3 The surface temperature of the pipe or surface,
scription of all variables used in the programs are given in the
8.1.4 The equations and constants selected for the thermal
listing of the program as comments.
conductivity versus mean temperature relationship,
7.4 Program Operation: 8.1.5 The ambient temperature and humidity, if applicable,
7.4.1 Log on procedures and any executive program for 8.1.6 The surface transfer conductance and condition of
execution of this program must be followed as needed. surface heat transfer,
FIG. 2 Thermal Conductivity vs. Mean Temperature
C680 − 23a
FIG. 3 Mean Temperature vs. Thermal Conductivity
FIG. 4 Thermal Conductivity vs. Mean Temperature
ASTM terminology on Precision and Bias.
8.1.6.1 If obtained from published information, the source
and limitations,
9.2 Many factors influence the accuracy of a calculative
8.1.6.2 If calculated or measured, the method and signifi-
procedure used for predicting heat flux results. These factors
cant parameters such as emittance, fluid velocity, etc.,
include accuracy of input data and the applicability of the
8.1.7 The resulting outer surface temperature, and
assumptions used in the method for the system under study.
8.1.8 The resulting heat loss or gain.
The system of mathematical equations used in this analysis has
been accepted as applicable for most systems normally insu-
8.2 Either tabular or graphical representation of the calcu-
lated results may be used. No recommendation is made for the lated with bulk type insulations. Applicability of this practice
to systems having irregular shapes, discontinuities and other
format in which results are presented.
variations from the one-dimensional heat transfer assumptions
9. Accuracy and Resolution
should be handled on an individual basis by professional
engineers familiar with those systems.
9.1 In many typical computers normally used, seven signifi-
cant digits are resident in the computer for calculations.
9.3 The computer resolution effect on accuracy is only
Adjustments to this level can be made through the use of
significant if the level of precision is less than that discussed in
“Double Precision;” however, for the intended purpose of this
9.1. Computers in use today are accurate in that they will
practice, standard levels of precision are adequate. The format-
reproduce the calculated results to resolution required if
ting of the output results, however, should be structured to
identical input data is used.
provide a resolution of 0.1 % for the typical expected levels of
9.4 The most significant factor influencing the accuracy of
heat flux and a resolution of 1°F (0.55°C) for surface tempera-
claims is the accuracy of the input thermal conductivity data.
tures.
NOTE 1—The term “double precision” should not be confused with The accuracy of applicability of these data is derived from two
C680 − 23a
n 1/2
factors. The first is the accuracy of the test method used to
S ] R
5 Δx (39)
S SS D D D
( i
generate the data. Since the test methods used to supply these R ] x
i51 i
data are typically Test Methods C177, C335, or C518, the
where:
reports should contain some statement of the estimates of error
S = estimate of the probable error of the procedure,
or estimates of uncertainty. The remaining factors influencing
R = result of the procedure,
the accuracy are the inherent variability of the product and the
x = ith variable in procedure,
i
variability of the installation practices. If the product variabil-
∂R/∂x = change in result with respect to change in ith
i
ity is large, the installation is poor, or both, serious differences
variable,
might exist between measured performance and predicted
Δx = uncertainty in value of variable, i, and
i
performance from this practice.
n = total number of variables in procedure.
10. Precision and Bias
10.2 ASTM Subcommittee C16.30, Task Group 5.2, which
is responsible for preparing this practice, has prepared Appen-
10.1 When concern exists with the accuracy of the input test
dix X1. The appendix provides a more complete discussion of
data, the recommended practice to evaluate the impact of
the precision and bias expected when using Practice C680 in
possible errors is to repeat the calculation for the range of the
the analysis of operating systems. While much of that discus-
uncertainty of the variable. This process yields a range in the
sion is relevant to this practice, the errors associated with its
desired output variable for a given uncertainty in the input
application to operating systems are beyond the primary
variable. Repeating this procedure for all the input variables
Practice C680 scope. Portions of this discussion, however,
would yield a measure of the contribution of each to the overall
were used in developing the Precision and Bias statements
uncertainty. Several methods exist for the combination of these
included in Section 10.
effects; however, the most commonly used is to take the square
root of the sum of the squares of the percentage errors induced
11. Keywords
by each variable’s uncertainty. Eq 39 from Theories of Engi-
neering Experimentation by H. Schenck gives the expression 11.1 computer program; heat flow; heat gain; heat loss;
in mathematical form: pipe; thermal insulation
ANNEX
(Mandatory Information)
A1. EQUATIONS DERIVED FROM THE NIST CIRCULAR 564
A1.1 Table A1.1 lists the equations derived from NBS
Circular 564 for the determination of the properties of air as
used in this practice.
A1.2 T is temperature in degrees Kelvin, T is temperature
k f
in degrees Farenheit.
C680 − 23a
TABLE A1.1 Equations and Polynomial Fits for the Properties of Air Between −100ºF and 1300ºF
(NBS Circular 564, Department of Commerce [1960])
Property Equation Units
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: C680 − 23 C680 − 23a
Standard Practice for
Estimate of the Heat Gain or Loss and the Surface
Temperatures of Insulated Flat, Cylindrical, and Spherical
Systems by Use of Computer Programs
This standard is issued under the fixed designation C680; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last reapproval. A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 This practice provides the algorithms and calculation methodologies for predicting the heat loss or gain and surface
temperatures of certain thermal insulation systems that can attain one dimensional, steady- or quasi-steady-state heat transfer
conditions in field operations.
1.2 This practice is based on the assumption that the thermal insulation systems can be well defined in rectangular, cylindrical or
spherical coordinate systems and that the insulation systems are composed of homogeneous, uniformly dimensioned materials that
reduce heat flow between two different temperature conditions.
1.3 Qualified personnel familiar with insulation-systems design and analysis should resolve the applicability of the methodologies
to real systems. The range and quality of the physical and thermal property data of the materials comprising the thermal insulation
system limit the calculation accuracy. Persons using this practice must have a knowledge of the practical application of heat
transfer theory relating to thermal insulation materials and systems.
1.4 The computer program that can be generated from the algorithms and computational methodologies defined in this practice
is described in Section 7 of this practice. The computer program is intended for flat slab, pipe and hollow sphere insulation systems.
1.5 The values stated in inch-pound units are to be regarded as standard. The values given in parentheses are mathematical
conversions to SI units that are provided for information only and are not considered standard.
1.6 This standard does not purport to address all of the safety concerns, if any, associated with its use. It is the responsibility
of the user of this standard to establish appropriate safety, health, and environmental practices and determine the applicability of
regulatory limitations prior to use.
1.7 This international standard was developed in accordance with internationally recognized principles on standardization
established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued
by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
2. Referenced Documents
2.1 ASTM Standards:
This practice is under the jurisdiction of ASTM Committee C16 on Thermal Insulation and is the direct responsibility of Subcommittee C16.30 on Thermal Measurement.
Current edition approved March 1, 2023Nov. 1, 2023. Published March 2023December 2023. Originally approved in 1971. Last previous edition approved in 20192023
as C680 – 19.C680 – 23. DOI: 10.1520/C0680-23.10.1520/C0680-23A.
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
C680 − 23a
C168 Terminology Relating to Thermal Insulation
C177 Test Method for Steady-State Heat Flux Measurements and Thermal Transmission Properties by Means of the
Guarded-Hot-Plate Apparatus
C335 Test Method for Steady-State Heat Transfer Properties of Pipe Insulation
C518 Test Method for Steady-State Thermal Transmission Properties by Means of the Heat Flow Meter Apparatus
C585 Practice for Inner and Outer Diameters of Thermal Insulation for Nominal Sizes of Pipe and Tubing
C1055 Guide for Heated System Surface Conditions that Produce Contact Burn Injuries
C1057 Practice for Determination of Skin Contact Temperature from Heated Surfaces Using a Mathematical Model and
Thermesthesiometer
2.2 Other Document:
NBS Circular 564 Tables of Thermodynamic and Transport Properties of Air, U.S. Dept of Commerce
3. Terminology
3.1 Definitions:
3.1.1 For definitions of terms used in this practice, refer to Terminology C168.
3.1.2 thermal insulation system—for this practice, a thermal insulation system is a system comprised of a single layer or layers
of homogeneous, uniformly dimensioned material(s) intended for reduction of heat transfer between two different temperature
conditions. Heat transfer in the system is steady-state. Heat flow for a flat system is normal to the flat surface, and heat flow for
cylindrical and spherical systems is radial.
3.2 Symbols:
3.2.1 The following symbols are used in the development of the equations for this practice. Other symbols will be introduced and
defined in the detailed description of the development.
where:
2 2
h = surface transfer conductance, Btu/(h·ft ·°F) (W/(m ·K)) h at inside surface; h at outside surface
i o
k = apparent thermal conductivity, Btu·in./(h·ft ·°F) (W/(m·K))
k = effective thermal conductivity over a prescribed temperature range, Btu·in./(h·ft ·°F) (W/(m·K))
e
2 2
q = heat flux, Btu/(h·ft ) (W/m )
q = time rate of heat flow per unit length of pipe, Btu/(h·ft) (W/m)
p
2 2
R = thermal resistance, °F·h·ft /Btu (K·m /W)
r = radius, in. (m); r − r = thickness
m+1 m
t = local temperature, °F (K)
t = inner surface temperature of the insulation, °F (K)
i
t = inner surface temperature of the system
t = temperature of ambient fluid and surroundings, °F (K)
o
x = distance, in. (m); x − x = thickness
m+1 m
ε = effective surface emittance between outside surface and the ambient surroundings, dimensionless
-8 2 4 -8 2 4
σ = Stefan-Boltzmann constant, 0.1714 × 10 Btu/(h·ft ·°R ) (5.6697 × 10 W/(m ·K ))
T = absolute surface temperature, °R (K)
s
T = absolute surroundings (ambient air if assumed the same) temperature, °R (K)
o
T = (T + T )/2
m s o
L = characteristic dimension for horizontal and vertical flat surfaces, and vertical cylinders
D = characteristic dimension for horizontal cylinders and spheres
c = specific heat of ambient fluid, Btu/(lb·°R) (J/(kg·K))
p
2 2
h = average convection conductance, Btu/(h·ft ·°F) (W/(m ·K))
c
k = thermal conductivity of ambient fluid, Btu/(h·ft·°F) (W/(m·K))
f
V = free stream velocity of ambient fluid, ft/h (m/s)
2 2
υ = kinematic viscosity of ambient fluid, ft /h (m /s)
2 2
g = acceleration due to gravity, ft/h (m ⁄s )
-1 -1
β = volumetric thermal expansion coefficient of ambient fluid, °R (K )
3 3
ρ = density of ambient fluid, lb/ft (kg ⁄m )
ΔT = absolute value of temperature difference between surface and ambient fluid, °R (K)
Nu = Nusselt number, dimensionless
Ra = Rayleith number, dimensionless
Re = Reynolds number, dimensionless
Pr = Prandtl number, dimensionless
C680 − 23a
4. Summary of Practice
4.1 The procedures used in this practice are based on standard, steady-state, one dimensional, conduction heat transfer theory as
outlined in textbooks and handbooks, Refs (1,2,3,4,5,6). Heat flux solutions are derived for temperature dependent thermal
conductivity in a material. Algorithms and computational methodologies for predicting heat loss or gain of single or multi-layer
thermal insulation systems are provided by this practice for implementation in a computer program. In addition, interested parties
can develop computer programs from the computational procedures for specific applications and for one or more of the three
coordinate systems considered in Section 6.
4.1.1 The computer program combines functions of data input, analysis and data output into an easy to use, interactive computer
program. By making the program interactive, little training for operators is needed to perform accurate calculations.
4.2 The operation of the computer program follows the procedure listed below:
4.2.1 Data Input—The computer requests and the operator inputs information that describes the system and operating
environment. The data includes:
4.2.1.1 Analysis identification.
4.2.1.2 Date.
4.2.1.3 Ambient temperature.
4.2.1.4 Surface transfer conductance or ambient wind speed, system surface emittance and system orientation.
4.2.1.5 System Description—Material and thickness for each layer (define sequence from inside out).
4.2.2 Analysis—Once input data is entered, the program calculates the surface transfer conductances (if not entered directly) and
layer thermal resistances. The program then uses this information to calculate the heat transfer and surface temperature. The
program continues to repeat the analysis using the previous temperature data to update the estimates of layer thermal resistance
until the temperatures at each surface repeat within 0.1°F between the previous and present temperatures at the various surface
locations in the system.
4.2.3 Program Output—Once convergence of the temperatures is reached, the program prints a table that presents the input data,
calculated thermal resistance of the system, heat flux and the inner surface and external surface temperatures.
5. Significance and Use
5.1 Manufacturers of thermal insulation express the performance of their products in charts and tables showing heat gain or loss
per unit surface area or unit length of pipe. This data is presented for typical insulation thicknesses, operating temperatures, surface
orientations (facing up, down, horizontal, vertical), and in the case of pipes, different pipe sizes. The exterior surface temperature
of the insulation is often shown to provide information on personnel protection or surface condensation. However, additional
information on effects of wind velocity, jacket emittance, ambient conditions and other influential parameters may also be required
to properly select an insulation system. Due to the large number of combinations of size, temperature, humidity, thickness, jacket
properties, surface emittance, orientation, and ambient conditions, it is not practical to publish data for each possible case, Refs
(7,8).
5.2 Users of thermal insulation faced with the problem of designing large thermal insulation systems encounter substantial
engineering cost to obtain the required information. This cost can be substantially reduced by the use of accurate engineering data
tables, or available computer analysis tools, or both. The use of this practice by both manufacturers and users of thermal insulation
will provide standardized engineering data of sufficient accuracy for predicting thermal insulation system performance. However,
it is important to note that the accuracy of results is extremely dependent on the accuracy of the input data. Certain applications
may need specific data to produce meaningful results.
5.3 The use of analysis procedures described in this practice can also apply to designed or existing systems. In the rectangular
coordinate system, Practice C680 can be applied to heat flows normal to flat, horizontal or vertical surfaces for all types of
enclosures, such as boilers, furnaces, refrigerated chambers and building envelopes. In the cylindrical coordinate system, Practice
C680 − 23a
C680 can be applied to radial heat flows for all types of piping circuits. In the spherical coordinate system, Practice C680 can be
applied to radial heat flows to or from stored fluids such as liquefied natural gas (LNG).
5.4 Practice C680 is referenced for use with Guide C1055 and Practice C1057 for burn hazard evaluation for heated surfaces.
Infrared inspection, in-situ heat flux measurements, or both are often used in conjunction with Practice C680 to evaluate insulation
system performance and durability of operating systems. This type of analysis is often made prior to system upgrades or
replacements.
5.5 All porous and non-porous solids of natural or man-made origin have temperature dependent thermal conductivities. The
change in thermal conductivity with temperature is different for different materials, and for operation at a relatively small
temperature difference, an average thermal conductivity may suffice. Thermal insulating materials (k < 0.85 {Btu·in}/{h·ft ·°F})
are porous solids where the heat transfer modes include conduction in series and parallel flow through the matrix of solid and
gaseous portions, radiant heat exchange between the surfaces of the pores or interstices, as well as transmission through
non-opaque surfaces, and to a lesser extent, convection within and between the gaseous portions. With the existence of radiation
and convection modes of heat transfer, the measured value should be called apparent thermal conductivity as described in
Terminology C168. The main reason for this is that the premise for pure heat conduction is no longer valid, because the other
modes of heat transfer obey different laws. Also, phase change of a gas, liquid, or solid within a solid matrix or phase change by
other mechanisms will provide abrupt changes in the temperature dependence of thermal conductivity. For example, the
condensation of the gaseous portions of thermal insulation in extremely cold conditions will have an extremely influential effect
on the apparent thermal conductivity of the insulation. With all of this considered, the use of a single value of thermal conductivity
at an arithmetic mean temperature will provide less accurate predictions, especially when bridging temperature regions where
strong temperature dependence occurs.
5.6 The calculation of surface temperature and heat loss or gain of an insulated system is mathematically complex, and because
of the iterative nature of the method, computers best handle the calculation. Computers are readily available to most producers and
consumers of thermal insulation to permit the use of this practice.
5.7 Computer programs are described in this practice as a guide for calculation of the heat loss or gain and surface temperatures
of insulation systems. The range of application of these programs and the reliability of the output is a primary function of the range
and quality of the input data. The programs are intended for use with an “interactive” terminal. Under this system, intermediate
output guides the user to make programming adjustments to the input parameters as necessary. The computer controls the terminal
interactively with program-generated instructions and questions, which prompts user response. This facilitates problem solution
and increases the probability of successful computer runs.
5.8 The user of this practice may wish to modify the data input and report sections of the computer programs presented in this
practice to fit individual needs. Also, additional calculations may be desired to include other data such as system costs or economic
thickness. No conflict exists with such modifications as long as the user verifies the modifications using a series of test cases that
cover the range for which the new method is to be used. For each test case, the results for heat flow and surface temperature must
be identical (within resolution of the method) to those obtained using the practice described herein.
5.9 This practice has been prepared to provide input and output data that conforms to the system of units commonly used by
United States industry. Although modification of the input/output routines could provide an SI equivalent of the heat flow results,
no such “metric” equivalent is available for some portions of this practice. To date, there is no accepted system of metric
dimensions for pipe and insulation systems for cylindrical shapes. The dimensions used in Europe are the SI equivalents of
American sizes (based on Practice C585), and each has a different designation in each country. Therefore, no SI version of the
practice has been prepared, because a standard SI equivalent of this practice would be complex. When an international standard
for piping and insulation sizing occurs, this practice can be rewritten to meet those needs. In addition, it has been demonstrated
that this practice can be used to calculate heat transfer for circumstances other than insulated systems; however, these calculations
are beyond the scope of this practice.
6. Method of Calculation
6.1 Approach:
6.1.1 The calculation of heat gain or loss and surface temperature requires: (1) The thermal insulation is homogeneous as outlined
by the definition of thermal conductivity in Terminology C168; (2) the system operating temperature is known; (3) the insulation
C680 − 23a
thickness is known; (4) the surface transfer conductance of the system is known, reasonably estimated or estimated from algorithms
defined in this practice based on sufficient information; and, (5) the thermal conductivity as a function of temperature for each
system layer is known in detail.
6.1.2 The solution is a procedure calling for (1) estimation of the system temperature distribution; (2) calculation of the thermal
resistances throughout the system based on that distribution; (3) calculation of heat flux; and (4) reestimation of the system
temperature distribution. The iterative process continues until a calculated distribution is in reasonable agreement with the previous
distribution. This is shown diagrammatically in Fig. 1. The layer thermal resistance is calculated each time with the effective
thermal conductivity being obtained by integration of the thermal conductivity curve for the layer being considered. This practice
uses the temperature dependence of the thermal conductivity of any insulation or multiple layer combination of insulations to
calculate heat flow.
6.2 Development of Equations—The development of the mathematical equations is for conduction heat transfer through
homogeneous solids having temperature dependent thermal conductivities. To proceed with the development, several precepts or
guidelines must be cited:
6.2.1 Steady-state Heat Transfer—For all the equations it is assumed that the temperature at any point or position in the solid is
invariant with time. Thus, heat is transferred solely by temperature difference from point to point in the solid.
6.2.2 One-dimensional Heat Transfer—For all equations it is assumed there is heat flow in only one dimension of the particular
coordinate system being considered. Heat transfer in the other dimensions of the particular coordinate system is considered to be
zero.
6.2.3 Conduction Heat Transfer—The premise here is that the heat flux normal to any surface is directly proportional to the
temperature gradient in the direction of heat flow, or
dt
q 52k (1)
dp
where the thermal conductivity, k, is the proportionality constant, and p is the space variable through which heat is flowing. For
steady-state conditions, one-dimensional heat flow, and temperature dependent thermal conductivity, the equation becomes
dt
q 52k~t! (2)
dp
where at all surfaces normal to the heat flux, the total heat flow through these surfaces is the same and changes in the thermal
conductivity must dictate changes in the temperature gradient. This will ensure that the total heat passing through a given surface
does not change from that surface to the next.
6.2.4 Solutions from Temperature Boundary Conditions—The temperature boundary conditions on a uniformly thick, homoge-
neous mth layer material are:
t 5 t at x 5 x ~r 5 r !; (3)
m m m
t 5 t at x 5 x r 5 r
~ !
m11 m11 m
For heat flow in the flat slab, let p = x and integrate Eq 2:
x t
m+1 m+1
q dx 52 k t dt (4)
* * ~ !
x t
m m
t 2 t
m m11
q 5 k
e,m
x 2 x
m11 m
For heat flow in the hollow cylinder, let p = r,q = Q/(2πrl) and integrate Eq 2:
C680 − 23a
FIG. 1 Flow Chart
C680 − 23a
r t
m+1 m+1
Q dr
52 k t dt (5)
* * ~ !
2πl r
r t
m m
t 2 t
m m11
Q 5 k 2πl
e,m
ln r /r
~ !
m11 m
Divide both sides by 2πrl
t 2 t
m m11
q 5 k
e,m
rln r /r
~ !
m11 m
For radial heat flow in the hollow sphere, let p = r,q = Q/(4πr ) and integrate Eq 2:
r t
m+1 m+1
Q dr
* 5 * k~t!dt (6)
4π r
r t
m m
t 2 t
m m11
Q 5 k 4π
e,m
1 1
r r
m m11
Divide both sides by 4πr and multiply both sides by r r /r r
m m11 m m11
r r t 2 t
m m11 m m11
q 5 k
e,m 2
r r 2 r
m11 m
Note that the effective thermal conductivity over the temperature range is:
t
m+1
* k~t!dt
t
m
k 5 (7)
e,m
t 2 t
m11 m
6.3 Case 1, Flat Slab Systems:
6.3.1 From Eq 4, the temperature difference across the mth layer material is:
t 2 t 5 qR (8)
m m11 m
x 2 x
~ !
m11 m
where R 5
m
k
e,m
Note that R is defined as the thermal resistance of the mth layer of material. Also, for a thermal insulation system of n layers,
m
m = 1,2.n, it is assumed that perfect contact exists between layers. This is essential so that continuity of temperature between
layers can be assumed.
6.3.2 Heat is transferred between the inside and outside surfaces of the system and ambient fluids and surrounding surfaces by the
relationships:
q 5 h ~t 2 t ! (9)
i i 1
q 5 h t 2 t
~ !
o n11 o
where h and h are the inside and outside surface transfer conductances. Methods for estimating these conductances are found
i o
in 6.7. Eq 9 can be rewritten as:
t 2 t 5 qR (10)
i 1 i
t 2 t 5 qR
n11 o o
1 1
where R 5 , R 5
i o
h h
i o
For the computer program, the inside surface transfer conductance, h , is assumed to be very large such that R = 0, and t = t
i i 1 i
is the given surface temperature.
6.3.3 Adding Eq 8 and Eq 10 yields the following equation:
C680 − 23a
t 2 t 5 q R 1R 1…1R 1R 1R (11)
~ !
i o 1 2 n i o
Eq
From the previous equation a value for q can be calculated from estimated values of the resistances, R. Then, by rewriting
8 to the following:
t 5 t 2 qR (12)
m11 m m
t 5 t 2 qR , for R .0
1 i i i
The temperature at the interface(s) and the outside surface can be calculated starting with m = 1. Next, from the calculated
temperatures, values of k (Eq 7) and R (Eq 8) can be calculated as well as R and R . Then, by substituting the calculated
e,m m o i
R-values back into Eq 11, a new value for q can be calculated. Finally, desired (correct) values can be obtained by repeating this
calculation methodology until all values agree with previous values.
6.4 Case 2, Cylindrical (Pipe) Systems:
6.4.1 From Eq 5, the heat flux through any layer of material is referenced to the outer radius by the relationship:
r t 2 t
m m11
q 5 q 5 k (13)
n m e,m
r r ln~r /r !
n11 n11 m11 m
and, the temperature difference can be defined by Eq 8, where:
r ln r /r
~ !
n11 m11 m
R 5 (14)
m
k
e,m
Utilizing the methodology presented in case 1 (6.3), the heat flux, q , and the surface temperature, t , can be found by
n n+1
successive iterations. However, one should note that the definition of R found in Eq 14 must be substituted for the one presented
m
in Eq 8.
6.4.2 For radial heat transfer in pipes, it is customary to define the heat flux in terms of the pipe length:
q 5 2πr q (15)
p n11 n
where q is the time rate of heat flow per unit length of pipe. If one chooses not to do this, then heat flux based on the interior
p
radius must be reported to avoid the influence of outer-diameter differences.
6.5 Case 3, Spherical Systems:
6.5.1 From Eq 6, the flux through any layer of material is referenced to the outer radius by the relationship:
r r r t 2 t
~ !
m m11 m m11
q 5 q 5 k (16)
n m 2 e,m 2
r r ~r 2 r !
n11 n11 m11 m
The temperature difference can be defined by Eq 8, where:
r r 2 r
~ !
n11 m11 m
R 5 (17)
m
k r r
e,m m m11
Again, utilizing the methodology presented in case 1 (6.3), the heat flux, q , and the surface temperature, t , can be found by
n n+1
successive iterations. However, one should note that the definition of R found in Eq 17 must be substituted for the one presented
m
in Eq 8.
6.6 Calculation of Effective Thermal Conductivity:
6.6.1 In the calculational methodologies of 6.3, 6.4, and 6.5, it is necessary to evaluate k as a function of the two surface
e,m
temperatures of each layer comprising the thermal insulating system. This is accomplished by use of Eq 7 where k(t) is defined
as a polynomial function or a piecewise continuous function comprised of individual, integrable functions over specific
temperature ranges. It is important to note that temperature can either be in °F (°C) or absolute temperature, because the thermal
conductivity versus temperature relationship is regression dependent. It is assumed for the programs in this practice that the user
regresses the k versus t functions using °F.
2 3
6.6.1.1 When k(t) is defined as a polynomial function, such as k(t) = a + bt + ct + dt , the expression for the effective thermal
conductivity is:
C680 − 23a
t
m11
2 3
a1bt1ct 1dt dt
* ~ !
m
k 5 (18)
e,m
t 2 t
~ !
m11 m
b c d
2 2 3 3 4 4
a~t 2 t !1 ~t 2 t !1 ~t 2 t !1 ~t 2 t !
m11 m m11 m m11 m m11 m
2 3 4
k 5
e,m
t 2 t
~ !
m11 m
b c d
2 2 3 2 2 3
k 5 a1 ~t 1t !1 ~t 1t t 1t !1 ~t 1t t 1t t 1t !
e,m m m11 m m m11 m11 m m m11 m m11 m11
2 3 4
It should be noted here that for the linear case, c = d = 0, and for the quadratic case, d = 0.
a+bt
6.6.1.2 When k(t) is defined as an exponential function, such as k(t) = e , the expression for the effective thermal conductivity
is:
t
m11
a1bt
e dt
*
m
k 5 (19)
e,m
t 2 t
~ !
m11 m
a1bt a1bt
m11 m
e 2 e
~ !
b
k 5
e,m
~t 2 t !
m11 m
a1bt a1bt
m11 m
~e 2 e !
k 5
e,m
b t 2 t
~ !
m11 m
6.6.1.3 The piece-wise continuous function may be defined as:
k t 5 k t t # t # t (20)
~ ! ~ !
1 bl l
5 k t t # t # t t # t and t # t
~ !
2 l u bl m m11 bu
5 k ~t! t # t # t
3 u bu
where t and t are the experimental lower and upper boundaries for the function. Also, each function is integrable, and k (t )
bl bu 1 l
= k (t ) and k (t ) = k (t ). In terms of the effective thermal conductivity, some items must be considered before performing the
2 l 2 u 3 u
integration in Eq 8. First, it is necessary to determine if t is greater than or equal to t . Next, it is necessary to determine which
m+1 m
temperature range t and t fit into. Once these two parameters are decided, the effective thermal conductivity can be determined
m m+1
using simple calculus. For example, if t ≤ t ≤ t and t ≤ t ≤ t then the effective thermal conductivity would be:
bl m l u m+1 bu
T T t
1 u m+1
* k ~t!dt1* k ~t!1* k ~t!
1 2 3
t T T
m l u
k 5 (21)
e,m
t 2 t
~ !
m11 m
It should be noted that other piece-wise functions exist, but for brevity, the previous is the only function presented.
6.6.2 It should also be noted that when the relationship of k with t is more complex and does not lend itself to simple mathematical
treatment, a numerical method might be used. It is in these cases that the power of the computer is particularly useful. There are
a wide variety of numerical techniques available. The most suitable will depend of the particular situation, and the details of the
factors affecting the choice are beyond the scope of this practice.
6.7 Surface Transfer Conductance:
6.7.1 The surface transfer conductance, h, as defined in Terminology C168, assumes that the principal surface is at a uniform
temperature and that the ambient fluid and other visible surfaces are at a different uniform temperature. The conductance includes
the combined effects of radiant, convective, and conductive heat transfer. The conductance is defined by:
h 5 h 1h (22)
r c
where h is the component due to radiation and h is the component due to convection and conduction. In subsequent sections,
r c
algorithms for these components will be presented.
C680 − 23a
6.7.1.1 The algorithms presented in this practice for calculating surface transfer conductances are used in the computer program;
however, surface transfer conductances may be estimated from published values or separately calculated from algorithms other
than the ones presented in this practice. One special note, care must be exercised at low or high surface temperatures to ensure
reasonable values.
6.7.2 Radiant Heat Transfer Conductance—The radiation conductance is simply based on radiant heat transfer and is calculated
from the Stefan-Boltzmann Law divided by the average difference between the surface temperature and the air temperature. In
other words:
4 4
σε T 2 T
~ !
s o
h 5 or (23)
r
T 2 T
s o
3 2 2 3
h 5 σε· T 1T T 1T T 1T or
~ !
r s s o s o o
T 2 T
s o
h 5 σε·4T 11
F S D G
r m
T 1T
s o
where:
ε = effective surface emittance between outside surface and the ambient surroundings, dimensionless,
-8 2 4 -8 2 4
σ = Stefan-Boltzman constant, 0.1714 × 10 Btu/(h·ft ·°R ) (5.6697 × 10 W/(m ·K )),
T = absolute surface temperature, °R (K),
s
T = absolute surroundings (ambient air if assumed the same) temperature, °R (K), and
o
T = (T + T )/2
m s o
6.7.3 Convective Heat Transfer Conductance—Certain conditions need to be identified for proper calculation of this component.
The conditions are: (a) Surface geometry—plane, cylinder or sphere; (b) Surface orientation—from vertical to horizontal including
flow dependency; (c) Nature of heat transfer in fluid—from free (natural) convection to forced convection with variation in the
direction and magnitude of fluid flow; (d) Condition of the surface—from smooth to various degrees of roughness (primarily a
concern for forced convection).
6.7.3.1 Modern correlation of the surface transfer conductances are presented in terms of dimensionless groups, which are defined
for fluids in contact with solid surfaces. These groups are:
H H
h L h D
c c
H H
Nusselt, Nu 5 or Nu 5 (24)
L D
k k
f f
3 3
g·β·ρ·c ΔT L g·β·ρ·c ΔT D
~ ! ~ !
p p
Rayleigh, Ra 5 or Ra 5 (25)
L D
ν·k ν·k
f f
VL VD
Reynolds, Re 5 or Re 5 (26)
L D
ν ν
ν·ρ·c
p
Prandtl, Pr 5 (27)
k
f
where:
L = characteristic dimension for horizontal and vertical flat surfaces, and vertical cylinders feet (m), in general, denotes height
of vertical surface or length of horizontal surface,
D = characteristic dimension for horizontal cylinders and spheres feet (m), in general, denotes the diameter,
c = specific heat of ambient fluid, Btu/(lb·°R) (J/(kg·K)),
p
2 2
h¯ = average convection conductance, Btu/(h·ft ·°F) (W/(m ·K)),
c
k = thermal conductivity of ambient fluid, Btu/(h·ft·°F) (W/(m·K)),
f
V = free stream velocity of ambient fluid, ft/h (m/s),
2 2
ν = kinematic viscosity of ambient fluid, ft /h (m /s),
2 2
g = acceleration due to gravity, ft/h (m/s ),
-1 -1
β = volumetric thermal expansion coefficient of ambient fluid, °R (K ),
3 3
ρ = density of ambient fluid, lb/ft (kg/m ), and
ΔT = absolute value of temperature difference between surface and ambient fluid, °R (K).
It needs to be noted here that (except for spheres–forced convection) the above fluid properties must be calculated at the film
temperature, T , which is the average of surface and ambient fluid temperatures. For this practice, it is assumed that the ambient
f
C680 − 23a
fluid is dry air at atmospheric pressure. The properties of air can be found in references such as Ref (9). This reference contains
equations for some of the properties and polynomial fits for others, and the equations are summarized in Table A1.1.
6.7.3.2 When a heated surface is exposed to flowing fluid, the convective heat transfer will be a combination of forced and free
convection. For this mixed convection condition, Churchill (10) recommends the following equation. For each geometric shape
and surface orientation the overall average Nusselt number is to be computed from the average Nusselt number for forced
convection and the average Nusselt number for natural convection. The film conductance, h, is then computed from Eq 24. The
relationship is:
j j j
H H H
~Nu 2 δ! 5~Nu 2 δ! 1~Nu 2 δ! (28)
f n
where the exponent, j, and the constant, δ, are defined based on the geometry and orientation.
6.7.3.3 Once the Nusselt number has been calculated, the surface transfer conductance is calculated from a rearrangement of Eq
24:
H
h 5 Nu ·k /L (29)
c L f
H H
h 5 Nu ·k /D
c D f
where L and D are the characteristic dimension of the system. The term k is the thermal conductivity of air determined at the
a
film temperature using the equation in Table A1.1.
6.7.4 Convection Conductances for Flat Surfaces:
6.7.4.1 From Heat Transfer by Churchill and Ozoe as cited in Fundamentals of Heat and Mass Transfer by Incropera and Dewitt,
the relation for forced convection by laminar flow over an isothermal flat surface is:
1/2 1/3
0.6774 Re Pr
L
H 5
Nu 5 Re ,5 ×10 (30)
f,L 2/3 1/4 L
@11~0.0468/Pr! #
For forced convection by turbulent flow over an isothermal flat surface, Incropera and Dewitt suggest the following:
H 4/5 1/3 5 8
Nu 5 0.037 Re 2 871 Pr 5 ×10 ,Re ,10 (31)
~ !
f,L L L
It should be noted that the upper bound for Re is an approximate value, and the user of the above equation must be aware of this.
L
6.7.4.2 In “Correlating Equations for Laminar and Turbulent Free Convection from a Vertical Plate” by Churchill and Chu, as cited
by Incropera and Dewitt, it is suggested for natural convection on isothermal, vertical flat surfaces that:
1/6 2
0.387 Ra
L
H
Nu 5 0.8251 All Ra (32)
H 9/16 8/27J
n,L L
@11~0.492/Pr! #
For slightly better accuracy in the laminar range, it is suggested by the same source (p. 493) that:
1/4
0.670 Ra
L
H 9
Nu 5 0.681 Ra ,10 (33)
n,L 9/16 4/9 L
@11~0.492/Pr! #
In the case of both vertical flat and cylindrical surfaces the characteristic dimension, L or D, is the vertical height (ft). To compute
the overall Nusselt number (Eq 28), set j = 3 and δ = 0. Also, it is important to note that the free convection correlations apply
to vertical cylinders in most cases.
6.7.4.3 For natural convection on horizontal flat surfaces, Incropera and Dewitt (p. 498) cite Heat Transmission by McAdams,
“Natural Convection Mass Transfer Adjacent to Horizontal Plates” by Goldstein, Sparrow and Jones, and “Natural Convection
Adjacent to Horizontal Surfaces of Various Platforms” for the following correlations:
H 1/4 4 7
Heat flow up: Nu 5 0.54 Ra 10 ,Ra ,10 (34)
n,L L L
H 1/3 7 11
Nu 5 0.15 Ra 10 ,Ra ,10
n,L L L
H 1/4 5 10
Heat flow down: Nu 5 0.27 Ra 10 ,Ra ,10
n,L L L
In the case of horizontal flat surfaces, the characteristic dimension, L, is the area of the surface divided by the perimeter of the
surface (ft). To compute the overall Nusselt number (Eq 28), set j = 3.5 and δ = 0.
6.7.5 Convection Conductances for Horizontal Cylinders:
C680 − 23a
6.7.5.1 For forced convection with fluid flow normal to a circular cylinder, Incropera and Dewitt (p. 370) cite Heat Transfer by
Churchill and Bernstein for the following correlation:
1/2 1/3 5/8 4/5
0.62Re Pr Re
D D
H
Nu 5 0.31 11 (35)
F S D G
f,D 2/3 1/4
11 0.4/Pr 282 000
@ ~ ! #
All Re ·Pr.0.2
D
In the case of horizontal cylinders, the characteristic dimension, D, is the diameter of the cylinder, (ft). In addition, this
correlation should be used for forced convection from vertical pipes.
6.7.5.2 For natural convection on horizontal cylinders, Incropera and Dewitt (p. 502) cite “Correlating Equations for Laminar and
Turbulent Free Convection from a Horizontal Cylinder” by Churchill and Chu for the following correlation:
1/6 2
0.387Ra
D
H
Nu 5 0.601 Ra ,10 (36)
H 9/16 8/27J
n,D D
11 0.559/Pr
@ ~ ! #
To compute the overall Nusselt number using Eq 28, set j = 4 and δ = 0.3.
6.7.6 Convection Conductances for Spheres:
6.7.6.1 For forced convection on spheres, Incropera and DeWitt cite S. Whitaker in AIChE J. for the following correlation:
1/4
μ
1/2 2/3 0.4
H
Nu 5 21 0.4 Re 10.06 Re Pr (37)
~ ! S D
f,D D D
μ
s
0.71,Pr,380
3.5,Re ,7.6 ×10
D
1.0,~μ/μ !,3.2
s
where μ and μ are the free stream and surface viscosities of the ambient fluid respectively. It is extremely important to note that
s
all properties need to be evaluated based on the free stream temperature of the ambient fluid, except for μ , which needs to be
s
evaluated based on the surface temperature.
6.7.6.2 For natural convection on spheres, Incropera and DeWitt cite “Free Convection Around Immersed Bodies” by S. W.
Churchill in Heat Exchange Design Handbook (Schlunder) for the following correlation:
1/4
0.589 Ra
D
H
Nu 5 21 (38)
n,D 9/16 4/9
@11~0.469/Pr! #
0.7 # Pr
Ra ,10
D
where all properties are evaluated at the film temperature. To compute the overall Nusselt number for spheres (Eq 28) set j =
4 and δ = 2.
7. Computer Program
7.1 General:
7.1.1 The computer program(s) can be written in any language, but Appendix X3 is written in C#.
7.1.2 The program consists of a main program that utilizes several subroutines. Other subroutines may be added to make the
program more applicable to the specific problems of individual users.
7.2 Functional Description of Program—The flow chart shown in Fig. 1 is a schematic representations of the operational
procedures for each coordinate system covered by the program. The flow chart presents the logic path for entering data, calculating
and recalculating system thermal resistances and temperatures, relaxing the successive errors in the temperature to within 0.1° of
the temperature, calculating heat loss or gain for the system and printing the parameters and solution in tabular form.
C680 − 23a
7.3 Computer Program Variable Descriptions—The description of all variables used in the programs are given in the listing of the
program as comments.
7.4 Program Operation:
7.4.1 Log on procedures and any executive program for execution of this program must be followed as needed.
7.4.2 The input for the thermal conductivity versus mean temperature parameters must be obtained as outlined in 6.6. The type
code determines the thermal conductivity versus temperature relationship applying to the insulation. The same type code may be
used for more than one insulation. As presented, the programs will operate on three functional relationships:
Type Functional Relationship
Quadratic k = a + bt + ct
where a,b, and c are constants
Linear k = a + b t; t < t
1 1 L
k = a + b t; t < t < t
2 2 L U
k = a + b t; t > t
3 3 U
where a1, a2, a3, b1, b2, b3 are constants, and
t and t are, respectively, the lower and upper
L U
inflection points of an S-shaped curve
Additional or different relationships may be used, but the main program must be modified.
8. Report
8.1 The results of calculations performed in accordance with this practice may be used as design data for specific job conditions,
or may be used in general form to represent the performance of a particular product or system. When the results will be used for
comparison of performance of similar products, it is recommended that reference be made to the specific constants used in the
calculations. These references should include:
8.1.1 Name and other identification of products or components,
8.1.2 Identification of the nominal pipe size or surface insulated, and its geometric orientation,
8.1.3 The surface temperature of the pipe or surface,
8.1.4 The equations and constants selected for the thermal conductivity versus mean temperature relationship,
FIG. 2 Thermal Conductivity vs. Mean Temperature
C680 − 23a
FIG. 3 Mean Temperature vs. Thermal Conductivity
FIG. 4 Thermal Conductivity vs. Mean Temperature
8.1.5 The ambient temperature and humidity, if applicable,
8.1.6 The surface transfer conductance and condition of surface heat transfer,
8.1.6.1 If obtained from published information, the source and limitations,
8.1.6.2 If calculated or measured, the method and significant parameters such as emittance, fluid velocity, etc.,
8.1.7 The resulting outer surface temperature, and
8.1.8 The resulting heat loss or gain.
8.2 Either tabular or graphical representation of the calculated results may be used. No recommendation is made for the format
in which results are presented.
9. Accuracy and Resolution
9.1 In many typical computers normally used, seven significant digits are resident in the computer for calculations. Adjustments
to this level can be made through the use of “Double Precision;” however, for the intended purpose of this practice, standard levels
C680 − 23a
of precision are adequate. The formatting of the output results, however, should be structured to provide a resolution of 0.1 % for
the typical expected levels of heat flux and a resolution of 1°F (0.55°C) for surface temperatures.
NOTE 1—The term “double precision” should not be confused with ASTM terminology on Precision and Bias.
9.2 Many factors influence the accuracy of a calculative procedure used for predicting heat flux results. These factors include
accuracy of input data and the applicability of the assumptions used in the method for the system under study. The system of
mathematical equations used in this analysis has been accepted as applicable for most systems normally insulated with bulk type
insulations. Applicability of this practice to systems having irregular shapes, discontinuities and other variations from the
one-dimensional heat transfer assumptions should be handled on an individual basis by professional engineers familiar with those
systems.
9.3 The computer resolution effect on accuracy is only significant if the level of precision is less than that discussed in 9.1.
Computers in use today are accurate in that they will reproduce the calculated results to resolution required if identical input data
is used.
9.4 The most significant factor influencing the accuracy of claims is the accuracy of the input thermal conductivity data. The
accuracy of applicability of these data is derived from two factors. The first is the accuracy of the test method used to generate
the data. Since the test methods used to supply these data are typically Test Methods C177, C335, or C518, the reports should
contain some statement of the estimates of error or estimates of uncertainty. The remaining factors influencing the accuracy are
the inherent variability of the product and the variability of the installation practices. If the product variability is large, the
installation is poor, or both, serious differences might exist between measured performance and predicted performance from this
practice.
10. Precision and Bias
10.1 When concern exists with the accuracy of the input test data, the recommended practice to evaluate the impact of possible
errors is to repeat the calculation for the range of the uncertainty of the variable. This process yields a range in the desired output
variable for a given uncertainty in the input variable. Repeating this procedure for all the input variables would yield a measure
of the contribution of each to the overall uncertainty. Several methods exist for the combination of these effects; however, the most
commonly used is to take the square root of the sum of the squares of the percentage errors induced by each variable’s uncertainty.
Eq 39 from Theories of Engineering Experimentation by H. Schenck gives the expression in mathematical form:
n 1/2
S ]R
5 Δx (39)
S SS D D D
( i
R ]x
i51
i
where:
S = estimate of the probable error of the procedure,
R = result of the procedure,
x = ith variable in procedure,
i
∂R/∂x = change in result with respect to change in ith variable,
i
Δx = uncertainty in value of variable, i, and
i
n = total number of variables in procedure.
10.2 ASTM Subcommittee C16.30, Task Group 5.2, which is responsible for preparing this practice, has prepared Appendix X1.
The appendix provides a more complete discussion of the precision and bias expected when using Practice C680 in the analysis
of operating systems. While much of that discussion is relevant to this practice, the errors associated with its application to
operating systems are beyond the primary Practice C680 scope. Portions of this discussion, however, were used in developing the
Precision and Bias statements included in Section 10.
11. Keywords
11.1 computer program; heat flow; heat gain; heat loss; pipe; thermal insulation
C680 − 23a
ANNEX
(Mandatory Information)
A1. EQUATIONS DERIVED FROM THE NIST CIRCULAR 564
A1.1 Table A1.1 lists the equations derived from the
...