ASTM E637-22

Standard Test Method for Calculation of Stagnation Enthalpy from Heat Transfer Theory and Experimental Measurements of Stagnation-Point Heat Transfer and Pressure

SIGNIFICANCE AND USE
3.1 The purpose of this test method is to provide a standard calculation of the stagnation enthalpy of an aerodynamic simulation device using the heat transfer theory and measured values of stagnation point heat transfer and pressure. A stagnation enthalpy obtained by this test method gives a consistent set of data, along with heat transfer and stagnation pressure for ablation computations.
SCOPE
1.1 This test method covers the calculation from heat transfer theory of the stagnation enthalpy from experimental measurements of the stagnation-point heat transfer and stagnation pressure.  
1.2 Advantages:  
1.2.1 A value of stagnation enthalpy can be obtained at the location in the stream where the model is tested. This value gives a consistent set of data, along with heat transfer and stagnation pressure, for ablation computations.  
1.2.2 This computation of stagnation enthalpy does not require the measurement of any arc heater parameters.  
1.3 Limitations and Considerations—There are many factors that may contribute to an error using this type of approach to calculate stagnation enthalpy, including:  
1.3.1 Turbulence—The turbulence generated by adding energy to the stream may cause deviation from the laminar equilibrium heat transfer theory.  
1.3.2 Equilibrium, Nonequilibrium, or Frozen State of Gas—The reaction rates and expansions may be such that the gas is far from thermodynamic equilibrium.  
1.3.3 Noncatalytic Effects—The surface recombination rates and the characteristics of the metallic calorimeter may give a heat transfer deviation from the equilibrium theory.  
1.3.4 Free Electric Currents—The arc-heated gas stream may have free electric currents that will contribute to measured experimental heat transfer rates.  
1.3.5 Nonuniform Pressure Profile—A nonuniform pressure profile in the region of the stream at the point of the heat transfer measurement could distort the stagnation point velocity gradient.  
1.3.6 Mach Number Effects—The nondimensional stagnation-point velocity gradient is a function of the Mach number. In addition, the Mach number is a function of enthalpy and pressure such that an iterative process is necessary.  
1.3.7 Model Shape—The nondimensional stagnation-point velocity gradient is a function of model shape.  
1.3.8 Radiation Effects—The hot gas stream may contribute a radiative component to the heat transfer rate.  
1.3.9 Heat Transfer Rate Measurement—An error may be made in the heat transfer measurement (see Method E469 and Test Methods E422, E457, E459, and E511).  
1.3.10 Contamination—The electrode material may be of a large enough percentage of the mass flow rate to contribute to the heat transfer rate measurement.  
1.4 Units—The values stated in SI units are to be regarded as standard. No other units of measurement are included in this standard.  
1.4.1 Exception—The values given in parentheses are for information only.  
1.5 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.  
1.6 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.

General Information

• Status: Published
• Publication Date: 31-Jul-2022
• ICS: 17.200.10 - Heat. Calorimetry
• Technical Committee: E21 - Space Simulation and Applications of Space Technology
• Drafting Committee: E21.08 - Thermal Protection

Relations

• Refers: ASTM E422-05(2011) - Standard Test Method for Measuring Heat Flux Using a Water-Cooled Calorimeter
• Effective Date: 01-Oct-2011
• Refers: ASTM E459-05(2011) - Standard Test Method for Measuring Heat Transfer Rate Using a Thin-Skin Calorimeter
• Effective Date: 01-Oct-2011
• Refers: ASTM E457-08 - Standard Test Method for Measuring Heat-Transfer Rate Using a Thermal Capacitance (Slug) Calorimeter
• Effective Date: 01-May-2008
• Refers: ASTM E341-08 - Standard Practice for Measuring Plasma Arc Gas Enthalpy by Energy Balance
• Effective Date: 01-May-2008
• Refers: ASTM E511-07 - Standard Test Method for Measuring Heat Flux Using a Copper-Constantan Circular Foil, Heat-Flux Transducer
• Effective Date: 01-Nov-2007
• Refers: ASTM E422-05 - Standard Test Method for Measuring Heat Flux Using a Water-Cooled Calorimeter
• Effective Date: 15-Sep-2005
• Refers: ASTM E459-05 - Standard Test Method for Measuring Heat Transfer Rate Using a Thin-Skin Calorimeter
• Effective Date: 15-Sep-2005
• Refers: ASTM E511-01 - Standard Test Method for Measuring Heat Flux Using a Copper-Constantan Circular Foil, Heat-Flux Gage
• Effective Date: 10-Oct-2001
• Refers: ASTM E511-73(1994)e1 - Standard Test Method for Measuring Heat Flux Using a Copper-Constantan Circular Foil, Heat-Flux Gage
• Effective Date: 10-Oct-2001
• Refers: ASTM E422-99 - Standard Test Method for Measuring Heat Flux Using a Water-Cooled Calorimeter
• Effective Date: 10-Oct-1999
• Refers: ASTM E459-97 - Standard Test Method for Measuring Heat Transfer Rate Using a Thin-Skin Calorimeter
• Effective Date: 01-Jan-1997
• Refers: ASTM E457-96 - Standard Test Method for Measuring Heat-Transfer Rate Using a Thermal Capacitance (Slug) Calorimeter
• Effective Date: 10-Oct-1996
• Refers: ASTM E341-96(2002) - Standard Practice for Measuring Plasma Arc Gas Enthalpy by Energy Balance
• Effective Date: 10-Oct-1996
• Refers: ASTM E341-96 - Standard Practice for Measuring Plasma Arc Gas Enthalpy by Energy Balance
• Effective Date: 10-Oct-1996
• Refers: ASTM E457-96(2002) - Standard Test Method for Measuring Heat-Transfer Rate Using a Thermal Capacitance (Slug) Calorimeter
• Effective Date: 10-Oct-1996

Overview

ASTM E637-22: Standard Test Method for Calculation of Stagnation Enthalpy from Heat Transfer Theory and Experimental Measurements of Stagnation-Point Heat Transfer and Pressure provides a consistent methodology for calculating stagnation enthalpy in aerodynamic simulation environments. Developed by ASTM International, this standard ensures reliable and accurate calculations using experimental data from stagnation-point heat transfer and pressure measurements-without requiring direct arc heater measurements. The resulting stagnation enthalpy data are critical for various aerospace applications, particularly for ablation and thermal protection system analysis.

Key Topics

• Stagnation Enthalpy Calculation: The standard utilizes heat transfer theory and measured values (stagnation-point heat transfer and pressure) to compute stagnation enthalpy, providing localized and relevant enthalpy values at the test model's location in the gas flow.
• Consistency for Ablation Computations: By linking heat transfer, stagnation pressure, and enthalpy, the method facilitates ablation and thermal response calculations for materials exposed to high-enthalpy flows.
• Error Considerations: The method acknowledges several potential sources of calculation uncertainty, such as:
  • Turbulence affecting heat transfer assumptions
  • Non-equilibrium or frozen gas states
  • Noncatalytic effects on heat transfer surfaces
  • Free electric currents influencing heat transfer rates
  • Nonuniform pressure profiles and Mach number effects
  • Model geometry and radiation effects
  • Measurement errors and contamination from electrode materials
• Measurement Units: The standard uses SI units, with alternate (parenthetical) values provided for informational purposes.

Applications

ASTM E637-22 serves critical functions in the aerospace sector where high-temperature, high-speed gas flows are simulated:

• Thermal Protection System (TPS) Validation: The calculation of stagnation enthalpy is essential for understanding material behavior under simulated atmospheric entry conditions, supporting TPS material selection and ablation studies.
• Aerothermal Facility Testing: Used in arc-jet and plasma wind tunnel testing, this method provides essential flow properties required for validating computational models and experimental results.
• Material and Component Qualification: Researchers and engineers utilize this standard to ensure enthalpy data are accurate and comparable across facilities and testing campaigns.
• Research and Development: Stagnation enthalpy data serve as a key input for developing new heat-shield materials and improving aerothermal prediction capabilities in hypersonics research.

Related Standards

Several other ASTM standards complement E637-22 and are often referenced in conjunction with stagnation enthalpy measurements:

• ASTM E341 - Practice for Measuring Plasma Arc Gas Enthalpy by Energy Balance
• ASTM E422 - Test Method for Measuring Net Heat Flux Using a Water-Cooled Calorimeter
• ASTM E457 - Test Method for Measuring Heat-Transfer Rate Using a Thermal Capacitance (Slug) Calorimeter
• ASTM E459 - Test Method for Measuring Heat Transfer Rate Using a Thin-Skin Calorimeter
• ASTM E511 - Test Method for Measuring Heat Flux Using a Copper-Constantan Circular Foil, Heat-Flux Transducer

By leveraging ASTM E637-22 alongside these related standards, facilities can attain a unified, rigorous approach to measuring, analyzing, and comparing stagnation enthalpy in high-enthalpy aerodynamic flows, maintaining the reliability and integrity of experimental data.

Keywords: stagnation enthalpy, heat transfer measurement, stagnation-point, aerodynamic simulation, ablation computations, arc-jet testing, aerospace standard, ASTM E637-22, thermal protection, experimental aerothermodynamics.

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: E637 − 22
Standard Test Method for
Calculation of Stagnation Enthalpy from Heat Transfer
Theory and Experimental Measurements of Stagnation-Point
Heat Transfer and Pressure
This standard is issued under the fixed designation E637; the number immediately following the designation indicates the year of
original adoption or, in the case of revision, the year of last revision.Anumber in parentheses indicates the year of last reapproval.A
superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
INTRODUCTION
The enthalpy (energy per unit mass) determination in a hot gas aerodynamic simulation device is
a difficult measurement. Even at temperatures that can be measured with thermocouples (1), there are
many corrections to be made at 600 K and above. Methods that are used for temperatures above the
range of thermocouples that give bulk or average enthalpy values are energy balance (see Practice
E341), sonic flow (2, 3), and the pressure rise method (4). Local enthalpy values (thus distribution)
may be obtained by using either an energy balance probe (see Method E470), or the spectrometric
technique described in Ref (5).
1. Scope 1.3.3 Noncatalytic Effects—Thesurfacerecombinationrates
and the characteristics of the metallic calorimeter may give a
1.1 This test method covers the calculation from heat
heat transfer deviation from the equilibrium theory.
transfer theory of the stagnation enthalpy from experimental
1.3.4 Free Electric Currents—The arc-heated gas stream
measurements of the stagnation-point heat transfer and stagna-
mayhavefreeelectriccurrentsthatwillcontributetomeasured
tion pressure.
experimental heat transfer rates.
1.2 Advantages:
1.3.5 Nonuniform Pressure Profile—Anonuniform pressure
1.2.1 Avalue of stagnation enthalpy can be obtained at the profile in the region of the stream at the point of the heat
location in the stream where the model is tested. This value transfer measurement could distort the stagnation point veloc-
gives a consistent set of data, along with heat transfer and ity gradient.
stagnation pressure, for ablation computations. 1.3.6 Mach Number Effects—The nondimensional
stagnation-point velocity gradient is a function of the Mach
1.2.2 This computation of stagnation enthalpy does not
number.Inaddition,theMachnumberisafunctionofenthalpy
require the measurement of any arc heater parameters.
and pressure such that an iterative process is necessary.
1.3 Limitations and Considerations—There are many fac-
1.3.7 Model Shape—The nondimensional stagnation-point
tors that may contribute to an error using this type of approach
velocity gradient is a function of model shape.
to calculate stagnation enthalpy, including:
1.3.8 Radiation Effects—The hot gas stream may contribute
1.3.1 Turbulence—The turbulence generated by adding en-
a radiative component to the heat transfer rate.
ergy to the stream may cause deviation from the laminar
1.3.9 Heat Transfer Rate Measurement—An error may be
equilibrium heat transfer theory.
made in the heat transfer measurement (see Method E469 and
1.3.2 Equilibrium, Nonequilibrium, or Frozen State of
Test Methods E422, E457, E459, and E511).
Gas—The reaction rates and expansions may be such that the
1.3.10 Contamination—The electrode material may be of a
gas is far from thermodynamic equilibrium.
large enough percentage of the mass flow rate to contribute to
the heat transfer rate measurement.
1.4 Units—The values stated in SI units are to be regarded
This test method is under the jurisdiction of ASTM Committee E21 on Space
asstandard.Nootherunitsofmeasurementareincludedinthis
Simulation andApplications of SpaceTechnology and is the direct responsibility of
standard.
Subcommittee E21.08 on Thermal Protection.
1.4.1 Exception—The values given in parentheses are for
Current edition approved Aug. 1, 2022. Published September 2022. Originally
approvedin1978.Lastpreviouseditionapprovedin2016asE637–05(2016).DOI:
information only.
10.1520/E0637-22.
1.5 This standard does not purport to address all of the
The boldface numbers in parentheses refer to the list of references appended to
this method. safety concerns, if any, associated with its use. It is the
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E637 − 22
responsibility of the user of this standard to establish appro- lytic theoretical laminar stagnation-point heat transfer rate for
priate safety, health, and environmental practices and deter- a hemispherical body is as follows (6):
mine the applicability of regulatory limitations prior to use.
R
1.6 This international standard was developed in accor-
q 5 K ~H 2 H ! (1)
Œ
i e w
P
t
dance with internationally recognized principles on standard-
ization established in the Decision on Principles for the
where:
Development of International Standards, Guides and Recom- 2 2
q = stagnation-pointheattransferrate,W/m (orBtu/ft ·s),
mendations issued by the World Trade Organization Technical
P = model stagnation pressure, Pa (or atm),
t
Barriers to Trade (TBT) Committee.
R = hemispherical nose radius, m (or ft),
H = stagnation enthalpy, J/kg (or Btu/lb),
e
2. Referenced Documents
H = wall enthalpy, J/kg (or Btu/lb), and
w
K = heat transfer computation constant.
2.1 ASTM Standards:
i
E341Practice for Measuring Plasma Arc Gas Enthalpy by
4.2 Low Mach Number Correction—Eq 1 is simple and
Energy Balance
convenient to use since K can be considered approximately
i
E422Test Method for Measuring Net Heat Flux Using a
constant(seeTable1).However,Eq1isbasedonastagnation-
Water-Cooled Calorimeter
point velocity gradient derived using “modified” Newtonian
E457Test Method for Measuring Heat-Transfer Rate Using
flow theory which becomes inaccurate for M <2. An im-
∞
a Thermal Capacitance (Slug) Calorimeter
proved Mach number dependence at lower Mach numbers can
E459Test Method for Measuring Heat Transfer Rate Using
beobtainedbyremovingthe“modified”Newtonianexpression
a Thin-Skin Calorimeter
andreplacingitwithamoreappropriateexpressionasfollows:
E469Measuring Heat Flux Using a Multiple-Wafer Calo-
0.5
β D/U
~ !
K q˙
` Eq 3
rimeter (Withdrawn 1982) M
H 2 H 5 F G (2)
e w 0.5
P /R ~β D/U !
E470Measuring Gas Enthalpy Using Calorimeter Probes ~ !
t `
2 x50
(Withdrawn 1982)
Where the “modified” Newtonian stagnation-point velocity
E511TestMethodforMeasuringHeatFluxUsingaCopper-
gradient is given by:
Constantan Circular Foil, Heat-Flux Transducer
2 0.5
4 @~γ 2 1! M 12#
`
β D/U 5 (3)
~ ! H J
` x50
3. Significance and Use
γ M
`
3.1 The purpose of this test method is to provide a standard
A potential problem exists when using Eq 3 to remove the
calculation of the stagnation enthalpy of an aerodynamic
“modified” Newtonian velocity gradient because of the singu-
simulation device using the heat transfer theory and measured
larity at M =0. The procedure recommended here should be
∞
values of stagnation point heat transfer and pressure. A
limited to M > 0.1.
∞
stagnation enthalpy obtained by this test method gives a
where:
consistent set of data, along with heat transfer and stagnation
−1
pressure for ablation computations.
β = stagnation-point velocity gradient, s ,
D = hemispherical diameter, m (or ft),
4. Enthalpy Computations
U = freestream velocity, m/s (or ft/s),
∞
(βD/U ) = dimensionless stagnation velocity gradient,
∞ x=0
4.1 This method of calculating the stagnation enthalpy is
K = enthalpy computation constant,
M
based on experimentally measured values of the stagnation-
1/2 1/2 3/2 1/2
(N ·m · s)/kg or (ft ·atm ·s)/lb,
pointheattransferrateandpressuredistributionandtheoretical
M = the freestream Mach number, and
∞
calculation of laminar equilibrium catalytic stagnation-point
γ = dimensionless ratio of gas specific heat at
heat transfer on a hemispherical body. The equilibrium cata-
constant pressure to its specific heat at con-
stant volume.
For subsonic Mach numbers, an expression for (βD/U )
For referenced ASTM standards, visit the ASTM website, www.astm.org, or ∞ x=0
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM for a hemisphere is given in Ref (7) as follows:
Standards volume information, refer to the standard’s Document Summary page on
βD
the ASTM website.
5 3 2 0.755 M ~M ,1! (4)
S D
4 ` `
The last approved version of this historical standard is referenced on
U
`
x50
www.astm.org.
TABLE 1 Heat Transfer and Enthalpy Computation Constants for
Various Gases
1/2 1/2 1/2 1/2
K , kg/(N ·m ·s) K ,(N ·m ·s)/kg
i M
Gas
3/2 1/2 3/2 1/2
(lb/(ft ·s·atm )) ((ft ·s·atm )/lb)
−4
Air 3.905 × 10 (0.0461) 2561 (21.69)
−4
Argon 5.513 × 10 (0.0651) 1814 (15.36)
−4
Carbon dioxide 4.337 × 10 (0.0512) 2306 (19.53)
−4
Hydrogen 1.287 × 10 (0.0152) 7768 (65.78)
−4
Nitrogen 3.650 × 10 (0.0431) 2740 (23.20)
E637 − 22
For a Mach number of 1 or greater, (βD/U ) for a 4.4 Velocity Gradient Calculation from Pressure
∞ x=0
hemisphere based on “classical” Newtonian flow theory is Distribution—The dimensionless stagnation-point velocity
presented in Ref (8) as follows:
gradient may be obtained from an experimentally measured
pressure distribution by using Bernoulli’s compressible flow
βD 8 γ 2 1 M 12
@~ ! #
`
5H · 1 equation as follows:
S D F
U ~γ 1 1!M
` `
x50
γ21 0.5
1 0.5
2 2 1 2 p/p γ
U @ #
~ t !
γ 2 1 γ 2 1 M 12 γ21
~ ! 2
`
5 (7)
1 · J (5) γ21 0.5
G
U
2 2γM 2 ~γ 2 1! `
@1 2 p /p γ #
` ~ !
` t
Avariationof(βD/U ) with M andγisshowninFig.1.
∞ x=0 ∞
where the velocity ratio may be calculated along the body
The value of the Newtonian dimensionless velocity gradient
from the stagnation point. Thus, the dimensionless stagnation-
approaches a constant value as the Mach number approaches
point velocity gradient, (βD/U ) , is the slope of the U/U
∞ x=0 ∞
infinity:
and the x/D curve at the stagnation point.
βD γ 2 1
4.5 Model Shape—Thenondimensionalstagnation-pointve-
5 4 (6)
S D ΠS D
U γ
`
x50,M→`
locity gradient is a function of the model shape and the Mach
and thus, sinceγ, the ratio of specific heats, is a function of number. For supersonic Mach numbers, the heat transfer
enthalpy, (βD/U ) is also a function of enthalpy.Again, an relationship between a hemisphere and other axisymmetric
∞ x=0
iteration is necessary. From Fig. 1, it can be seen that
blunt bodies is shown in Fig. 5 (11).In Fig. 5, r is the corner
c
(βD/U ) for a hemisphere is approximately 1 for large radius, r isthebodyradius, r isthenoseradius,and q˙ isthe
∞ x=0
b n s,h
Mach numbers and γ=1.2. K is tabulated in Table 1 using
stagnation-point heat transfer rate on a hemisphere. For sub-
M
(βD/U ) =1 and K from Ref (6).
sonic Mach numbers, the same type of variation is shown in
∞ x=0 i
Fig. 6 (7).
4.3 Mach Number Determination:
4.3.1 TheMachnumberofastreamisafunctionofthetotal
4.6 Radiation Effects:
enthalpy, the ratio of freestream pressure to the total pressure,
4.6.1 As this test method depends on the accurate determi-
p/p ,thetotalpressure, p ,andtheratiooftheexitnozzlearea
t t
1 1
nation of the convective stagnation-point heat transfer, any
totheareaofthenozzlethroat, A/A'.Fig.2(a)andFig.2(b)are
radiant energy absorbed by the calorimeter surface and incor-
reproduced from Ref (9) for the reader’s convenience in
rectly attributed to the convective mode will directly affect the
determining Mach numbers for supersonic flows.
overall accuracy of the test method. Generally, the sources of
4.3.2 The subsonic Mach number may be determined from
radiant energy are the hot gas stream itself or the gas heating
Fig.3(seealsoTestMethodE511).Aniterationisnecessaryto
device, or both. For instance, arc heaters operated at high
determine the Mach number since the ratio of specific heats,γ,
pressure (10 atm or higher) can produce significant radiant
is also a function of enthalpy and pressure.
fluxes at the nozzle exit plane.
4.3.3 Theratioofspecificheats,γ,isshownasafunctionof
4.6.2 The proper application requires some knowledge of
entropy and enthalpy for air in Fig. 4 from Ref (10). S/R is the
dimensionless entropy, and H/RT is the dimensionless en- the radiant environment in the stream at the desired operating
thalpy. conditions. Usually, it is necessary to measure the radiant heat
FIG. 1 Dimensionless Velocity Gradient as a Function of Mach Number and Ratio of Specific Heats
E637 − 22
FIG. 2 (a) Variation of Area Ratio with Mach Numbers
FIG. 2 (b) Variation of Area Ratio with Mach Numbers (continued)
E637 − 22
FIG. 3 Subsonic Pressure Ratio as a Function of Mach Number and γ
FIG. 4 Isentropic Exponent for Air in Equilibrium
transfer rate either directly or indirectly.The following is a list from affecting the calorimeter while transmitting the radiant
of suggested methods by which the necessary measurements energy.Thewindowisusuallymadeofquartzorsapphire.The
can be made. sensing surface is at the stagnation point of a test probe and is
4.6.2.1 Direct Measurement with Radiometer—Radiometers located in such a manner that the view angle is not restricted.
are available for the measurement of the incident radiant flux The basic radiometer view angle should be 120° or greater.
while excluding the convective heat transfer. In its simplest This technique allows for immersion of the radiometer in the
form, the radiometer is a slug, thin-skin, or circular foil test stream and direct measurement of the radiant heat transfer
calorimeter with a sensing area with a coating of known rate. There is a major limitation to this technique, however, in
absorptance and covered with some form of window. The that even with high-pressure water cooling of the radiometer
purpose of the window is to prevent convective heat transfer enclosure, the window
...

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: E637 − 05 (Reapproved 2016) E637 − 22
Standard Test Method for
Calculation of Stagnation Enthalpy from Heat Transfer
Theory and Experimental Measurements of Stagnation-Point
Heat Transfer and Pressure
This standard is issued under the fixed designation E637; 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.
INTRODUCTION
The enthalpy (energy per unit mass) determination in a hot gas aerodynamic simulation device is
a difficult measurement. Even at temperatures that can be measured with thermocouples,
thermocouples (1), there are many corrections to be made at 600 K and above. Methods that are used
for temperatures above the range of thermocouples that give bulk or average enthalpy values are
energy balance (see Practice E341), sonic flow (12, 23), and the pressure rise method (34). Local
enthalpy values (thus distribution) may be obtained by using either an energy balance probe (see
Method E470), or the spectrometric technique described in Ref (45).
1. Scope
1.1 This test method covers the calculation from heat transfer theory of the stagnation enthalpy from experimental measurements
of the stagnation-point heat transfer and stagnation pressure.
1.2 Advantages:
1.2.1 A value of stagnation enthalpy can be obtained at the location in the stream where the model is tested. This value gives a
consistent set of data, along with heat transfer and stagnation pressure, for ablation computations.
1.2.2 This computation of stagnation enthalpy does not require the measurement of any arc heater parameters.
1.3 Limitations and Considerations—There are many factors that may contribute to an error using this type of approach to
calculate stagnation enthalpy, including:
1.3.1 Turbulence—The turbulence generated by adding energy to the stream may cause deviation from the laminar equilibrium
heat transfer theory.
1.3.2 Equilibrium, Nonequilibrium, or Frozen State of Gas—The reaction rates and expansions may be such that the gas is far from
thermodynamic equilibrium.
This test method is under the jurisdiction of ASTM Committee E21 on Space Simulation and Applications of Space Technology and is the direct responsibility of
Subcommittee E21.08 on Thermal Protection.
Current edition approved April 1, 2016Aug. 1, 2022. Published April 2016September 2022. Originally approved in 1978. Last previous edition approved in 20112016 as
E637 – 05 (2011).(2016). DOI: 10.1520/E0637-05R16.10.1520/E0637-22.
The boldface numbers in parentheses refer to the list of references appended to this method.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E637 − 22
1.3.3 Noncatalytic Effects—The surface recombination rates and the characteristics of the metallic calorimeter may give a heat
transfer deviation from the equilibrium theory.
1.3.4 Free Electric Currents—The arc-heated gas stream may have free electric currents that will contribute to measured
experimental heat transfer rates.
1.3.5 Nonuniform Pressure Profile—A nonuniform pressure profile in the region of the stream at the point of the heat transfer
measurement could distort the stagnation point velocity gradient.
1.3.6 Mach Number Effects—The nondimensional stagnation-point velocity gradient is a function of the Mach number. In addition,
the Mach number is a function of enthalpy and pressure such that an iterative process is necessary.
1.3.7 Model Shape—The nondimensional stagnation-point velocity gradient is a function of model shape.
1.3.8 Radiation Effects—The hot gas stream may contribute a radiative component to the heat transfer rate.
1.3.9 Heat Transfer Rate Measurement—An error may be made in the heat transfer measurement (see Method E469 and Test
Methods E422, E457, E459, and E511).
1.3.10 Contamination—The electrode material may be of a large enough percentage of the mass flow rate to contribute to the heat
transfer rate measurement.
1.4 Units—The values stated in SI units are to be regarded as standard. No other units of measurement are included in this
standard.
1.4.1 Exception—The values given in parentheses are for information only.
1.5 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 safety, health, and healthenvironmental practices and determine the
applicability of regulatory limitations prior to use.
1.6 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:
E341 Practice for Measuring Plasma Arc Gas Enthalpy by Energy Balance
E422 Test Method for Measuring Net Heat Flux Using a Water-Cooled Calorimeter
E457 Test Method for Measuring Heat-Transfer Rate Using a Thermal Capacitance (Slug) Calorimeter
E459 Test Method for Measuring Heat Transfer Rate Using a Thin-Skin Calorimeter
E469 Measuring Heat Flux Using a Multiple-Wafer Calorimeter (Withdrawn 1982)
E470 Measuring Gas Enthalpy Using Calorimeter Probes (Withdrawn 1982)
E511 Test Method for Measuring Heat Flux Using a Copper-Constantan Circular Foil, Heat-Flux Transducer
3. Significance and Use
3.1 The purpose of this test method is to provide a standard calculation of the stagnation enthalpy of an aerodynamic simulation
device using the heat transfer theory and measured values of stagnation point heat transfer and pressure. A stagnation enthalpy
obtained by this test method gives a consistent set of data, along with heat transfer and stagnation pressure for ablation
computations.
4. Enthalpy Computations
4.1 This method of calculating the stagnation enthalpy is based on experimentally measured values of the stagnation-point heat
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.
The last approved version of this historical standard is referenced on www.astm.org.
E637 − 22
transfer rate and pressure distribution and theoretical calculation of laminar equilibrium catalytic stagnation-point heat transfer on
a hemispherical body. The equilibrium catalytic theoretical laminar stagnation-point heat transfer rate for a hemispherical body is
as follows (56):
R
q 5 K ~H 2 H ! (1)
Œ
i e w
P
t
where:
2 2
q = stagnation-point heat transfer rate, W/m (or Btu/ft ·s),
P = model stagnation pressure, Pa (or atm),
t
R = hemispherical nose radius, m (or ft),
H = stagnation enthalpy, J/kg (or Btu/lb),
e
H = wall enthalpy, J/kg (or Btu/lb), and
w
K = heat transfer computation constant.
i
4.2 Low Mach Number Correction—Eq 1 is simple and convenient to use since K can be considered approximately constant (see
i
Table 1). However, Eq 1 is based on a stagnation-point velocity gradient derived using “modified” Newtonian flow theory which
becomes inaccurate for M <2. An improved Mach number dependence at lower Mach numbers can be obtained by removing
oo∞
the “modified” Newtonian expression and replacing it with a more appropriate expression as follows:
0.5
β D/U
~ !
K q˙ oo
Eq 3
M
H 2 H 5 (2)
F G
e w 0.5
P /R β D/U
~ !
~ t ! oo
x50
0.5
β D/U
~ !
K q˙ `
Eq 3
M
H 2 H 5 (2)
F G
0.5
e w
P /R β D/U
~ !
~ !
t ` x50
Where the “modified” Newtonian stagnation-point velocity gradient is given by:
2 0.5
4 @~γ2 1! M 12#
oo
β D/U 5 (3)
~ ! F G
oo 2
x50
γ M
oo
2 0.5
4 @ γ2 1 M 12#
~ !
`
~β D/U ! 5 (3)
H J
` x50 2
γ M
`
A potential problem exists when using Eq 3 to remove the “modified” Newtonian velocity gradient because of the singularity
at M = 0. The procedure recommended here should be limited to M > 0.10.1.
oo∞ oo∞
where:
−1
β = stagnation-point velocity gradient, s ,
D = hemispherical diameter, m (or ft),
U = freestream velocity, m/s (or ft/s),
∞
(βD/U ) = dimensionless stagnation velocity gradient,
∞ x = 0
K = enthalpy computation constant,
M
1/2 1/2 3/2 1/2
(N ·m · s)/kg or (ft ·atm ·s)/lb, and
K = enthalpy computation constant,
M
1/2 1/2 3/2 1/2
(N ·m · s)/kg or (ft ·atm ·s)/lb,
M = the freestream Mach number, and
∞
M∞ = the freestream Mach number.
γ = dimensionless ratio of gas specific heat at constant pressure to its specific heat at constant volume.
For subsonic Mach numbers, an expression for (βD/U ) for a hemisphere is given in Ref (67) as follows:
∞ x = 0
TABLE 1 Heat Transfer and Enthalpy Computation Constants for
Various Gases
1/2 1/2 1/2 1/2
K , kg/(N ·m ·s) K , (N ·m ·s)/kg
i M
Gas
3/2 1/2 3/2 1/2
(lb/(ft ·s·atm )) ((ft ·s·atm )/lb)
−4
Air 3.905 × 10 (0.0461) 2561 (21.69)
−4
Argon 5.513 × 10 (0.0651) 1814 (15.36)
−4
Carbon dioxide 4.337 × 10 (0.0512) 2306 (19.53)
−4
Hydrogen 1.287 × 10 (0.0152) 7768 (65.78)
−4
Nitrogen 3.650 × 10 (0.0431) 2740 (23.20)
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βD
5 32 0.755 M ~M ,1! (4)
S D
x50 ` `
U
`
For a Mach number of 1 or greater, (βD/U ) for a hemisphere based on “classical” Newtonian flow theory is presented in
∞ x= 0
Ref (78) as follows:
1 0.5
γ2 1 γ21
8 γ2 1 M 2 12 2
βD @~ ! #
`
5 (5)
S D
x50
~γ2 1!M 2 12
U ~γ11!M 2 @ #
`
` `
5 6
3 4
2γM 2 2 ~γ2 1!
`
A variation of (βD/U ) with M and γ is shown in Fig. 1. The value of the Newtonian dimensionless velocity gradient
∞ x= 0 ∞
approaches a constant value as the Mach number approaches infinity:
βD γ2 1
5 4 (6)
S D ΠS D
x50,M→`
U γ
`
and thus, since γ, the ratio of specific heats, is a function of enthalpy, (βD/U ) is also a function of enthalpy. Again, an
∞ x= 0
iteration is necessary. From Fig. 1, it can be seen that (βD/U ) for a hemisphere is approximately 1 for large Mach numbers
∞ x = 0
and γ = 1.2. K is tabulated in Table 1 using (βD/U ) = 1 and K from Ref (56).
M ∞ x = 0 i
4.3 Mach Number Determination:
4.3.1 The Mach number of a stream is a function of the total enthalpy, the ratio of freestream pressure to the total pressure, p/p ,
t
the total pressure, p , and the ratio of the exit nozzle area to the area of the nozzle throat, A/A'.Fig. 2(a) and Fig. 2(b) are reproduced
t
from Ref (89) for the reader’s convenience in determining Mach numbers for supersonic flows.
4.3.2 The subsonic Mach number may be determined from Fig. 3 (see also Test Method E511). An iteration is necessary to
determine the Mach number since the ratio of specific heats, γ, is also a function of enthalpy and pressure.
4.3.3 The ratio of specific heats, γ, is shown as a function of entropy and enthalpy for air in Fig. 4 from Ref (910).S/R is the
dimensionless entropy, and H/RT is the dimensionless enthalpy.
4.4 Velocity Gradient Calculation from Pressure Distribution—The dimensionless stagnation-point velocity gradient may be
obtained from an experimentally measured pressure distribution by using Bernoulli’s compressible flow equation as follows:
γ21 0.5
12 p/p γ
U @ #
~ t !
5 (7)
γ21 0.5
U
`
@12 p /p γ #
~ !
` t
FIG. 1 Dimensionless Velocity Gradient as a Function of Mach Number and Ratio of Specific Heats
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FIG. 2 (a) Variation of Area Ratio with Mach Numbers
FIG. 2 (b) Variation of Area Ratio with Mach Numbers (continued)
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FIG. 3 Subsonic Pressure Ratio as a Function of Mach Number and γ
FIG. 4 Isentropic Exponent for Air in Equilibrium
γ21 0.5
@12 p/p γ #
U ~ !
t
5 (7)
γ21 0.5
U
` 12 p /p γ
@ #
~ ` t !
where the velocity ratio may be calculated along the body from the stagnation point. Thus, the dimensionless stagnation-point
velocity gradient, (βD/U ) , is the slope of the U/U and the x/D curve at the stagnation point.
∞ x= 0 ∞
4.5 Model Shape—The nondimensional stagnation-point velocity gradient is a function of the model shape and the Mach number.
For supersonic Mach numbers, the heat transfer relationship between a hemisphere and other axisymmetric blunt bodies is shown
in Fig. 5 (1011). In Fig. 5, r is the corner radius, r is the body radius, r is the nose radius, and q˙ is the stagnation-point heat
c b n s,h
transfer rate on a hemisphere. For subsonic Mach numbers, the same type of variation is shown in Fig. 6 (67).
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FIG. 5 Stagnation-Point Heating-Rate Parameters on Hemispherical Segments of Different Curvatures for Varying Corner-Radius Ratios
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FIG. 6 Stagnation-Point Heat Transfer Ratio to a Blunt Body and a Hemisphere as a Function of the
Body-to-Nose Radius in a Subsonic Stream
4.6 Radiation Effects:
4.6.1 As this test method depends on the accurate determination of the convective stagnation-point heat transfer, any radiant energy
absorbed by the calorimeter surface and incorrectly attributed to the convective mode will directly affect the overall accuracy of
the test method. Generally, the sources of radiant energy are the hot gas stream itself or the gas heating device, or both. For
instance, arc heaters operated at high pressure (10 atm or higher) can produce significant radiant fluxes at the nozzle exit plane.
4.6.2 The proper application requires some knowledge of the radiant environment in the stream at the desired operating conditions.
Usually, it is necessary to measure the radiant heat transfer rate either directly or indirectly. The following is a list of suggested
methods by which the necessary measurements can be made.
4.6.2.1 Direct Measurement with Radiometer—Radiometers are available for the measurement of the incident radiant flux while
excluding the convective heat transfer. In its simplest form, the radiometer is a slug, thin-skin, or circular foil calorimeter with a
sensing area with a coating of known absorptance and covered with some form of window. The purpose of the window is to prevent
convective heat transfer from affecting the calorimeter while transmitting the radiant energy. The window is usually made of quartz
or sapphire. The sensing surface is at the stagnation point of a test probe and is located in such a manne
...