ASTM E511-07(2015)
(Test Method)Standard Test Method for Measuring Heat Flux Using a Copper-Constantan Circular Foil, Heat-Flux Transducer
Standard Test Method for Measuring Heat Flux Using a Copper-Constantan Circular Foil, Heat-Flux Transducer
SIGNIFICANCE AND USE
3.1 Fig. 1 is a sectional view of an example circular foil heat-flux transducer. It consists of a circular Constantan foil attached by a metallic bonding process to a heat sink of oxygen-free high conductivity copper (OFHC), with copper leads attached at the center of the circular foil and at any point on the heat-sink body. The transducer impedance is usually less than 1 V. To minimize current flow, the data acquisition system (DAS) should be a potentiometric system or have an input impedance of at least 100 000 Ω.
3.2 As noted in 2.3, an approximately linear output (versus heat flux) is produced when the body and center wire of the transducer are constructed of copper and the circular foil is constantan. Other metal combinations may be employed for use at higher temperatures, but most (4) are nonlinear.
3.3 Because the thermocouple junction at the edge of the foil is the reference for the center thermocouple, no cold junction compensation is required with this instrument. The wire leads used to convey the signal from the transducer to the readout device are normally made of stranded, tinned copper, insulated with TFE-fluorocarbon and shielded with a braid over-wrap that is also TFE-fluorocarbon-covered.
3.4 Transducers with a heat-sink thermocouple can be used to indicate the foil center temperature. Once the edge temperature is known, the temperature difference from the foil edge to its center may be directly read from the copper-constantan (Type T) thermocouple table. This temperature difference then is added to the body temperature, indicating the foil center temperature.
3.5 Water-Cooled Transducer:
3.5.1 A water-cooled transducer should be used in any application where the copper heat-sink would rise above 235°C (450°F) without cooling. Examples of cooled transducers are shown in Fig. 2. The coolant flow must be sufficient to prevent local boiling of the coolant inside the transducer body, with its characteristic pulsations (“chugging”) of the ...
SCOPE
1.1 This test method describes the measurement of radiative heat flux using a transducer whose sensing element (1, 2)2 is a thin circular metal foil. These sensors are often called Gardon Gauges.
1.2 The values stated in SI units are to be regarded as the standard. The values stated in parentheses are provided for information only.
1.3 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 and health practices and determine the applicability of regulatory limitations prior to use.
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Designation: E511 − 07 (Reapproved 2015)
Standard Test Method for
Measuring Heat Flux Using a Copper-Constantan Circular
Foil, Heat-Flux Transducer
This standard is issued under the fixed designation E511; 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.
1. Scope under steady-state conditions. The center–perimeter tempera-
ture difference produces a thermoelectric potential, E, that will
1.1 Thistestmethoddescribesthemeasurementofradiative
2 varyinproportiontotheabsorbedheatflux, q'.Withprescribed
heat flux using a transducer whose sensing element (1, 2) is a
foildiameter,thickness,andmaterials,thepotential Eisalmost
thin circular metal foil. These sensors are often called Gardon
linearlyproportionaltotheaverageheatflux q'absorbedbythe
Gauges.
foil. This relationship is described by the following equation:
1.2 The values stated in SI units are to be regarded as the
E 5 Kq' (1)
standard. The values stated in parentheses are provided for
information only.
where:
1.3 This standard does not purport to address all of the K = a sensitivity constant determined experimentally.
safety concerns, if any, associated with its use. It is the
2.3 For nearly linear response, the heat sink and the center
responsibility of the user of this standard to establish appro-
wire of the transducer are made of high purity copper and the
priate safety and health practices and determine the applica-
foil of thermocouple grade Constantan. This combination of
bility of regulatory limitations prior to use.
materials produces a nearly linear output over a gauge tem-
perature range from –45 to 232°C (–50 to 450°F). The linear
2. Summary of Test Method
range results from the basically offsetting effects of
2.1 The purpose of this test method is to facilitate measure-
temperature-dependent changes in the thermal conductivity
ment of a radiant heat flux. Although the sensor will measure
and the Seebeck coefficient of the Constantan (3). All further
heat fluxes from mixed radiative – convective or pure convec-
discussion is based on the use of these two metals, since
tive sources, the uncertainty will increase as the convective
engineering practice has demonstrated they are commonly the
fraction of the total heat flux increases.
most useful.
2.2 The circular foil heat flux transducer generates a milli-
3. Description of the Instrument
Volt output in response to the rate of thermal energy absorbed
3.1 Fig. 1 is a sectional view of an example circular foil
(see Fig. 1). The perimeter of the circular metal foil sensing
heat-flux transducer. It consists of a circular Constantan foil
element is mounted in a metal heat sink, forming a reference
attached by a metallic bonding process to a heat sink of
thermocouple junction due to their different thermoelectric
oxygen-free high conductivity copper (OFHC), with copper
potentials. A differential thermocouple is created by a second
leads attached at the center of the circular foil and at any point
thermocouple junction formed at the center of the foil using a
ontheheat-sinkbody.Thetransducerimpedanceisusuallyless
fine wire of the same metal as the heat sink. When the sensing
than1V.Tominimizecurrentflow,thedataacquisitionsystem
element is exposed to a heat source, most of the heat energy
(DAS) should be a potentiometric system or have an input
absorbedatthesurfaceofthecircularfoilisconductedradially
impedance of at least 100 000 Ω.
to the heat sink. If the heat flux is uniform and heat transfer
down the center wire is neglected, a parabolic temperature
3.2 As noted in 2.3, an approximately linear output (versus
profile is established between the center and edge of the foil
heat flux) is produced when the body and center wire of the
transducer are constructed of copper and the circular foil is
constantan. Other metal combinations may be employed for
This test method is under the jurisdiction of ASTM Committee E21 on Space
use at higher temperatures, but most (4) are nonlinear.
Simulation andApplications of SpaceTechnology and is the direct responsibility of
3.3 Because the thermocouple junction at the edge of the
Subcommittee E21.08 on Thermal Protection.
Current edition approved May 1, 2015. Published June 2015. Originally
foil is the reference for the center thermocouple, no cold
approved in 1973. Last previous edition approved in 2007 as E511–07. DOI:
junction compensation is required with this instrument. The
10.1520/E0511-07R15.
wire leads used to convey the signal from the transducer to the
The boldface numbers in parentheses refer to the list of references at the end of
this standard. readout device are normally made of stranded, tinned copper,
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E511 − 07 (2015)
FIG. 1 Heat Drain—Either by Water Cooling the Body with a Surrounding Water Jacket or Conducting the Heat Away with Sufficient
Thermal Mass
insulated with TFE-fluorocarbon and shielded with a braid water channel. Velocities of 15 to 30 m/s (49 to 98 ft/s) are
over-wrap that is also TFE-fluorocarbon-covered. produced by water at 3.4 to 6.9 MPa (500 to 1000 psi). For
such thin shells, zirconium-copper may be used for its combi-
3.4 Transducers with a heat-sink thermocouple can be used
nation of strength and high thermal conductivity.
to indicate the foil center temperature. Once the edge tempera-
ture is known, the temperature difference from the foil edge to
NOTE 1—Changing the heat sink from pure copper to zirconium copper
may change the sensitivity and the linearity of the response.
its center may be directly read from the copper-constantan
(Type T) thermocouple table. This temperature difference then
3.6 Foil Coating:
is added to the body temperature, indicating the foil center
3.6.1 High-absorptance coatings are used when radiant
temperature.
energyistobemeasured.Ideally,thehigh-absorptancecoating
should provide a nearly diffuse absorbing surface, where
3.5 Water-Cooled Transducer:
3.5.1 A water-cooled transducer should be used in any absorptionisindependentoftheangleofincidenceofradiation
on the coating. Such a coating is said to be Lambertian and the
applicationwherethecopperheat-sinkwouldriseabove235°C
(450°F) without cooling. Examples of cooled transducers are sensor output is proportional to the cosine of the angle of
incidence with respect to normal.An ideal coating also would
shown in Fig. 2.The coolant flow must be sufficient to prevent
local boiling of the coolant inside the transducer body, with its have no dependency of absorption with wavelength, approxi-
mating a gray-body. Only a few coatings approach these ideal
characteristic pulsations (“chugging”) of the exit flow indicat-
ing that boiling is occurring. Water-cooled transducers can use characteristics.
3.6.2 Most high absorptivity coatings have different absorp-
brass water tubes and sides for better machinability and
mechanical strength. tivities when exposed to hemispherically-incident or narrower-
angle, incident radiation. For five coatings, measurements by
3.5.2 The water pressure required for a given transducer
design and heat-flux level depends on the flow resistance and Alpert,etal,showedthenear-normalabsorptivitywas3to5%
higher than the hemispherical absorptivity (5). This work also
the shape of the internal passages. Rarely will a transducer
require more than a few litres of water per minute. Most showed that commercial heat flux gauge coatings generally
maintain Lambertian (Cosine Law) behavior out to incidence
require only a fraction of litres per minute.
2 2
3.5.3 Heat fluxes in excess of 3400 W/cm (3000 Btu/ft /s) angles 60° to 70º off-normal.
may require transducers with thin internal shells for efficient 3.6.3 Acetylene soot (total absorptanceα =0.99) and cam-
T
transfer of heat from the foil/heat sink into a high-velocity phor soot (α =0.98) have the disadvantages (4) of low
T
E511 − 07 (2015)
FIG. 2 Cross-Sectional View of Water-Cooled Heat-Flux Gages
oxidation resistance and poor adhesion to the transducer τ'ρcR /4k (2)
surface. Colloidal graphite coatings dried from acetone or
where the foil properties and dimensions are:
alcoholsolutions(α = 0.83)arecommonlyusedbecausethey
T
τ = radial coordinate,
adhere well to the transducer surface over a wide temperature
ρ = density,
range. Spray black lacquer paints (α =0.94 to 0.98), some of
T
c = specific heat,
whichmayrequirebaking,alsoareused.Theyareintermediate
R = radius, and
in oxidation resistance and adhesion between the colloidal
k = conductivity.
graphites and soots. Colloidal graphite is commonly used as a
4.2 Foil diameters and thicknesses are limited by typical
primer for other, higher-absorptance coatings.
3.6.4 Low-absorptance metallic coatings, such as highly manufacturingconstraints.Maximumoptimumfoildiameterto
polished gold or nickel, may be used to reduce a transducer’s thicknessratiois4to1forsensorslessthan2.54mmdiameter.
response to radiant heat. Because these coatings effectively
Foil diameters range from 25.4 to 0.254 mm, with most gages
increase the foil thickness, they reduce the transducer sensitiv-
between 1.02 and 6.35 mm. The time constants, τ, for a
ity. Gold coating also makes the transducer response nonlinear
25.4-mm and 0.254-mm diameter foil are 6 s and 0.0006 s,
because the thermal conductivity of this metal changes more
respectively.Forconstantan,thetimeconstantisapproximated
rapidly with temperature than that of constantan or nickel; the
by τ = 0.0094 d , where d is in mm. The effects of foil
coating must be thin to avoid changing the Seebeck Coeffi-
dimensions on the nominal time constant are shown in Fig. 3.
cient.
KeltnerandWildinprovideadetailedanalysisofthesensitivity
3.6.5 Exothermic reactions occurring at the foil surface will
and dynamic response that includes the effect of heat transfer
cause additional heating of the transducer. This effect may be
down the center wire (7).
highly dependent on the catalytic properties of the foil surface.
4.3 The radiative sensitivity of commercially available
Catalysis can be controlled by surface coatings (3).
2 2
transducers is limited to about 2 mV/W/cm (1.76 BTU⁄ft /s).
4. Characteristics and Limitations
Higher sensitivities can be achieved, but the foils of more
sensitive transducers are extremely fragile. The range of
4.1 The principal response characteristics of a circular foil
commercial transducers may be up to 10000 W/cm (~8800
heat flux transducer are sensitivity, full-scale range, and the
BTU/ ft /s), and typically is limited by the capacity of the heat
nominal time constant, which are established by the foil
sink for heat removal. The full-scale range is normally speci-
material, diameter and thickness. For a given heat flux, the
fied as that which produces 10 mV of output. This is the
transducer sensitivity is proportional to the temperature differ-
potential produced by a copper-constantan transducer with a
ence between the center and edge of the circular foil. To
temperature difference between the foil center and edge of
increase sensitivity, the foil is made thinner or its diameter is
190°C(374°F).Thesetransducersmaybeusedtomeasureheat
increased.Thefull-scalerangeofatransducerislimitedbythe
fluxesexceedingthefull-scale(10mVoutput)rating;however,
maximum allowed temperature at the center of the foil. The
more than 50% over-ranging will shorten the life and possibly
rangemaybeincreasedbymakingthefoilsmallerindiameter,
or thicker.An approximate transducer time constant is propor- change the transducer characteristics. If a transducer is used
beyond 200% of its full-scale rating, it should be returned to
tionaltothesquareofthefoilradius,andischaracterizedby (1,
3, 6): themanufacturerforinspectionandrecalibrationbeforefurther
E511 − 07 (2015)
FIG. 3 Chart for Design of Copper-Constantan Circular Foil Heat-Flow Meters (SI Units)
use. Care should be taken not to exceed recommended tem- 4.8 The circular foil transducer should be used with great
peraturelimitstoensurelinearresponse.Thisisdesignedforin
care for convective heat-flux measurements because (a) there
two ways: active cooling and by providing a heat sink with the
are no standardized calibration methods; (b) the uncertainty
copper body. The effects of foil dimensions on the transducer
increases rapidly for free-stream temperatures below 1000ºC,
sensitivity are shown in Fig. 4. Refs (7-9) provide more
although proper range selection can minimize the increase;
detailed analysis of the sensitivity that includes the effects of
and, (c) the uncertainty varies with the free-stream velocity
heat transfer down the center wire.
vector (10,11).Inshearflows,thesensorscandisplaynonlinear
4.4 Water-cooled sensors are recommended for any appli- response and high uncertainty (12,13).
cation in which the sensor body would otherwise rise above
4.9 Error Sources:
235°C (450°F). When applying a liquid-cooled transducer in a
4.9.1 Radiative Heat Transfer—If there is a uniform inci-
hot environment, it may be important to insulate the body of
dentheatfluxoverthefoil,convectiveandradiativeheatlosses
the transducer from the surrounding structure if it is also hot.
fromthefoilsurfacesarenegligible,andheattransferdownthe
This will improve the effectiveness of cooling and reduce the
center wire is neglected, then the foil temperature distribution
required liquid flow rate.
is parabolic:
4.5 The temperature of the gage body normally is low in
q
comparison to the heat source. The resulting heat flux mea-
r
2 2
T~r! 5 ~R 2 r ! (3)
sured by the gage is known as a “cold wall” heat flux. 4kδ
4.6 For measurements of purely radiant heat flux, the
where:
transducer output signal is a direct response to the energy
q = absorbed radiant heat flux,
r
absorbed by the foil; the absorptivity of the surface of the
δ = foil thickness,
coating must be known to correctly calculate the incident
R = foil radius, and
radiation flux (5).
k = foil conductivity.
4.7 The circular foil transducer cannot be used for conduc-
and the center to edge temperature difference is:
tion heat-flux measurements.
E511 − 07 (2015)
FIG. 4 Chart for Design of Copper-Constantan Circular Foil Heat-Flow Meters (U.S. Customary Units)
q R 4.9.1.2 Vacuum Operation—A circular foil transducer can
r
∆T 5 (4)
4kδ be used in a vacuum for radiant heat flux measurements. In
general, the back of the gauge should be ven
...
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: E511 − 07 E511 − 07 (Reapproved 2015)
Standard Test Method for
Measuring Heat Flux Using a Copper-Constantan Circular
Foil, Heat-Flux Transducer
This standard is issued under the fixed designation E511; 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 test method describes the measurement of radiative heat flux using a transducer whose sensing element (1, 2) is a thin
circular metal foil. These sensors are often called Gardon Gauges.
1.2 The values stated in SI units are to be regarded as the standard. The values stated in parentheses are provided for information
only.
1.3 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 and health practices and determine the applicability of regulatory
limitations prior to use.
2. Summary of Test Method
2.1 The purpose of this test method is to facilitate measurement of a radiant heat flux. Although the sensor will measure heat
fluxes from mixed radiative – convective or pure convective sources, the uncertainty will increase as the convective fraction of
the total heat flux increases.
2.2 The circular foil heat flux transducer generates a milliVolt output in response to the rate of thermal energy absorbed (see
Fig. 1). The perimeter of the circular metal foil sensing element is mounted in a metal heat sink, forming a reference thermocouple
junction due to their different thermoelectric potentials. A differential thermocouple is created by a second thermocouple junction
formed at the center of the foil using a fine wire of the same metal as the heat sink. When the sensing element is exposed to a heat
source, most of the heat energy absorbed at the surface of the circular foil is conducted radially to the heat sink. If the heat flux
is uniform and heat transfer down the center wire is neglected, a parabolic temperature profile is established between the center
and edge of the foil under steady-state conditions. The center – perimeter temperature difference produces a thermoelectric
potential, E, that will vary in proportion to the absorbed heat flux, q'. With prescribed foil diameter, thickness, and materials, the
potential E is almost linearly proportional to the average heat flux q' absorbed by the foil. This relationship is described by the
following equation:
E 5 Kq' (1)
where:
K = a sensitivity constant determined experimentally.
2.3 For nearly linear response, the heat sink and the center wire of the transducer are made of high purity copper and the foil
of thermocouple grade Constantan. This combination of materials produces a nearly linear output over a gauge temperature range
from –45 to 232°C (–50 to 450°F). The linear range results from the basically offsetting effects of temperature-dependent changes
in the thermal conductivity and the Seebeck coefficient of the Constantan (3). All further discussion is based on the use of these
two metals, since engineering practice has demonstrated they are commonly the most useful.
3. Description of the Instrument
3.1 Fig. 1 is a sectional view of an example circular foil heat-flux transducer. It consists of a circular Constantan foil attached
by a metallic bonding process to a heat sink of oxygen-free high conductivity copper (OFHC), with copper leads attached at the
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 Nov. 1, 2007May 1, 2015. Published December 2007June 2015. Originally approved in 1973. Last previous edition approved in 20012007 as
E511 – 01.E511 – 07. DOI: 10.1520/E0511-07.10.1520/E0511-07R15.
The boldface numbers in parentheses refer to the list of references at the end of this standard.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
E511 − 07 (2015)
FIG. 1 Heat Drain—Either by Water Cooling the Body with a Surrounding Water Jacket or Conducting the Heat Away with Sufficient
Thermal Mass
center of the circular foil and at any point on the heat-sink body. The transducer impedance is usually less than 1 V. To minimize
current flow, the data acquisition system (DAS) should be a potentiometric system or have an input impedance of at least 100 000
Ω.
3.2 As noted in 2.3, an approximately linear output (versus heat flux) is produced when the body and center wire of the
transducer are constructed of copper and the circular foil is constantan. Other metal combinations may be employed for use at
higher temperatures, but most (4) are nonlinear.
3.3 Because the thermocouple junction at the edge of the foil is the reference for the center thermocouple, no cold junction
compensation is required with this instrument. The wire leads used to convey the signal from the transducer to the readout device
are normally made of stranded, tinned copper, insulated with TFE-fluorocarbon and shielded with a braid over-wrap that is also
TFE-fluorocarbon-covered.
3.4 Transducers with a heat-sink thermocouple can be used to indicate the foil center temperature. Once the edge temperature
is known, the temperature difference from the foil edge to its center may be directly read from the copper-constantan (Type T)
thermocouple table. This temperature difference then is added to the body temperature, indicating the foil center temperature.
3.5 Water-Cooled Transducer:
3.5.1 A water-cooled transducer should be used in any application where the copper heat-sink would rise above 235°C (450°F)
without cooling. Examples of cooled transducers are shown in Fig. 2. The coolant flow must be sufficient to prevent local boiling
of the coolant inside the transducer body, with its characteristic pulsations (“chugging”) of the exit flow indicating that boiling is
occurring. Water-cooled transducers can use brass water tubes and sides for better machinability and mechanical strength.
3.5.2 The water pressure required for a given transducer design and heat-flux level depends on the flow resistance and the shape
of the internal passages. Rarely will a transducer require more than a few litres of water per minute. Most require only a fraction
of litres per minute.
2 2
3.5.3 Heat fluxes in excess of 3400 W/cm (3000 Btu/ft /s) may require transducers with thin internal shells for efficient transfer
of heat from the foil/heat sink into a high-velocity water channel. Velocities of 15 to 30 m/s (49 to 98 ft/s) are produced by water
at 3.4 to 6.9 MPa (500 to 1000 psi). For such thin shells, zirconium-copper may be used for its combination of strength and high
thermal conductivity.
NOTE 1—Changing the heat sink from pure copper to zirconium copper may change the sensitivity and the linearity of the response.
E511 − 07 (2015)
FIG. 2 Cross-Sectional View of Water-Cooled Heat-Flux Gages
3.6 Foil Coating:
3.6.1 High-absorptance coatings are used when radiant energy is to be measured. Ideally, the high-absorptance coating should
provide a nearly diffuse absorbing surface, where absorption is independent of the angle of incidence of radiation on the coating.
Such a coating is said to be Lambertian and the sensor output is proportional to the cosine of the angle of incidence with respect
to normal. An ideal coating also would have no dependency of absorption with wavelength, approximating a gray-body. Only a
few coatings approach these ideal characteristics.
3.6.2 Most high absorptivity coatings have different absorptivities when exposed to hemispherically-incident or narrower-angle,
incident radiation. For five coatings, measurements by Alpert, et al, showed the near-normal absorptivity was 3 to 5 % higher than
the hemispherical absorptivity (5). This work also showed that commercial heat flux gauge coatings generally maintain Lambertian
(Cosine Law) behavior out to incidence angles 60° to 70º off-normal.
3.6.3 Acetylene soot (total absorptance α = 0.99) and camphor soot (α = 0.98) have the disadvantages (4) of low oxidation
T T
resistance and poor adhesion to the transducer surface. Colloidal graphite coatings dried from acetone or alcohol solutions (α =
T
0.83) are commonly used because they adhere well to the transducer surface over a wide temperature range. Spray black lacquer
paints (α = 0.94 to 0.98), some of which may require baking, also are used. They are intermediate in oxidation resistance and
T
adhesion between the colloidal graphites and soots. Colloidal graphite is commonly used as a primer for other, higher-absorptance
coatings.
3.6.4 Low-absorptance metallic coatings, such as highly polished gold or nickel, may be used to reduce a transducer’s response
to radiant heat. Because these coatings effectively increase the foil thickness, they reduce the transducer sensitivity. Gold coating
also makes the transducer response nonlinear because the thermal conductivity of this metal changes more rapidly with temperature
than that of constantan or nickel; the coating must be thin to avoid changing the Seebeck Coefficient.
3.6.5 Exothermic reactions occurring at the foil surface will cause additional heating of the transducer. This effect may be highly
dependent on the catalytic properties of the foil surface. Catalysis can be controlled by surface coatings (3).
4. Characteristics and Limitations
4.1 The principal response characteristics of a circular foil heat flux transducer are sensitivity, full-scale range, and the nominal
time constant, which are established by the foil material, diameter and thickness. For a given heat flux, the transducer sensitivity
is proportional to the temperature difference between the center and edge of the circular foil. To increase sensitivity, the foil is made
thinner or its diameter is increased. The full-scale range of a transducer is limited by the maximum allowed temperature at the
center of the foil. The range may be increased by making the foil smaller in diameter, or thicker. An approximate transducer time
constant is proportional to the square of the foil radius, and is characterized by (1, 3, 6):
τ'ρcR /4k (2)
where the foil properties and dimensions are:
τ = radial coordinate,
ρ = density,
c = specific heat,
R = radius, and
E511 − 07 (2015)
k = conductivity.
4.2 Foil diameters and thicknesses are limited by typical manufacturing constraints. Maximum optimum foil diameter to
thickness ratio is 4 to 1 for sensors less than 2.54 mm diameter. Foil diameters range from 25.4 to 0.254 mm, with most gages
between 1.02 and 6.35 mm. The time constants, τ, for a 25.4-mm and 0.254-mm diameter foil are 6 s and 0.0006 s, respectively.
For constantan, the time constant is approximated by τ = 0.0094 d , where d is in mm. The effects of foil dimensions on the nominal
time constant are shown in Fig. 3. Keltner and Wildin provide a detailed analysis of the sensitivity and dynamic response that
includes the effect of heat transfer down the center wire (7).
2 2
4.3 The radiative sensitivity of commercially available transducers is limited to about 2 mV/W/cm (1.76 BTU ⁄ft /s). Higher
sensitivities can be achieved, but the foils of more sensitive transducers are extremely fragile. The range of commercial transducers
2 2
may be up to 10 000 W/cm (~8800 BTU/ ft /s), and typically is limited by the capacity of the heat sink for heat removal. The
full-scale range is normally specified as that which produces 10 mV of output. This is the potential produced by a
copper-constantan transducer with a temperature difference between the foil center and edge of 190°C (374°F). These transducers
may be used to measure heat fluxes exceeding the full-scale (10 mV output) rating; however, more than 50 % over-ranging will
shorten the life and possibly change the transducer characteristics. If a transducer is used beyond 200 % of its full-scale rating, it
should be returned to the manufacturer for inspection and recalibration before further use. Care should be taken not to exceed
recommended temperature limits to ensure linear response. This is designed for in two ways: active cooling and by providing a
heat sink with the copper body. The effects of foil dimensions on the transducer sensitivity are shown in Fig. 4. Refs (7-9) provide
more detailed analysis of the sensitivity that includes the effects of heat transfer down the center wire.
4.4 Water-cooled sensors are recommended for any application in which the sensor body would otherwise rise above 235°C
(450°F). When applying a liquid-cooled transducer in a hot environment, it may be important to insulate the body of the transducer
from the surrounding structure if it is also hot. This will improve the effectiveness of cooling and reduce the required liquid flow
rate.
4.5 The temperature of the gage body normally is low in comparison to the heat source. The resulting heat flux measured by
the gage is known as a “cold wall” heat flux.
FIG. 3 Chart for Design of Copper-Constantan Circular Foil Heat-Flow Meters (SI Units)
E511 − 07 (2015)
FIG. 4 Chart for Design of Copper-Constantan Circular Foil Heat-Flow Meters (U.S. Customary Units)
4.6 For measurements of purely radiant heat flux, the transducer output signal is a direct response to the energy absorbed by
the foil; the absorptivity of the surface of the coating must be known to correctly calculate the incident radiation flux (5).
4.7 The circular foil transducer cannot be used for conduction heat-flux measurements.
4.8 The circular foil transducer should be used with great care for convective heat-flux measurements because (a) there are no
standardized calibration methods; (b) the uncertainty increases rapidly for free-stream temperatures below 1000ºC, although proper
range selection can minimize the increase; and, (c) the uncertainty varies with the free-stream velocity vector (10,11). In shear
flows, the sensors can display nonlinear response and high uncertainty (12,13).
4.9 Error Sources:
4.9.1 Radiative Heat Transfer—If there is a uniform incident heat flux ove
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